Every question traces back to a concept

Dimensional analysis and formulas

Understand dimensional formulas, the principle of homogeneity, dimensional consistency, and how to use dimensional analysis to check equations and derive relationships.

Exam level15 questions

Physical quantities and units

Learn about fundamental and derived physical quantities, SI units, supplementary units, and how units relate to the measurement of physical quantities.

Foundation7 questions

Errors in measurement

Study the types of errors in physical measurements, their classification, and how errors propagate in calculations and experiments.

Exam level4 questions

Measurement tools and techniques

Explore the use and principles of common measuring instruments like vernier calipers and screw gauge, including least count, zero error, and practical measurement methods.

Exam level4 questions

Significant figures and calculations

Learn the rules for determining significant figures and applying them correctly in mathematical operations like multiplication and division.

Foundation3 questions

Applications of units and dimensions

Apply your knowledge of units and dimensions to solve problems involving physical constants, mechanical and electromagnetic properties, and real-world scenarios like thermal expansion.

Challenging1 question

Absolute, Relative, and Percentage Error

For a nonzero measured value x with absolute uncertainty Delta x, the relative uncertainty is Delta x/|x| and the percentage uncertainty is 100 Delta x/|x|; the uncertainty magnitude is reported nonnegative.

MixedJEE Physics · Original

Accuracy, Precision, and Resolution

Accuracy is agreement between a measured result and an accepted reference, precision is agreement among repeated results under stated conditions, and resolution or least count is the smallest indicated increment of the measuring system.

MixedJEE Physics · Original

Arguments of Transcendental Functions

For a physically meaningful expression, the complete argument of exp, log, sin, cos, or any power-series-defined transcendental function must be dimensionless because terms in its series expansion must be addable.

MixedJEE Physics · Original

Arithmetic Rules for Significant Results

For measured-number arithmetic, a product or quotient is rounded to the smallest significant-figure count among factors, whereas a sum or difference is rounded to the least precise decimal place among terms.

MixedJEE Physics · Original

Atomic, Nuclear, and Astronomical Units

A non-SI unit used with SI is a defined multiple of an SI unit for the same quantity; conversion preserves the quantity while replacing the specialised scale by its exact or stated SI equivalent.

MixedJEE Physics · Original

Buckingham Pi and Similarity Groups

The Buckingham Pi theorem states that a dimensionally homogeneous relation among n dimensional variables involving r independent base dimensions can be rewritten using n-r independent dimensionless products, though physics must select their functional relation.

MixedJEE Physics · Original

Coherent Derived Units

A coherent SI derived unit is obtained from a quantity equation without an extra numerical factor, so its expression in base units is a product of integer or rational powers of SI base units.

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Combining Repeated and Instrument Uncertainty

When repeated observations and instrument resolution both matter, the stated uncertainty must use the experiment's prescribed rule and cannot be smaller than the defensible resolution contribution; statistical and bounded uncertainties must not be mixed silently.

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Conversion Between Unit Systems

If Q has dimensions M^aL^bT^c..., numerical measures n_1 and n_2 in two coherent systems satisfy n_1 M_1^aL_1^bT_1^c... = n_2 M_2^aL_2^bT_2^c..., expressing invariance of Q.

MixedJEE Physics · Original

Dimensional Homogeneity

The principle of dimensional homogeneity requires the dimensions of the left and right sides of a physical equation, and of every term connected by addition or subtraction, to be identical for all admissible values.

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Dimensional Representation

The dimensional formula [Q] = M^a L^b T^c I^d Theta^e N^f J^g specifies the powers with which base quantities combine to form Q and is independent of the system of units.

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Dimensionless Ratios and Numbers

A quantity is dimensionless when its dimensional formula is M^0L^0T^0I^0Theta^0N^0J^0; plane angle and solid angle are dimensionless derived quantities with the special units radian and steradian.

MixedJEE Physics · Original

Dimensions from Graphs and Slopes

For a graph of y against x, the slope Delta y/Delta x has dimensions [y][x]^-1 and the signed area integral y dx has dimensions [y][x], subject to the plotted variables and scale factors.

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Dimensions in Differential and Integral Relations

If y has dimensions [Y] and x has dimensions [X], then [dy/dx]=[Y][X]^-1 and [integral y dx]=[Y][X], provided the derivative or integral is taken with respect to the stated physical variable.

MixedJEE Physics · Original

Dimensions of Material and Transport Coefficients

A constitutive coefficient's dimensions are defined by the physical law relating flux, stress, strain, gradient, or response; the same descriptive word can denote different coefficients, so the defining equation is essential.

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Dimensions of Physical Constants

The dimensions of a physical constant are determined by algebraically isolating it in a dimensionally homogeneous defining equation; only constants multiplying like-dimensional quantities without conversion are dimensionless.

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Electromagnetic Quantity Dimensions

In the M-L-T-I system, electromagnetic quantities are dimensionally derived while electric current I remains independent; for example [q]=IT, [V]=ML^2T^-3I^-1, and [R]=ML^2T^-3I^-2.

MixedJEE Physics · Original

Errors in Products, Quotients, and Powers

For z = C x^a y^b... with exact C and small independent bounded errors, the maximum fractional uncertainty is Delta z/|z| = |a|Delta x/|x| + |b|Delta y/|y| + ... to first order.

MixedJEE Physics · Original

Errors in Sums and Differences

If z = a x plus or minus b y with exact coefficients and bounded input uncertainties, the maximum absolute uncertainty is Delta z = |a|Delta x + |b|Delta y to first order.

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Geometric Scaling of Units

For geometrically similar measures, area has dimensions L^2 and volume L^3; therefore replacing a length unit by a factor k changes the corresponding numerical area and volume conversion factors by k^2 and k^3.

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Graphical Measurement and Best-Fit Slope

For a linearised relation Y = mX + c, the slope m = Delta Y/Delta X and intercept c carry dimensions set by the plotted axes; a defensible best-fit line balances residuals rather than being forced through arbitrary points.

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Instrument Calibration and Scale Design

Calibration is the documented comparison of an instrument's indications with traceable reference values to establish a correction relation, sensitivity, usable range, resolution, and associated uncertainty.

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Limits of Dimensional Analysis

Dimensional analysis cannot determine pure numerical coefficients, distinguish quantities sharing dimensions, select additive alternatives, or derive dependence through arbitrary dimensionless functions; dimensional correctness is therefore necessary but not sufficient.

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Maximum and Minimum Bounds

If measured inputs lie in stated intervals, output bounds are obtained by evaluating the extrema permitted by the model; for monotonic positive formulas this means selecting endpoint combinations according to whether each variable increases or decreases the result.

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Mean Value and Mean Absolute Error

For n repeated observations x_i, the arithmetic mean is x_bar = sum x_i/n and the mean absolute error is Delta x_mean = sum |x_i-x_bar|/n under the elementary JEE reporting convention.

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Measurement Planning and Defect-Aware Interpretation

Defect-aware interpretation preserves missing answers, ambiguous figures, damaged notation, and stated review flags while teaching only the governing measurement method; unsupported values or options must not be reconstructed as verified facts.

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Mechanical Quantity Dimensions

Mechanical dimensions are obtained by substituting [x]=L, [m]=M, and [t]=T into defining equations such as F=ma, p=mv, W=Fs, pressure=F/A, and power=W/t.

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Parallax and Observational Technique

Parallax is an observational systematic error caused by viewing separated reference planes from an oblique direction; it is minimised by aligning the eye normal to the scale and, where provided, superposing pointer and mirror image.

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Pendulum and Timing Experiments

For small angular amplitude and negligible damping, a simple pendulum obeys T = 2 pi sqrt(L/g); measuring total time t_N for N cycles gives T=t_N/N, and g can be inferred from 4 pi^2 L/T^2.

MixedJEE Physics · Original

Prefixes and Powers of Ten

An SI prefix denotes an exact factor 10^n attached to a unit symbol, and conversion requires compensating the numerical value so that the product of number and unit represents the same quantity.

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Random and Systematic Errors

Random error varies unpredictably among repeated observations under nominally identical conditions, whereas systematic error produces a reproducible bias through calibration, method, environment, or observer effects.

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Rayleigh Dimensional Method

Rayleigh's method assumes Q = k x^a y^b ... with dimensionless k, substitutes dimensional formulas, and equates independent base exponents to solve the unknown powers, provided the stated variables form a sufficient independent set.

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Rounding and Reporting Results

A reported uncertainty is normally rounded to one significant digit, sometimes two when the first digit is small, and the central value is rounded to the same place value; guard digits are retained during calculation.

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Same Dimensions, Different Quantities

Dimensional equality means two quantities scale identically with base quantities, but it does not establish equality of their tensor character, operational definition, physical role, or permissible algebraic combination.

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Screw Gauge and Pitch

The pitch is the axial distance moved in one complete rotation, and the screw-gauge least count is pitch divided by the number of equal circular-scale divisions; the corrected reading includes the signed zero correction.

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SI Base Quantities and Base Units

An SI base unit is the conventionally chosen reference unit for one of the seven base quantities; a derived unit is an algebraic product of powers of base units defined by a quantity equation.

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Significant Figures

Significant figures are all certain digits in a measured value plus the first uncertain digit; leading zeros are not significant, embedded zeros are significant, and trailing zeros are significant when decimal notation makes them explicit.

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Uncertainty in Derived Experimental Results

For a differentiable result f(x_1,...,x_n), first-order worst-case uncertainty is bounded by Delta f approximately sum |partial f/partial x_i| Delta x_i, which reduces to standard sum and power rules in common cases.

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Vernier Caliper Least Count

For a direct vernier, least count equals one main-scale division minus one vernier-scale division; the reading is main-scale reading plus coincident division times least count, followed by zero correction.

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Zero Error and Zero Correction

Zero error is the instrument's indicated value when the true input is zero; zero correction is the negative of zero error, and corrected reading equals observed reading plus zero correction.

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Distance, Displacement, Speed, and Velocity

Understand the differences between distance and displacement, and between speed and velocity, including how to calculate average speed and average velocity in various scenarios.

Foundation6 questions

Equations of Motion and Uniform Acceleration

Learn the kinematic equations for uniformly accelerated motion, including applications to free fall, retardation, and calculation of distance in specific time intervals.

Exam level5 questions

Projectile Motion

Study the motion of projectiles, including the independence of horizontal and vertical components, trajectory equations, maximum height, and the effect of initial conditions.

Exam level4 questions

Relative Velocity and Motion Analysis

Explore how to determine the velocity of one object relative to another and analyze motion from different reference frames, including periodic motion.

Exam level1 question

Acceleration as a Function of Position

Where v(x) is differentiable along a one-dimensional trajectory, a = dv/dt = (dv/dx)(dx/dt) = v dv/dx. Equivalently, a = (1/2)d(v²)/dx.

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Acceleration-Time Graphs

The velocity change over an interval is Delta v = integral(a dt). Velocity follows from v(t) = v0 + integral_0^t a(tau)d tau, and position then requires a second integration.

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Advanced Projectile Diagnostics and Modified Motion

The ideal formulas apply only for uniform gravity and negligible drag. More generally use m dv/dt = sum F, rho = v cubed over |v cross a| for planar curvature, and impulse Delta p = integral F dt.

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Average Speed and Average Velocity

Over elapsed time Delta t, average speed equals total distance/Delta t and average velocity equals Delta x/Delta t. For equal distances the relevant mean of speeds is harmonic, not arithmetic.

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Braking and Stopping Distance

For initial speed u and constant acceleration a opposite the motion, setting v = 0 in v² = u² + 2as gives stopping displacement s_stop = u²/(2|a|). Reaction distance, if present, must be added separately.

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Constant-Acceleration Equations

For constant signed acceleration a over an interval, v = u + at, s = ut + at²/2, v² = u² + 2as, and s = (u+v)t/2. These equations are not valid unchanged for variable acceleration.

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Constant-Acceleration Kinematics in a Plane

For constant vector acceleration a, v(t) = v_0 + at and r(t) = r_0 + v_0 t + one-half at squared; eliminating time or applying components gives the required geometry.

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Cross Product, Area, and Scalar Triple Product

A cross B has magnitude AB sin(theta) and right-hand-rule direction. The scalar triple product A dot (B cross C) is zero exactly when the three vectors are coplanar, apart from degenerate cases.

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Crossing Trains and Extended Bodies

For collinear bodies of lengths L1 and L2, complete crossing time is (L1+L2)/|v1-v2| with signed velocities. Passing a point or pole uses only the moving body's length.

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Differential and Integral Kinematics

For sufficiently smooth one-dimensional motion, v = dx/dt and a = dv/dt = d²x/dt². Conversely, integrating acceleration and velocity determines v and x up to constants fixed by initial conditions.

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Distance in the Nth Second

For constant acceleration, displacement during the nth second of unit duration is s_n = u + (a/2)(2n-1). More generally, use x(n)-x(n-1), and distinguish signed displacement from distance if reversal occurs.

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Dot Product and Vector Projection

For nonzero vectors A and B, A dot B = AB cos(theta). The scalar component of A along B is (A dot B)/|B|, while the vector projection is [(A dot B)/|B|^2]B.

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Free Fall and Downward Motion

For vertical motion near Earth's surface with negligible drag, acceleration is constant and downward with magnitude g. Choosing upward positive gives a = -g; choosing downward positive gives a = +g.

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Horizontal Projection and Dropped Bodies

For horizontal launch speed u_x from height h over level ground, t = sqrt(2h/g), horizontal displacement is u_x t, and impact velocity combines u_x with gt downward.

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Horizontal Range and Complementary Angles

For ideal projection on level ground, R = u squared sin 2theta divided by g. The maximum is u squared over g at 45 degrees, and theta and 90 degrees minus theta have equal ranges.

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Maximum Height and the Highest Point

For ideal launch and height measured above the launch level, H = u squared sin squared theta divided by 2g, reached at t = u sin theta/g.

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Meeting, Catch-Up, and Pursuit

For particles A and B in one dimension, encounter times solve x_A(t) = x_B(t) within the allowed time domain. Multiple valid roots represent multiple crossings; tangency can represent contact without passing.

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Piecewise and Multi-Stage Motion

For piecewise kinematics, x and v remain continuous across a boundary without an impulsive event, while acceleration may jump. Total elapsed time, displacement, and distance are assembled from the stage intervals with their signs.

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Position-Time Graphs

For a differentiable x-t curve, the tangent slope is v = dx/dt; a secant slope gives average velocity. A physically valid single-particle graph assigns only one position to each instant.

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Position, Path Length, and Displacement

For one-dimensional motion, displacement over an interval is Delta x = x_f - x_i and may be positive, negative, or zero. Distance is the non-negative path length, so distance is always at least |Delta x|.

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Position, Velocity, and Acceleration Vectors

For a differentiable position r(t), v = dr/dt and a = dv/dt = d^2r/dt^2. If velocity is given as a spatial field, acceleration along a trajectory is the material derivative dv/dt.

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Projectile Collisions, Explosions, and Centre of Mass

For a short internal event, total momentum is conserved. During subsequent flight, the centre of mass obeys M a_CM = sum F_external and therefore follows the gravity-driven path fixed by its state at the event.

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Projectile Components and Instantaneous Velocity

For launch speed u at angle theta above horizontal with no air resistance, v_x = u cos theta and v_y = u sin theta - gt; position follows by integrating these components.

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Projectile Motion in Inclined or Accelerating Frames

In a translating frame with acceleration A_frame, the projectile has effective acceleration a_rel = g_vector - A_frame. For an incline, resolve displacement parallel and normal to the plane and impose zero normal separation at impact.

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Projectile Trajectory Equation and Geometric Constraints

For launch from the origin under uniform downward gravity, y = x tan theta - gx squared divided by 2u squared cos squared theta. A shifted origin adds the initial coordinates.

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Rain, Wind, and Moving-Observer Velocity

If rain has ground velocity v_R and the observer has velocity v_O, then v_R/O = v_R - v_O. The observed angle follows from the ratio of horizontal and vertical components.

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Relative Position, Velocity, and Acceleration

For particles A and B in a nonrotating frame, r_AB = r_A - r_B, v_AB = v_A - v_B, and a_AB = a_A - a_B. Closest approach occurs when relative position is perpendicular to relative velocity.

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Relative Velocity on a Line

In Galilean one-dimensional motion, v_A/B = v_A - v_B in a common coordinate system. Relative acceleration similarly satisfies a_A/B = a_A - a_B.

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Release from a Moving Platform

At detachment, the object's inertial-frame position and velocity are continuous: x_object(t0) = x_platform(t0) and v_object(t0) = v_platform(t0), unless an additional throw impulse is specified.

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River Crossing and Swimmer-Boat Problems

The bank-frame velocity is v_ground = v_relative_water + v_river. For heading normal to the bank, crossing time is width divided by the cross-stream component; for zero drift, the along-stream components cancel.

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Speed and Velocity

Instantaneous velocity is the signed rate v = dx/dt, while instantaneous speed is |v|. In one dimension the sign of v identifies motion along or opposite the positive axis.

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Successive Drops at Regular Intervals

If drops are released every tau and the first has fallen for T, the kth released drop has fall time T-(k-1)tau when non-negative. From rest, its fall distance is (1/2)g[T-(k-1)tau]².

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Tangential-Normal Acceleration and Curved Paths

For instantaneous speed v and radius of curvature rho, a = a_t t_hat + a_n n_hat, with a_t = dv/dt and a_n = v squared over rho.

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Time of Flight and Vertical Timing

The flight time is obtained from y_f - y_0 = u_y T - one-half gT squared. When y_f = y_0, the nonzero root is T = 2u sin theta/g.

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Turning Points and Direction Reversal

At an interior extremum of differentiable x(t), v(t*) = 0 is necessary. A reversal requires a sign change of v across t*; acceleration need not be zero there.

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Uniform Circular Motion

For radius R and constant speed v, angular speed omega = v/R and centripetal acceleration has magnitude v squared over R directed radially inward.

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Vector Addition, Subtraction, and Resultants

For vectors A and B separated by angle theta, the resultant is R = A + B and its squared magnitude follows the cosine rule; A - B is obtained by reversing B.

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Vector Components and Unit Vectors

In Cartesian coordinates, A = A_x i_hat + A_y j_hat + A_z k_hat, its magnitude is the square root of the sum of squared components, and its direction is specified by component ratios or direction cosines.

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Velocity-Time Graphs

For v(t), acceleration is dv/dt and displacement is the signed integral integral(v dt). Distance is integral(|v| dt), so negative graph regions contribute positively to distance.

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Vertical Projection and Maximum Height

For upward launch speed u with negligible drag, time to the top is u/g and rise is u²/(2g). At equal heights on ascent and descent, speed magnitudes are equal while velocity signs are opposite.

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Newton's laws of motion

Understand and apply Newton's first, second, and third laws to analyze forces, motion, and interactions in various physical situations.

Foundation9 questions

Circular motion and centripetal force

Analyze the forces involved in uniform and non-uniform circular motion, including centripetal force, tension, and friction.

Exam level5 questions

Friction and its applications

Study the types of friction (static and kinetic), their effects on motion, and how friction interacts with other forces in different scenarios.

Exam level3 questions

Impulse, momentum, and conservation laws

Explore the principles of linear momentum, impulse, and their conservation in collisions and explosions.

Challenging1 question

Accelerating Frames and Pseudo Force

For translational frame acceleration A, m a_rel=sum F_real-mA; this simple form excludes rotating-frame terms.

MixedJEE Physics · Original

Angled Applied Force and Minimum Force

At impending horizontal motion under an upward pull, F cos alpha=mu_s(mg-F sin alpha); tan alpha_opt=mu_s.

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Banked Roads and Safe Speed

Without friction tan theta=v0^2/(rg); with friction resolve N and limiting friction vertically and radially.

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Conical Pendulum and Turning Deflection

For steady height, T cos theta=mg and T sin theta=ma_h, so tan theta=a_h/g.

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Free-Body Diagrams and Equilibrium

Particle equilibrium requires sum F_x = sum F_y = 0; rigid bodies may additionally require sum tau = 0.

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Friction in Connected and Stacked Bodies

For no slip, bodies share acceleration and |f_required| <= mu_s N; after slip use kinetic friction.

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Horizontal Circular Motion and Friction

a_r=v^2/r=omega^2 r inward; on a level rough surface mv^2/r <= mu_s mg.

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Impulse and Momentum Change

J = integral F dt = p_f - p_i; signs are essential for rebounds and reversals.

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Inertia and Inertial Frames

In an inertial frame, sum F_ext = 0 implies a = 0. Earth-fixed frames are approximate when rotational effects are negligible.

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Kinetic Friction and Rough Inclines

For Coulomb sliding f_k=mu_k N; down a fixed incline a=g(sin theta-mu_k cos theta).

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Momentum Conservation, Recoil, and Mass Flow

For an isolated system sum p_i = sum p_f; steady exhaust thrust is u_rel |dm/dt| with a consistent sign convention.

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Newton’s Second Law in Vector Form

In an inertial frame sum F_ext = dp/dt; for constant mass this becomes m dv/dt = ma.

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Newton’s Third Law and Contact Pairs

For interacting A and B, F_AB = -F_BA. Partners have the same interaction type and act on different objects.

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Normal Reaction and Apparent Weight

With upward positive, N-mg=ma; contact requires N >= 0.

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Pulley and String Constraints

For an inextensible string, L=sum l_i is constant; differentiating gives velocity and acceleration constraints.

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Spring Force and Coupled Motion

For an ideal spring operating within its linear elastic limit, the restoring force is F_s = -kx, where x is measured from the natural length, and the stored elastic potential energy is U_s = kx^2/2.

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State-Dependent Forces and Differential Dynamics

In one dimension a=dv/dt=v dv/dx; insert the useful form into m a=F(x,v,t).

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Static Friction and Limiting Equilibrium

Static friction obeys |f_s| <= mu_s N and opposes the impending relative motion.

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Tension in Strings, Ropes, Rods, and Chains

An ideal string over an ideal pulley has uniform T; a cut through a massive connector exposes the local carried force.

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Vertical Circles and Contact Loss

At every point sum F_inward=mv^2/r; for a smooth fixed track mechanical energy is conserved.

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Conservation of energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed, including cases with energy loss and efficiency considerations.

Foundation3 questions

Conservative and non-conservative forces

Conservative forces, like gravity and spring force, conserve mechanical energy, while non-conservative forces, like friction, dissipate energy as heat.

Exam level2 questions

Elastic potential energy

Elastic potential energy is the energy stored in a stretched or compressed spring, proportional to the square of its displacement.

Foundation2 questions

Work and its calculation

Work is the energy transferred by a force acting over a distance, and can be calculated using the dot product, area under a force-displacement graph, or for variable and constant forces.

Foundation2 questions

Power

Power is the rate at which work is done or energy is transferred, and can be calculated as the product of force and velocity at any instant.

Foundation1 question

Work-energy theorem

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy, and applies to both constant and variable forces.

Foundation1 question

Average Power, Machines, and Efficiency

Average power is Delta W/Delta t; efficiency is useful output divided by input and cannot exceed one for an ordinary passive conversion accounting.

MixedJEE Physics · Original

Ballistic Pendulum and Collision-Then-Energy

For a projectile embedding in a pendulum bob, conserve momentum across the brief collision, then conserve mechanical energy during the subsequent frictionless rise; never bridge both phases with one energy equation.

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Coefficient of Restitution and Rebounds

The coefficient of restitution is the relative speed of separation divided by the relative speed of approach along the line of impact, with 0<=e<=1 for ordinary passive impacts.

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Collision Impulse and Momentum Balance

Impulse is the time integral of force and equals momentum change; for a two-body collision, internal impulses are equal and opposite and total momentum is conserved when external impulse is negligible.

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Conservative Forces and Path Independence

A force is conservative in a simply connected region when its line integral is path independent, equivalently its work around every closed path is zero and a single-valued potential exists.

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Constant-Power Motion

For one-dimensional constant-mass motion under constant net power, P=mv(dv/dt), giving v^2=v_0^2+2Pt/m and displacement from integrating that speed.

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Energy in Vertical Circles and Curved Tracks

For frictionless vertical circular motion, mechanical energy relates speeds at different heights and the radial force equation sum F_in=mv^2/R determines tension or normal reaction and contact conditions.

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Gravitational Energy and Mechanical Conservation

Near Earth, U_g=mgy relative to a chosen datum; when only conservative forces do work, the sum of kinetic and potential energies remains constant between states.

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Inelastic Collisions and Energy Loss

In a perfectly inelastic collision the bodies coalesce and momentum is conserved when external impulse is negligible, while kinetic-energy loss is the positive difference K_i-K_f.

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Instantaneous Power from Force and Velocity

Instantaneous mechanical power delivered by a force is dW/dt=F dot v; the net power also equals dK/dt for a constant-mass particle.

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Kinetic Energy and Momentum Relations

For a nonrelativistic particle, K=p^2/(2m) and p=sqrt(2mK), so ratio questions must retain the relevant mass as well as momentum or speed.

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Momentum Conservation, Recoil, and Explosions

If external impulse during a short interaction is negligible, total linear momentum of the chosen system is conserved even when kinetic energy changes or internal energy is released.

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Nonconservative Work and Energy Loss

For a chosen system, the change in K+U equals work by external nonconservative forces; a stated energy loss must remain distinct from momentum changes during an impact.

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One-Dimensional Elastic Collisions

A perfectly elastic head-on collision is determined by conservation of linear momentum together with relative speed of separation equal to relative speed of approach.

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Potential Energy, Force, and Equilibrium

For one-dimensional conservative motion, F_x is minus dU/dx; equilibrium occurs where dU/dx is zero, with stability determined by local curvature or nearby restoring direction.

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Spring Force and Elastic Energy

For an ideal Hooke-law spring measured from natural length, F_s=-kx and U_s=(1/2)kx^2; spring constants combine according to their physical connection.

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Two-Dimensional Elastic Collisions and Scattering

For an isolated two-dimensional elastic collision, both components of total momentum and total translational kinetic energy are conserved, subject to the collision geometry.

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Variable Force and Work Integrals

Work by a position-dependent force along a specified path is the line integral of F dot dr; in one dimension it is the signed integral of F(x) with respect to x.

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Work by a Constant Force

For a constant force, work is the scalar product of force and displacement, so its sign is set by the angle between those vectors and not by force magnitude alone.

MixedJEE Physics · Original

Work-Energy Theorem

For a particle of constant mass, the total work by all forces between two states equals K_f minus K_i, whether or not individual forces are conservative.

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Conservation of momentum and angular momentum

The total linear and angular momentum of a system remains constant in the absence of external forces or torques, including during collisions and rotational motion.

Exam level7 questions

Moment of inertia and radius of gyration

Moment of inertia quantifies how mass is distributed with respect to an axis of rotation, and the radius of gyration is a measure related to this distribution.

Foundation7 questions

Torque and rotational equilibrium

Torque is the rotational analogue of force, causing angular acceleration, and equilibrium occurs when the net torque on a body is zero.

Exam level6 questions

Center of mass: definition and calculation

The center of mass is the point representing the mean position of the mass in a system, and can be calculated for discrete particles or continuous bodies.

Foundation5 questions

Rotational kinematics and dynamics

Rotational kinematics describes the motion of rotating bodies, while dynamics relates torque, angular acceleration, and rotational kinetic energy.

Foundation3 questions

Acceleration Down an Incline

For a symmetric rigid body rolling without slipping down a fixed incline, its centre acceleration is g sin theta divided by one plus I_cm/(MR squared), provided static friction can enforce the constraint.

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Angular Impulse

Angular impulse about an origin is the time integral of external torque about that origin and equals the corresponding change in angular momentum, including impulsive forces during short collisions.

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Angular Momentum of a Particle

The angular momentum of a particle about origin O is L_O = r cross p, so its magnitude is mvr_perpendicular and can change even at constant speed if the direction or line of motion changes.

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Angular Momentum of a Rigid Body

For rotation about a fixed principal axis, rigid-body angular momentum along that axis is L = I omega; in general three-dimensional rotation, angular momentum need not be parallel to angular velocity.

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Angular Position, Velocity, and Acceleration

For rotation about a fixed axis, angular velocity is the time derivative of angular displacement and angular acceleration is the derivative of angular velocity, with signs set by one chosen rotational sense.

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Angular-Linear Kinematic Relations

For a point at perpendicular distance r from a fixed rotation axis, tangential displacement, speed, and acceleration scale with r, while centripetal acceleration points toward the axis.

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Central Forces and Orbital Angular Momentum

When force is always parallel or antiparallel to the radius vector from a fixed centre, its torque about that centre is zero, so particle angular momentum and the rate at which radius sweeps area remain constant.

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Centre of Gravity and Stability

The centre of gravity is the point through which the resultant gravitational force acts; in a uniform gravitational field it coincides with the centre of mass, while stability requires its vertical projection to remain inside the support base.

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Centre of Mass of Continuous Bodies

For a continuous mass distribution, the centre of mass is defined by integrating position against the mass element and dividing by total mass, with symmetry used only when the density and geometry justify it.

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Centre of Mass of Discrete Particles

For discrete particles, the centre-of-mass position is the vector sum of each mass times its position divided by total mass; this point governs translation even when the particles move relative to one another.

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Centre-of-Mass Shift and Internal Rearrangement

When external force is zero, total momentum and centre-of-mass velocity remain constant; therefore mass displacements relative to a fixed frame obey the mass-weighted displacement constraint.

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Composite and Cutout Moments of Inertia

The moment of inertia of a composite body is the signed sum of component inertias about the same final axis, with holes represented by negative mass and shifted using axis theorems when necessary.

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Composite Bodies, Holes, and Cavities

The centre of mass of a composite body follows the ordinary mass-weighted formula when added pieces have positive mass and removed pieces are represented by negative mass of the same density.

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Conservation of Angular Momentum

For a chosen system and origin, angular momentum is conserved over an interval if the net external angular impulse about that origin vanishes, even though internal torques may redistribute spin.

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Constant and Variable Angular Kinematics

For fixed-axis rotation with constant angular acceleration, angular displacement, angular velocity, and time satisfy the standard uniformly accelerated equations; for variable acceleration they must be integrated instead.

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Coupled Rolling Bodies, Strings, and Spools

For rolling bodies linked to strings, pulleys, planks, or other bodies, each component obeys its own translation and rotation equations while inextensible-string and no-slip constraints relate their accelerations.

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Impact and Capture about a Pivot

For a short collision involving a pivoted rigid body, angular momentum about the pivot is conserved when other external angular impulses are negligible, while mechanical energy is generally not conserved for sticking or capture.

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Impulsive Rolling and Angular Momentum at Contact

During a brief impulsive interaction, linear and angular impulse relations determine post-impact translation and spin; choosing the contact point as angular-momentum origin can eliminate an unknown contact impulse only when its line passes through that point.

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Kinetic Energy of Pure Rolling

A rigid body in pure rolling has total kinetic energy one-half M v_cm squared plus one-half I_cm omega squared, with omega = v_cm/R only when the no-slip condition holds.

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Moment of Inertia from Nonuniform Density

For a body whose line, area, or volume density varies with position, moment of inertia is obtained from I = integral r_perpendicular squared dm using the stated density law and geometric Jacobian.

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Moment of Inertia: Definition and Discrete Systems

The moment of inertia of particles about an axis is the sum of m_i r_perpendicular,i squared and quantifies resistance to angular acceleration about that exact axis.

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Parallel-Axis Theorem

For two parallel axes separated by perpendicular distance d, one through the centre of mass, the moment of inertia is I = I_cm + M d squared; the theorem cannot directly connect arbitrary nonparallel axes.

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Perpendicular-Axis Theorem

For a thin lamina lying in a plane, the moment of inertia about an axis perpendicular to the plane through a point equals the sum about any two mutually perpendicular in-plane axes through that point.

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Physical and Torsional Pendulums

For small angular displacement, a physical pendulum has period determined by pivot inertia and centre-of-mass offset, while a torsional pendulum is restored by torque proportional to twist angle.

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Principle of Moments

For a rigid body in static equilibrium, the vector sum of torques is zero; in a planar problem this is equivalent to equality of total clockwise and anticlockwise moments about any selected origin.

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Pure Rolling Constraint

For a rigid circular body rolling without slipping on a stationary surface, the centre-of-mass speed and angular speed satisfy v_cm = R omega, with corresponding tangential accelerations when the constraint remains valid.

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Radius of Gyration

The radius of gyration k of a body about a specified axis is defined by I = M k squared, so k is the root-mean-square perpendicular distance of mass from that axis.

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Rigid-Body Static Equilibrium and Supports

Static equilibrium of a rigid body requires simultaneous translational and rotational equilibrium, so all external force components sum to zero and the total torque about any point is zero.

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Rolling Energy on Slopes and Tracks

For a rigid body rolling without slipping on a fixed surface with negligible dissipative resistance, gravitational potential energy converts into both translational and rotational kinetic energy.

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Rotating Frames and Centrifugal Effects

In a frame rotating with angular velocity Omega, equations of motion include centrifugal and Coriolis terms; for a point stationary in a uniformly rotating frame, the outward centrifugal term balances real inward forces when relative equilibrium holds.

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Rotational Dynamics about a Fixed Axis

For a rigid body rotating about a fixed principal axis, the algebraic net external torque about that axis equals the moment of inertia about the same axis times angular acceleration.

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Rotational Kinetic Energy and Power

For fixed-axis rigid rotation, rotational kinetic energy is K = one half I omega squared, and instantaneous power delivered by a torque is the scalar product tau dot omega.

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Standard Moments of Inertia

A standard moment-of-inertia formula is valid only for its stated ideal body and axis, such as a ring, disc, cylinder, sphere, rod, or rectangular lamina of uniform density.

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Static Friction in Pure Rolling

In pure rolling, contact friction is static and its direction follows the required translational and rotational accelerations rather than automatically opposing centre-of-mass motion; its magnitude must satisfy |f| <= mu_s N.

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Toppling and Step Climbing

At the threshold of toppling, the normal reaction shifts to the edge that becomes the pivot and moments about that edge determine the required force; step-climbing analysis similarly uses the corner contact as instantaneous pivot.

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Torque from Distributed Forces

When force is spread over a line, area, or volume, the torque about an origin is the integral of r cross dF; a single resultant force may replace the distribution only when its line of action is also preserved.

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Torque Vector and Lever Arm

The torque of a force about an origin is tau = r cross F, with magnitude equal to force times perpendicular lever arm and direction fixed by the right-hand rule.

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Transition from Sliding to Rolling

While a body slips on a rough surface, kinetic friction produces linear acceleration and angular acceleration independently; pure rolling begins at the first instant v_cm = R omega in the surface frame.

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Velocities of Points on a Rolling Body

For a rigid body in planar motion, the velocity of point P equals v_cm plus omega cross r from the centre of mass to P; in pure rolling the contact point is instantaneously at rest and the top point has speed 2v_cm.

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Work-Energy in Fixed-Axis Rotation

For a rigid body constrained to rotate about a fixed axis, the work of all external torques equals the change in one-half I omega squared; conservative-force work may instead be handled through potential energy.

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Kepler's laws of planetary motion

Kepler's laws describe the motion of planets and satellites: orbits are ellipses (first law), equal areas are swept in equal times (second law), and the square of the period is proportional to the cube of the semi-major axis (third law).

Exam level7 questions

Gravitational potential energy and work

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, and work is required to move it against gravity.

Exam level4 questions

Satellite motion and orbital parameters

Satellite motion involves understanding orbital velocity, time period, escape velocity, and the specific conditions for geostationary orbits.

Exam level4 questions

Acceleration due to gravity and its variation

Acceleration due to gravity (g) is the acceleration experienced by a body due to Earth's gravity, and it varies with height, depth, and planetary properties.

Exam level3 questions

Newton's law of universal gravitation

Newton's law of universal gravitation states that every two masses attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Exam level3 questions

Conservative nature of gravitational force

Gravitational force is a conservative force, meaning the work done in moving an object between two points is independent of the path taken.

Foundation1 question

Gravitational field and potential

The gravitational field describes the force per unit mass at a point in space due to one or more masses, while gravitational potential is the work done per unit mass to bring a mass from infinity to that point.

Foundation1 question

Binary and Many-Body Gravitational Motion

For two isolated masses M1 and M2 separated by a, the relative orbit obeys T^2=4 pi^2 a^3/[G(M1+M2)] and M1 r1=M2 r2; symmetric many-body circular motion requires the vector sum of mutual forces to equal each body's centripetal force.

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Cavities by Negative-Mass Superposition

Because Newtonian field and potential are linear in source mass, a removed region may be represented by the same region carrying density -rho and superposed with the original complete distribution.

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Central-Force Angular Momentum and Kepler's Second Law

For a central force tau = r cross F = 0, hence L=m r cross v is constant and dA/dt=L/(2m); at apsides this gives r_p v_p = r_a v_a.

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Circular-Orbit Speed, Period, and Angular Momentum

For a satellite mass m in a circular orbit of radius r about mass M with m much smaller than M, GMm/r^2 = mv^2/r, so v=sqrt(GM/r), T=2 pi sqrt(r^3/GM), and L=m sqrt(GMr).

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Effective Gravity on a Rotating Planet

In the rotating Earth frame the effective gravitational acceleration is the vector sum of true gravity and centrifugal acceleration; at the equator its magnitude is approximately g_eff = g - omega^2 R.

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Energy of a Circular Orbit

Combining circular-force balance with U=-GMm/r gives K=-U/2 and E=K+U=U/2=-GMm/(2r), under the point-mass or spherical-source approximation.

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Escape Speed and Escape Energy

Ignoring drag and other bodies, setting K+U=0 at launch radius r gives v_e=sqrt(2GM/r); the required kinetic energy from rest is GMm/r and v_e=sqrt(2) v_c at the same radius.

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Geostationary and Relative Satellite Motion

A geostationary orbit has T equal to Earth's sidereal rotation period in a circular prograde equatorial orbit; observed conjunction rates use relative angular speed omega_rel=|omega1-omega2| for the same sense and omega1+omega2 for opposite senses.

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Gravitational Field, Potential, and Gradient

With V(infinity)=0, gravitational potential satisfies g = -grad V and V(B)-V(A) = -integral_A^B g dot dl; a particle of mass m has potential energy U=mV.

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Gravitational Potential Energy and Work

For point masses U=-GMm/r with zero at infinity, gravitational work is -Delta U, and in an isolated conservative motion K+U remains constant.

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Gravity and Weight Above a Planet

Outside a spherical planet, g(h) = GM/(R+h)^2 = g0[R/(R+h)]^2; for h much smaller than R, the first-order approximation is g(h) approximately g0(1-2h/R).

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Gravity Inside a Uniform Planet

At centre distance r inside a uniform sphere, M(r)=M r^3/R^3 and g(r)=GM r/R^3; at depth d below the surface this becomes g_d=g0(1-d/R).

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Gravity of Continuous Mass Distributions

For a continuous density, dm is written from the appropriate linear, surface, or volume density and d g = -G dm r_hat/r^2 is integrated over the complete source with geometric limits.

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Kepler's Third Law and Period Scaling

For a Keplerian ellipse of semi-major axis a around total gravitating mass M_tot, T^2 = 4 pi^2 a^3/(G M_tot); circular-orbit radius is the special case a=r.

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Orbital Transfer, Impulse, and Elliptic Orbits

Between impulses, specific orbital energy epsilon=v^2/2-GM/r and angular momentum are conserved; for an ellipse epsilon=-GM/(2a), giving the vis-viva equation v^2=GM(2/r-1/a).

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Potential and Field Graphs

For a finite nonsingular spherical density, V(r) is continuous at the surface and g_r=-dV/dr; for a uniform solid sphere the interior potential is quadratic in r and joins -GM/r smoothly at r=R.

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Radially Varying Density and Enclosed Mass

For spherical symmetry M(r)=4 pi integral_0^r rho(s)s^2 ds and g(r)=GM(r)/r^2; a circular test orbit then satisfies v^2=GM(r)/r and T=2 pi r/v.

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Shell Theorem and Spherical Symmetry

Newton's shell theorem states that a spherically symmetric mass distribution produces the external field -GM r_hat/r^2, while a thin uniform spherical shell produces zero field and constant potential at every interior point.

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Surface Gravity and Planet Scaling

For a spherical planet of mass M and radius R, surface gravitational acceleration is GM/R^2; if its mean density is rho, substituting M = 4 pi rho R^3/3 gives g = 4 pi G rho R/3.

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Universal Gravitation and Vector Superposition

For point masses separated by r, Newtonian gravity has magnitude Gm1m2/r^2 along the line of centres; the net force or field is the vector sum of every source contribution.

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Mechanical properties of solids

Study of how solids respond to applied forces, including stress, strain, elastic moduli (Young's modulus, bulk modulus), and breaking stress.

Exam level15 questions

Fluid dynamics and Bernoulli's theorem

Covers the motion of fluids, including Bernoulli's theorem, Torricelli's law, dynamic lift, and viscous flow.

Exam level5 questions

Surface tension and surface energy

Examines the molecular forces at liquid surfaces, including surface tension, surface energy, and related equations.

Challenging4 questions

Capillarity and contact angle

Focuses on capillary action, meniscus formation, and the role of contact angle in wetting phenomena.

Exam level3 questions

Pressure in fluids and hydrostatics

Explores how pressure is transmitted in fluids, including hydrostatic pressure, Pascal's law, and related phenomena.

Foundation3 questions

Thermal properties of matter

Deals with heat transfer, thermal conductivity, calorimetry, specific heat, and thermal expansion of solids.

Exam level2 questions

Apparent Expansion, Density, and Buoyancy

For fixed mass and small temperature change, density varies as rho approximately rho_0(1-gamma Delta T); an apparent liquid expansion coefficient is the liquid's real volume coefficient minus that of its container.

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Archimedes' Principle and Apparent Weight

A body wholly or partly immersed in a static fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the immersed portion of the body.

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Bernoulli's Equation and Pressure-Speed Conversion

For steady, incompressible, non-viscous flow with no shaft work or dissipative loss, P + rho v^2/2 + rho g h is constant along a streamline.

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Bulk Modulus and Fluid Compressibility

Bulk modulus is the negative pressure increment divided by the resulting fractional volume change in the small-deformation limit, with the sign chosen so stable materials have positive modulus.

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Bulk Modulus and Volume Compression

Bulk modulus is K = -Delta P/(Delta V/V) for small hydrostatic compression, where the minus sign records that increasing external pressure decreases volume.

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Calorimetry and Thermal Equilibrium

Calorimetry applies energy conservation to thermally interacting bodies: when external heat and work are negligible, the algebraic sum of all sensible and latent heat changes from the initial states to equilibrium is zero.

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Capillary Rise, Depression, and Tube Geometry

In static capillarity, the vertical component of surface tension around the contact perimeter balances the hydrostatic weight associated with rise or depression relative to the external free surface.

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Composite and Segmented Wires

For axially connected linear-elastic segments, compatibility gives total extension as the algebraic sum of segment extensions, Delta L_total = sum(F_i L_i/(A_i Y_i)).

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Continuity and Volume Flow Rate

Mass conservation in steady one-dimensional flow gives a constant mass-flow rate; if density is constant, the volume-flow rate is also constant along a stream tube.

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Convection and Bulk Fluid Heat Transfer

Convection is heat transfer associated with macroscopic motion of a fluid and is classified as natural when buoyancy drives the flow or forced when an external device drives it.

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Coupled Fluid Models and Conservation Workflow

A coupled fluid model applies mass, momentum, energy, and force or torque balance together with an appropriate material law and explicit boundary conditions, retaining only assumptions supported by the source.

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Density Change under Pressure

With fixed mass and small hydrostatic compression, Delta rho/rho is approximately minus Delta V/V, so the density increase is rho Delta P/K to first order.

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Density, Relative Density, and Fluid Columns

Mass density is mass per unit volume, and relative density is the dimensionless ratio of a substance's density to the density of water at the stated reference condition.

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Dynamic Tension and Stress

Dynamic tensile stress equals the internal tension obtained from Newton's laws divided by cross-sectional area, with centripetal and tangential contributions evaluated at the stated position.

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Efflux, Jet Trajectory, and Tank Drainage

Torricelli's law is the Bernoulli result for ideal discharge from a small orifice in a large open tank; unsteady drainage additionally requires volume conservation as the head falls.

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Elastic Energy and Energy Density

For quasistatic linear elastic loading from zero, stored strain energy is U = one-half F Delta L, and energy density is u = one-half sigma epsilon.

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Elastic Energy Converted to Motion

For an ideal massless linear elastic launcher released from extension x, the stored energy one-half kx squared is equated to the recipient's kinetic-energy gain, subject to any stated efficiency.

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Elastic Limit, Breaking Stress, and Safety

Breaking stress is the maximum tensile stress sustained before fracture, while an elastic limit bounds recoverable behavior; safe design requires the working stress not to exceed the applicable limit.

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Emissivity, Absorptivity, and Kirchhoff's Law

Kirchhoff's law of thermal radiation states that for a body in thermodynamic equilibrium, spectral emissivity equals spectral absorptivity at each wavelength and direction; a blackbody has both equal to one.

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Floating Equilibrium and Layered Liquids

Static floating equilibrium requires zero net force and, when orientation matters, zero net torque; buoyant contributions from distinct immiscible fluids are summed over their displaced volumes.

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Fluids in Accelerating and Rotating Frames

For a fluid stationary relative to a uniformly accelerating frame, hydrostatic balance uses the effective body acceleration g_eff = g - a_frame; rigid rotation adds a centrifugal potential.

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Fourier Conduction and Heat Flux

Fourier's law states that conductive heat flux is q_vector = -k grad T; for a uniform one-dimensional slab in steady state, heat current is H = kA(T_hot-T_cold)/L.

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Geometry-Coupled Elastic Equilibrium

In statically balanced elastic assemblies, vector equilibrium determines member forces and geometric compatibility relates displacement to member extension, after which each member obeys its constitutive law.

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Heat Capacity and Specific Heat

Heat capacity is C = dQ/dT for a specified process, specific heat is c = C/m, and molar heat is C_m = C/n; over a range with variable capacity the heat is the integral of C(T)dT.

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Heating Power, Losses, and Variable Thermal Response

For a lumped system with heat capacity C(T), the transient energy balance is C(T)dT/dt = P_in(t)-P_loss(T), and steady temperature occurs when input and loss powers are equal.

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Hydrostatic Pressure and Communicating Vessels

Hydrostatic equilibrium requires the pressure gradient to balance body force; for constant density in uniform gravity, pressure increases linearly with downward vertical distance.

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Hydrostatic Pressure and Compression with Depth

For approximately constant fluid density, gauge pressure at depth h is rho g h, and small fractional volume compression is rho g h divided by the relevant bulk modulus.

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Ideal-Gas Thermal Coefficients and Heat Capacities

For a gas the volume-expansion coefficient is beta = (1/V)(partial V/partial T) under the stated constraint, while ideal-gas molar heat capacities satisfy C_p-C_v=R and depend on allowed molecular degrees of freedom.

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Laplace Pressure in Drops and Bubbles

The Young-Laplace relation connects pressure jump across an interface to surface tension times the sum of its principal curvatures; spherical interfaces give inverse-radius laws.

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Latent Heat, Phase Change, and Heating Curves

At a first-order phase transition under fixed pressure, heat Q = mL changes phase at the transition temperature; a complete thermal path is divided into sensible-heat and latent-heat segments.

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Linear, Areal, and Volumetric Expansion

The linear expansion coefficient is alpha = (1/L)(dL/dT); for small changes a free isotropic solid obeys Delta L = alpha L Delta T, with areal and volumetric coefficients approximately 2 alpha and 3 alpha.

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Material Modulus, Geometry, and Temperature

Young's modulus is the local ratio of longitudinal stress to longitudinal strain in the linear regime, so specimen geometry does not change it, whereas material state and temperature may.

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Measurement Uncertainty in Young's Modulus

For Y = 4FL/(pi d squared Delta L), the maximum fractional uncertainty is the sum of the fractional input uncertainties weighted by their absolute exponents.

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Mechanical and Electrical Energy Converted to Heat

When macroscopic mechanical or electrical energy is irreversibly converted to thermal energy, conservation gives Q_received = eta Delta E_available, followed by the appropriate sensible-heat and phase-change relations.

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Momentum Flux and Force of Jets

For a steady control volume, the net external force equals the net outward flux of linear momentum, with pressure and body forces included consistently when they act on the control surface.

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Natural Length from Multiple Loads

For a fixed elastic wire in its linear regime, loaded length obeys l = L0 + T/k; subtracting two observations determines compliance and then the natural length L0.

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Newton Cooling and Radiative Cooling Dynamics

For a lumped body of heat capacity C, C dT/dt = -P_loss(T); Newton cooling gives P_loss=hA(T-T_s), while pure radiation gives P_loss=e sigma A(T^4-T_s^4).

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Pascal's Law and Hydraulic Machines

Pascal's law states that a pressure change applied to a confined incompressible fluid is transmitted undiminished to every part of the fluid and its containing walls.

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Poiseuille Flow and Hydraulic Resistance

Hagen-Poiseuille flow is steady, fully developed, incompressible Newtonian laminar flow through a long straight circular tube driven by a pressure difference.

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Poisson's Ratio and Lateral Strain

Poisson's ratio is minus transverse strain divided by longitudinal strain for small uniaxial deformation, so measured lateral strain can determine axial strain and elastic energy.

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Radial Conduction in Cylinders and Spheres

Integrating Fourier's law over concentric surfaces gives R_cyl = ln(r2/r1)/(2 pi k L) for a cylindrical shell and R_sph = (1/(4 pi k))(1/r1-1/r2) for a spherical shell.

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Relations among Elastic Moduli

In homogeneous isotropic linear elasticity, Young's modulus Y, bulk modulus K, shear modulus G, and Poisson's ratio nu obey fixed algebraic relations; these relations do not apply unchanged to anisotropic media.

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Self-Weight Elongation and Maximum Hanging Length

For a uniform vertical member of density rho and area A, local stress at distance x above the lower end is rho g x, giving total self-extension rho g L squared divided by 2Y.

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Shear Modulus and Shear Deformation

For small simple shear, shear modulus G is tangential stress divided by shear strain, with shear strain approximately lateral displacement divided by the perpendicular thickness.

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Steady-State Junctions and Thermal Networks

In a steady thermal network without internal energy storage at a node, energy conservation requires the algebraic sum of branch heat currents H_i=(T_node-T_i)/R_i to be zero.

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Stefan-Boltzmann Radiation and Net Radiant Power

A diffuse gray surface of area A and emissivity e emits P=e sigma A T^4; in a large isothermal enclosure its net radiative loss is e sigma A(T^4-T_s^4).

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Stokes Drag on a Sphere

Stokes' law gives the viscous drag on an isolated rigid sphere moving slowly relative to an unbounded Newtonian fluid when the particle Reynolds number is much less than one.

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Stress-Strain and Load-Extension Graphs

In the linear region, slope of stress versus strain is Young's modulus, slope of load versus extension is axial stiffness, and the area under stress versus strain is strain-energy density.

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Stress, Strain, and Hooke's Law

Normal stress is force per perpendicular area and longitudinal strain is fractional length change; in the proportional elastic region they satisfy sigma = Y epsilon.

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Surface Energy in Thermally Coupled Processes

Surface tension is surface free energy per unit area and force per unit length; for an interface of tension S, a quasistatic area change requires dW=S dA, with a factor equal to the number of created surfaces.

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Surface Energy, Drops, and Coalescence

At fixed temperature and composition, the reversible change in surface energy equals surface tension times the change in interfacial area, with each physical interface counted once.

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Surface Tension, Contact Angle, and Wetting

Surface tension is tangential force per unit length, equivalently reversible surface free energy per unit area for a liquid interface; contact angle is measured through the liquid at equilibrium.

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Temperature Scales and Linear Thermometry

For a property that varies linearly with temperature, equal fractions of the interval between two reproducible fixed points represent equal temperature intervals, independent of the numerical scale labels.

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Terminal Velocity and Viscous Transients

Terminal velocity is the constant relative speed reached when the resultant external force on a body moving through a fluid becomes zero; under Stokes drag it scales with radius squared.

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Thermal Resistance and Equivalent Conductivity

Thermal resistance is R_th = Delta T/H; for planar steady conduction R_th=L/(kA), series resistances add, and parallel conductances add under common node temperatures.

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Thermal Strain with Mechanical Loading

In one-dimensional linear thermoelasticity, total strain is alpha Delta T plus sigma/Y with a consistent tensile sign convention; constraints determine the stress needed for compatibility.

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Thermal Stress, Composite Rods, and Bimetallic Bending

In one-dimensional linear thermoelasticity the total strain is sigma/Y + alpha Delta T; constraints and interface compatibility determine sigma, while a bimetallic strip curves because its bonded layers have unequal free thermal strains.

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Viscosity, Shear Stress, and Flow Regime

For a Newtonian fluid, dynamic viscosity is the proportionality between shear stress and the transverse velocity gradient; Reynolds number compares inertial and viscous effects.

MixedJEE Physics · Original

Wien Displacement and Blackbody Spectra

Wien's displacement law states lambda_max T=b for an ideal blackbody, relating absolute temperature to the wavelength of maximum spectral radiance while the complete curve is governed by Planck's distribution.

MixedJEE Physics · Original

Young's Modulus and Axial Elongation

For a uniform prismatic member in small linear tension or compression, Young's modulus gives Delta L = FL/(AY), with force, area, and material properties constant along the segment.

MixedJEE Physics · Original

Thermodynamic Processes and P-V Diagrams

Study different thermodynamic processes (isothermal, isobaric, isochoric, adiabatic, polytropic, cyclic), their definitions, characteristics, and graphical representation on P-V diagrams.

Exam level9 questions

Internal Energy, Heat, and Work

Explore the concepts of internal energy, heat, and work, including their definitions, relationships, and how they change during various thermodynamic processes.

Exam level7 questions

Laws of Thermodynamics

Understand the zeroth and first laws of thermodynamics, including their statements, implications, and applications to physical systems.

Exam level7 questions

Ideal Gas Law and Equation

Learn the ideal gas equation, its relation to physical quantities like pressure, volume, temperature, and density, and its use in describing the behavior of ideal gases.

Exam level3 questions

Carnot Cycle and Reversible Performance Limit

For reservoirs at absolute temperatures T_H > T_C, a reversible Carnot engine has Q_C/Q_H = T_C/T_H and η_C = 1 − T_C/T_H; every irreversible engine between the same reservoirs has η < η_C.

MixedJEE Physics · Original

Coupled Pistons, Partitions, and Multistage Processes

For gases separated by a frictionless piston or subjected to staged constraints, each compartment obeys its own equation of state and first law, while piston equilibrium imposes force balance and shared boundary displacement with equal-and-opposite intercompartment work.

MixedJEE Physics · Original

Entropy and the Second Law

Entropy is a state function defined for a reversible differential transfer by dS = δQ_rev/T; for any process, ΔS_universe = ΔS_system + ΔS_surroundings is nonnegative, with equality only for a reversible process.

MixedJEE Physics · Original

First Law and Thermodynamic Sign Convention

Using the convention Q positive into the system and W positive when done by the system, the first law is ΔU = Q − W; reversing the work convention requires changing the written sign consistently rather than mixing conventions.

MixedJEE Physics · Original

Heat Capacities, Degrees of Freedom, and Mayer Relation

For a calorically perfect ideal gas with f active quadratic degrees of freedom, C_V = fR/2, C_P = C_V + R, and γ = C_P/C_V; vibrational modes are included only when thermally active as stated.

MixedJEE Physics · Original

Heat Engines and Efficiency

For a cyclic heat engine absorbing magnitude Q_H from a hot reservoir and rejecting magnitude Q_C to a cold reservoir, W = Q_H − Q_C and η = W/Q_H = 1 − Q_C/Q_H, with 0 ≤ η < 1.

MixedJEE Physics · Original

Internal Energy of an Ideal Gas

For a fixed amount of ideal gas with temperature-independent molar heat capacity, internal energy is a state function satisfying ΔU = nC_VΔT; therefore any isothermal ideal-gas process has ΔU = 0.

MixedJEE Physics · Original

Irreversible, Sudden, and Free Adiabatic Changes

For an irreversible adiabatic process Q = 0 but generally PV^γ is not constant; boundary work is evaluated from the external pressure, and ideal-gas free expansion into vacuum has W = 0, ΔU = 0, and ΔT = 0.

MixedJEE Physics · Original

Isobaric Processes

For a quasistatic isobaric ideal-gas process at constant external pressure, W_by = PΔV = nRΔT and Q = nC_PΔT, assuming only pressure-volume boundary work.

MixedJEE Physics · Original

Isochoric Processes

An isochoric process has dV = 0 and hence quasistatic pressure-volume work W = ∫P dV = 0; for an ideal gas with constant C_V, Q = ΔU = nC_VΔT.

MixedJEE Physics · Original

Isothermal Ideal-Gas Processes

For a fixed ideal gas undergoing a quasistatic isothermal process at absolute temperature T, PV is constant, ΔU = 0, and W_by = Q = nRT ln(V_f/V_i), positive for expansion under the stated convention.

MixedJEE Physics · Original

Phase Change, Latent Heat, and Internal Energy

At fixed transition temperature and pressure, latent heat satisfies Q = mL; the internal-energy change is ΔU = Q − PΔV for quasistatic pressure-volume work, so latent heat and internal-energy change need not be identical.

MixedJEE Physics · Original

Polytropic and General Quasistatic Paths

For a quasistatic ideal-gas path described by P(V) or PV^k = constant, work is ∫P(V)dV and heat follows Q = ΔU + W; for constant k ≠ 1, W_by = (P_fV_f − P_iV_i)/(1−k).

MixedJEE Physics · Original

Pressure-Volume Work and Path Dependence

For quasistatic compression or expansion with work by the system positive, W_by = ∫P dV; geometrically it is signed area under the P-V path, while ΔU depends only on endpoint states.

MixedJEE Physics · Original

Reading and Transforming Thermodynamic Diagrams

For a fixed ideal gas, every plotted state satisfies PV = nRT; a process curve is transformed by combining this equation with its constraint, while orientation and endpoint ordering must remain physically consistent.

MixedJEE Physics · Original

Refrigerators and Heat Pumps

For a cyclic refrigerator, W = Q_H − Q_C, COP_R = Q_C/W, and COP_HP = Q_H/W = COP_R + 1; reversible values are T_C/(T_H−T_C) and T_H/(T_H−T_C), respectively, in kelvin.

MixedJEE Physics · Original

Reversible Adiabatic Ideal-Gas Processes

For a quasistatic reversible adiabatic process of a calorically perfect ideal gas, Q = 0 and PV^γ, TV^(γ−1), and T^γP^(1−γ) remain constant, with γ = C_P/C_V.

MixedJEE Physics · Original

Thermodynamic Bulk Modulus and Compressibility

The process bulk modulus is B = −V(dP/dV) along the stated path; an ideal gas has B_T = P isothermally and B_S = γP for a reversible adiabatic change.

MixedJEE Physics · Original

Thermodynamic Cycles and Enclosed Area

For any closed thermodynamic cycle, ΔU_cycle = 0 and Q_net = W_net under the heat-in/work-by convention; on a P-V plot W_net = ∮P dV, positive for a clockwise loop.

MixedJEE Physics · Original

Thermodynamic Systems, Equilibrium, and State Variables

A thermodynamic system is a specified quantity of matter or region separated by a boundary; at equilibrium its macroscopic state variables are time-independent, with intensive properties independent of size and extensive properties additive with system size.

MixedJEE Physics · Original

Ideal gas law and gas laws

The ideal gas law and related gas laws describe the relationships between pressure, volume, temperature, and number of moles for ideal gases.

Foundation8 questions

Degrees of freedom and thermal properties

Degrees of freedom determine the distribution of energy among molecules, affecting internal energy, specific heats, and the ratio of specific heats (γ).

Foundation5 questions

Kinetic theory and molecular motion

The kinetic theory explains the behavior of gases in terms of the motion and collisions of their molecules, relating properties like pressure, temperature, and kinetic energy.

Foundation3 questions

Mean free path and collisions

Mean free path is the average distance a molecule travels between collisions, and depends on molecular size and number density.

Foundation3 questions

RMS speed and temperature dependence

The root mean square (rms) speed of gas molecules depends on temperature and molar mass, and is a key measure of molecular motion in gases.

Exam level2 questions

Adiabatic Ideal-Gas Processes

For a quasistatic reversible adiabatic process of an ideal gas with constant gamma, PV^gamma, TV^(gamma-1), and T^gamma P^(1-gamma) are constants.

MixedJEE Physics · Original

Collision Frequency and Mean Collision Time

In the dilute hard-sphere model, z = v_avg/lambda and tau = lambda/v_avg; scaling must account for speed, number density, and collision diameter.

MixedJEE Physics · Original

Connected Vessels and Final Gas Equilibrium

For connected ideal-gas compartments at final common pressure P_f, conservation of amount gives sum(P_iV_i/RT_i) = P_f sum(V_j/RT_j), unless matter enters or leaves the system.

MixedJEE Physics · Original

Degrees of Freedom and Molecular Models

In the classical active-mode model, a monoatomic molecule has f=3, a rigid linear molecule usually f=5, and a rigid nonlinear molecule f=6; each active vibration adds two quadratic terms.

MixedJEE Physics · Original

Equipartition and Internal Energy

For an ideal gas with f active quadratic degrees of freedom, U = (f/2)nRT and Delta U = nC_V Delta T, assuming f remains unchanged across the temperature interval.

MixedJEE Physics · Original

Gas Laws, Process Constraints, and State Graphs

For a fixed amount of ideal gas, P1V1/T1 = P2V2/T2; isothermal, isobaric, and isochoric laws follow by holding T, P, or V constant, respectively.

MixedJEE Physics · Original

Gas Pressure Gradients in Body-Force Fields

Mechanical equilibrium of a fluid element gives grad P = rho g_eff; with an isothermal ideal gas, density depends on pressure and the relation integrates exponentially.

MixedJEE Physics · Original

Heat Capacities and Gamma of Gas Mixtures

For nonreacting ideal gases at common temperature, C_V,mix = sum(n_i C_Vi)/sum n_i and C_P,mix = C_V,mix+R; gamma_mix=C_P,mix/C_V,mix.

MixedJEE Physics · Original

Ideal-Gas Equation and Molecular Form

For a dilute ideal gas in thermal equilibrium, the state variables satisfy PV = nRT = Nk_B T, where intermolecular potential energy and molecular volume are neglected.

MixedJEE Physics · Original

Ideal-Gas Mixtures, Partial Pressure, and Composition

Dalton's law states P = sum_i P_i with P_iV = n_iRT for a nonreacting ideal-gas mixture at common T and V; mole fraction gives P_i = x_iP.

MixedJEE Physics · Original

Isobaric Heating and First-Law Energy Split

With work by the gas positive, the first law is Q=Delta U+W; for a fixed ideal-gas sample heated isobarically, W=nRDelta T and Q=nC_PDelta T.

MixedJEE Physics · Original

Maxwell Speed Measures and Ordering

For a classical ideal gas in Maxwell-Boltzmann equilibrium, v_p = sqrt(2k_BT/m), v_avg = sqrt(8k_BT/(pi m)), and v_rms = sqrt(3k_BT/m).

MixedJEE Physics · Original

Mean Free Path

For hard-sphere molecules in a dilute equilibrium gas, lambda = 1/(sqrt(2) pi d^2 n_number) = k_BT/(sqrt(2) pi d^2 P).

MixedJEE Physics · Original

Microscopic Origin of Gas Pressure

For an isotropic dilute ideal gas, elastic wall collisions give P = (1/3)rho v_rms^2 = (2/3)(translational kinetic-energy density).

MixedJEE Physics · Original

Mixed Kinetic Applications and Model Limits

Ideal-gas kinetic relations remain valid only within their assumptions; mixed applications must add the relevant conservation law or constitutive relation without silently treating real-gas or phase effects as ideal.

MixedJEE Physics · Original

Pistons, Partitions, Springs, and Mechanical Equilibrium

For a quasistatic frictionless piston of area A, mechanical equilibrium requires (P_left-P_right)A plus all other signed external forces to vanish.

MixedJEE Physics · Original

RMS-Speed Scaling with Temperature and Molar Mass

For an ideal gas, v_rms = sqrt(3RT/M); hence speed ratios depend on sqrt(T/M) and do not directly depend on gas pressure.

MixedJEE Physics · Original

Specific Heats, Mayer Relation, and Gamma

For an ideal gas with fixed f, C_V=fR/2, C_P=C_V+R, and gamma=C_P/C_V=1+2/f; these are molar heat capacities.

MixedJEE Physics · Original

Speed of Sound in an Ideal Gas

For small adiabatic disturbances in an ideal gas, c = sqrt(gamma P/rho) = sqrt(gamma RT/M), with equilibrium properties evaluated locally.

MixedJEE Physics · Original

Temperature and Translational Kinetic Energy

Equipartition assigns (1/2)k_BT to each independent quadratic translational coordinate, so every ideal-gas molecule has mean translational energy (3/2)k_BT.

MixedJEE Physics · Original

Standing Waves and Harmonics

Understand the formation of standing waves, harmonics in strings and organ pipes (open and closed), and the relationship between displacement and pressure in stationary waves.

Exam level5 questions

Superposition, Beats, and Resonance

Explore the principles of superposition, formation of beats, beat frequency, resonance, and phase difference in wave phenomena.

Exam level5 questions

Velocity and Acceleration in SHM

Explore how velocity and acceleration vary with displacement and time in SHM, including maximum values and their physical significance.

Exam level5 questions

Wave Motion and Wave Equation

Learn the fundamentals of wave motion, including the wave equation, its parameters, direction, and the distinction between particle and wave velocities.

Exam level4 questions

Basics of Simple Harmonic Motion (SHM)

Understand the defining features of SHM, including its standard equations, conditions, and the relationship between displacement, velocity, and acceleration.

Exam level3 questions

Simple Pendulum and Spring Systems

Study the time period formulas, energy considerations, and behavior of simple pendulums and spring-mass systems, including effects of non-inertial frames and spring combinations.

Exam level3 questions

Energy in Simple Harmonic Motion

Learn how kinetic and potential energy are distributed and conserved in SHM, including the concept of energy equipartition.

Foundation2 questions

Relation between Circular Motion and SHM

Understand how SHM can be viewed as the projection of uniform circular motion and how perpendicular SHMs combine.

Exam level2 questions

Periodic and Non-Periodic Motion

Distinguish between periodic and non-periodic motions, and recognize examples such as exponential functions.

Foundation1 question

Beats and Slowly Varying Envelopes

When two sinusoids of frequencies f1 and f2 are superposed at one point, their resultant has an amplitude envelope varying at half the angular-frequency difference and successive loudness maxima occur at beat frequency absolute(f1-f2).

MixedJEE Physics · Original

Buoyancy and Fluid-Column Oscillations

For small vertical displacement x of a prismatic floater, the incremental buoyant force is -rho g A x, giving omega^2 = rho g A/M; analogous hydrostatic stiffness applies to liquid columns.

MixedJEE Physics · Original

Closed-Pipe Odd Harmonics

For an ideal pipe closed at one end and open at the other, a displacement node and antinode bound the column, so allowed frequencies are f_m=(2m-1)v/(4L_eff), with m=1,2,3,... and even harmonics absent.

MixedJEE Physics · Original

Contact, Friction, Detachment, and Sudden Changes

Across an ideal instantaneous event, position is continuous; momentum is conserved only for an isolated collision, while the post-event amplitude follows from the new equilibrium, frequency, position, and velocity.

MixedJEE Physics · Original

Doppler Effect for Moving Source and Observer

For sound in a stationary medium, the classical Doppler frequency is f_obs=f_src (v plus signed observer velocity)/(v minus signed source velocity), with signs chosen to increase frequency for mutual approach.

MixedJEE Physics · Original

Energy Partition in SHM

For an ideal linear oscillator, E = kA^2/2 is constant, U = kx^2/2, and K = k(A^2 - x^2)/2, provided x is measured from equilibrium.

MixedJEE Physics · Original

Equivalent Stiffness and Elastic Elements

The effective stiffness is defined by the quadratic elastic energy U = k_eff q^2/2 or generalized force Q = -k_eff q; series compliances add and parallel stiffnesses add.

MixedJEE Physics · Original

Fixed-String Harmonics and Sonometer Relations

For a uniform ideal string of length L fixed at both ends, allowed modes satisfy L=n lambda_n/2 and have frequencies f_n=n sqrt(T/mu)/(2L), where n is a positive integer.

MixedJEE Physics · Original

Interference, Phase Difference, and Path Difference

Interference is the position-dependent redistribution of intensity produced by coherent superposition, with phase difference delta equal to initial phase difference plus 2pi times path difference divided by wavelength.

MixedJEE Physics · Original

Longitudinal Sound: Pressure and Displacement

For a small-amplitude plane longitudinal wave, excess pressure is minus the bulk modulus times the spatial derivative of particle displacement, making pressure amplitude Bk times displacement amplitude and placing pressure and displacement ninety degrees out of phase.

MixedJEE Physics · Original

Normal Modes in Plates and Higher Dimensions

For separable small transverse oscillations of a rectangular membrane or plate idealized by the wave equation, normal modes have spatial factors X(x)Y(y) chosen to satisfy all edge conditions, with squared modal frequency proportional to kx squared plus ky squared.

MixedJEE Physics · Original

Open-Pipe Harmonics

For an ideal uniform air column of effective length L_eff open at both ends, displacement antinodes occur at both boundaries and allowed frequencies are f_n=n v/(2L_eff) for every positive integer n.

MixedJEE Physics · Original

Oscillation Through a Uniform Earth

For uniform density and radius R, the interior field is g(r) = g_surface r/R toward the centre; its component along a straight frictionless tunnel is linear in tunnel displacement.

MixedJEE Physics · Original

Pendulum in Effective Gravity and Fluids

For small pendulum motion in a uniformly accelerating frame, T = 2 pi sqrt(l/g_eff), where g_eff is the magnitude of g minus frame acceleration; buoyancy modifies the restoring weight separately.

MixedJEE Physics · Original

Period Scaling, Clocks, and Measurement

Oscillator periods scale as T proportional to sqrt(inertia/restoring coefficient); for a simple pendulum T proportional to sqrt(l/g), enabling clock-rate and graph analysis.

MixedJEE Physics · Original

Phase Intervals, Travel Time, and Distance

For x = A cos theta with theta advancing uniformly at omega, elapsed time is Delta theta/omega; total distance must be accumulated across every turning point crossed.

MixedJEE Physics · Original

Phase Space of an Oscillator

For a harmonic oscillator, p^2/(m^2 omega^2 A^2) + x^2/A^2 = 1, so constant-energy trajectories are ellipses and their orientation shows time flow.

MixedJEE Physics · Original

Physical and Compound Pendulums

A rigid body pivoted a distance d from its centre of mass has small-angle equation I_O theta double-dot + Mgd theta = 0 and period 2 pi sqrt(I_O/(Mgd)).

MixedJEE Physics · Original

Potential Curvature and Nonlinear Oscillations

If U has a stable minimum at x0, then U is approximately U(x0) + U''(x0)(x-x0)^2/2 and omega^2 = U''(x0)/m; higher terms govern nonlinear corrections.

MixedJEE Physics · Original

Progressive-Wave Equation and Parameters

A one-dimensional harmonic progressive wave is a field y(x,t)=A sin(kx minus-or-plus omega t plus phi), where constant phase points translate with speed omega/k without requiring the medium particles to travel with the wave.

MixedJEE Physics · Original

Reflected-Sound Doppler Shift and Echo Beats

For specular sound reflection in the classical low-speed regime, the reflector receives a Doppler-shifted frequency and reradiates that received frequency, so the return must be calculated by two sequential source-observer transformations.

MixedJEE Physics · Original

Reflection, Transmission, and Wave Impedance

At a junction, continuity conditions determine reflection and transmission amplitudes through the characteristic impedances of the two media; reflected displacement reverses phase when the second medium presents the larger effective impedance.

MixedJEE Physics · Original

Resonance Column and End Correction

In a narrow resonance tube closed by water, the effective air-column length is L+e and resonances obey L_n+e=(2n-1)lambda/4, so consecutive observed lengths differ by lambda/2.

MixedJEE Physics · Original

Rigid Bodies and Constrained Spring Systems

For a small generalized displacement q, a constrained elastic system satisfies I_eff q double-dot + k_q q = 0, where k_q follows from U = k_q q^2/2.

MixedJEE Physics · Original

SHM Criterion and Restoring Dynamics

For one-dimensional ideal SHM about x = 0, the net force obeys F = -kx, equivalently d2x/dt2 = -omega^2 x, with constant positive k and omega.

MixedJEE Physics · Original

Simple Pendulum and Small-Angle Motion

For a point bob on a light inextensible string of length l and small angular displacement, theta double-dot + (g/l)theta = 0 and T = 2 pi sqrt(l/g).

MixedJEE Physics · Original

Single Mass-Spring Oscillators

For a mass m attached to an ideal Hookean spring of effective stiffness k, oscillations about equilibrium have omega = sqrt(k/m), independent of amplitude in the linear regime.

MixedJEE Physics · Original

Sinusoidal State and Initial Phase

An ideal SHM may be written x = A sin(omega t + phi), where A is the nonnegative amplitude and phi is fixed by the simultaneous initial displacement and velocity.

MixedJEE Physics · Original

Sound Properties, Temperature, and Refraction

In linear acoustics, pitch is primarily associated with frequency, loudness with received intensity, and quality with harmonic waveform; at a stationary boundary frequency remains continuous while wavelength and propagation direction adjust to the new speed.

MixedJEE Physics · Original

Standing Waves, Nodes, and Antinodes

A standing wave is the superposition of equal-frequency, equal-speed counter-propagating waves, expressible as a product of a spatial amplitude factor and a temporal oscillation factor, with adjacent nodes separated by lambda/2.

MixedJEE Physics · Original

Superposition and Phasor Addition

The superposition principle states that in a linear medium the net disturbance equals the algebraic sum of individual disturbances; equal-frequency sinusoids combine through vector addition of their complex amplitudes.

MixedJEE Physics · Original

Superposition, Phasors, and Lissajous Motion

Linear SHMs of common angular frequency superpose to another SHM whose complex amplitude is the vector sum; perpendicular components may instead trace a Lissajous curve.

MixedJEE Physics · Original

Torsional Oscillations

For angular displacement theta and torsion constant kappa, the restoring torque is -kappa theta and a body of moment of inertia I has omega = sqrt(kappa/I).

MixedJEE Physics · Original

Transverse Waves on a Uniform String

For a small-slope transverse disturbance on a uniform flexible string under constant tension T and linear density mu, the propagation speed is sqrt(T/mu), independent of frequency in the ideal nondispersive model.

MixedJEE Physics · Original

Two-Body Relative Oscillation and Collision

For masses m1 and m2 connected by a spring k on a force-free line, the relative coordinate obeys mu r double-dot = -kr with reduced mass mu = m1m2/(m1+m2).

MixedJEE Physics · Original

Variable-Tension and Composite Strings

For a slowly varying string the local speed is v(x)=sqrt(T(x)/mu(x)) and travel time is the integral of dx/v(x); at ideal junctions frequency is common while wavelength and mode count adjust to each section's speed.

MixedJEE Physics · Original

Velocity, Acceleration, and Extrema

Differentiating sinusoidal displacement gives v = A omega cos(theta) and a = -omega^2 x, so v^2 = omega^2(A^2 - x^2) for ideal SHM.

MixedJEE Physics · Original

Wave Energy, Power, and Intensity

Wave intensity is time-averaged power per area normal to propagation; for a linear harmonic wave in a fixed medium it is proportional to amplitude squared, and a lossless spherical wave has intensity proportional to 1/r squared.

MixedJEE Physics · Original

Wave Pulses and Kinematic Translation

Any differentiable pulse y(x,t)=F(x minus vt) is a right-moving solution and y(x,t)=F(x plus vt) is a left-moving solution of the one-dimensional wave equation in a uniform nondispersive medium.

MixedJEE Physics · Original

Wave Speed in Elastic Media

For small disturbances in a homogeneous elastic medium, wave speed is the square root of the appropriate elastic modulus divided by mass density, with the modulus chosen for the actual deformation mode.

MixedJEE Physics · Original

Energy stored in capacitors and conservation of charge

Examine how energy is stored, transferred, or lost in capacitors, including during charging, discharging, and redistribution, and the principle of charge conservation.

Exam level7 questions

Capacitors, capacitance, and combinations

Learn about parallel plate capacitors, series and parallel combinations, and how capacitance changes with geometry and dielectrics.

Foundation6 questions

Electric dipoles and molecular dipole moments

Study the properties of electric dipoles, their fields and potentials, behavior in electric fields, and the distinction between polar and non-polar molecules.

Exam level6 questions

Gauss's law and its applications

Learn Gauss's law, electric flux, and how to use symmetry to find electric fields of charged spheres, shells, and other symmetric objects.

Foundation6 questions

Dielectrics and their effect on capacitors

Study how inserting dielectrics affects capacitance, electric field, and potential difference in various capacitor setups.

Exam level4 questions

Electrostatic potential and potential energy

Explore the concept of electric potential, its relation to electric field, work done in moving charges, and potential energy in electrostatics.

Foundation4 questions

Conductors, charge distribution, and electrostatic shielding

Understand how charges distribute on conductors, the concept of electrostatic shielding, and the minimization of potential energy in conductors.

Exam level3 questions

Coulomb's law and electric field of point charges

Understand Coulomb's law, the principle of superposition, and how to calculate the electric field due to point charges and simple charge distributions.

Exam level3 questions

Axial Field of Rings and Discs

A uniformly charged ring has axial field kQx/(x^2+R^2)^(3/2). A disc follows by integrating rings; formulas assume uniform charge and an on-axis point.

MixedJEE Physics · Original

Capacitance and the Parallel-Plate Model

For a linear capacitor C=Q/V. An ideal parallel-plate capacitor neglects fringing and has uniform dielectric, plate area much larger than separation squared, and C=epsilon A/d.

MixedJEE Physics · Original

Capacitor Current and Time-Varying Capacitance

For a lumped capacitor q=Cv, differentiating gives i=C dv/dt+v dC/dt; the familiar i=C dv/dt assumes constant capacitance and a consistent passive sign convention.

MixedJEE Physics · Original

Capacitor Energy Formula Selection

For a linear quasistatically charged capacitor, stored field energy is one half QV; substituting Q=CV gives equivalent fixed-V and fixed-Q forms.

MixedJEE Physics · Original

Capacitors in Parallel

For capacitors connected between the same two nodes, voltage is common, total charge is the sum of branch charges, and equivalent capacitance is the direct sum.

MixedJEE Physics · Original

Capacitors in Series

For initially neutral internal nodes in a series chain, each capacitor has the same charge magnitude and reciprocal capacitances add.

MixedJEE Physics · Original

Charge Quantization, Conservation, and Mass

For ordinary charging, q = ne with integer n. Total charge of an isolated system is constant, while a body's mass changes by the transferred electron mass when that effect is asked.

MixedJEE Physics · Original

Charge Sharing by Contact

Charge is conserved during contact. Identical spheres at large separation acquire q_f = (q_1 + q_2)/2; unequal conductors require equal-potential rather than equal-charge reasoning.

MixedJEE Physics · Original

Charge Sharing with Like Polarity

For capacitors connected in parallel with like polarity after sources are removed, algebraic total charge is conserved and V_f=(C_1V_1+C_2V_2)/(C_1+C_2).

MixedJEE Physics · Original

Charged Drops and Coalescence

For n identical isolated conducting drops of radius r and charge q coalescing without charge loss, R=n^(1/3)r and Q=nq before computing the new spherical potential.

MixedJEE Physics · Original

Charged-Particle Motion by Energy Conservation

If only electrostatic forces do work, K+qV is conserved between points; additional forces or radiation invalidate the simple relation.

MixedJEE Physics · Original

Charged-Particle Motion in a Uniform Field

With constant E and no other force, a = qE/m. Apply constant-acceleration kinematics component-wise, reversing acceleration for negative q.

MixedJEE Physics · Original

Conductors as Equipotential Bodies

In electrostatic equilibrium E=0 within conducting material, potential is constant throughout a connected conductor, and excess free charge resides on surfaces subject to cavity conditions.

MixedJEE Physics · Original

Connected Conducting Spheres and Charge Sharing

For well-separated spherical conductors joined by a thin wire, V_1=V_2 and Q_1+Q_2 is conserved; mutual influence is neglected in the simple radius rule.

MixedJEE Physics · Original

Coulomb Force Between Point Charges

For stationary point charges in a homogeneous medium, the force magnitude is k|q_1q_2|/r^2 with k = 1/(4 pi epsilon); its direction lies on the separation vector.

MixedJEE Physics · Original

Cylindrical Charge Distributions

For an infinite solid cylinder of density rho and radius R, E_inside = rho r/(2 epsilon_0) and E_outside = rho R^2/(2 epsilon_0 r).

MixedJEE Physics · Original

Dielectric in an Isolated Capacitor

For an isolated ideal capacitor with no leakage, free plate charge Q is conserved; complete dielectric insertion gives C=KC_0, V=Q/C, and U=Q^2/(2C).

MixedJEE Physics · Original

Dielectric with Battery Connected

For complete insertion of a linear dielectric K while an ideal battery remains connected, V is fixed, C becomes KC_0, and battery exchange changes Q and U.

MixedJEE Physics · Original

Dielectrics and Polarization

For a linear homogeneous dielectric, epsilon = K epsilon_0 and Coulomb interactions are reduced by K. Polarization P is dipole moment per unit volume; random permanent dipoles have zero average P at zero field.

MixedJEE Physics · Original

Dipole Energy, Torque, and Rotation Work

For an ideal electric dipole in a uniform external field, torque is p cross E and orientation energy is -p dot E; quasistatic external work equals the energy change.

MixedJEE Physics · Original

Dipole Potential Energy

In a uniform field, U = -p dot E = -pE cos theta, up to the conventional zero at 90 degrees. Quasistatic external work equals Delta U.

MixedJEE Physics · Original

Dipole Rotational Dynamics

The rotational equation is I theta_ddot = -pE sin theta. For small theta, omega = sqrt(pE/I); I must be computed about the actual rotation axis.

MixedJEE Physics · Original

Dipole-Dipole Interaction

For ideal dipoles separated by r, U = k[p_1 dot p_2 - 3(p_1 dot r_hat)(p_2 dot r_hat)]/r^3. Force follows from the spatial gradient of U.

MixedJEE Physics · Original

Electric Dipole Moment

For charges -q and +q separated by vector d from negative to positive, p = qd. The ideal-dipole approximation requires observation distance much larger than |d|.

MixedJEE Physics · Original

Electric Field from Potential

For a differentiable electrostatic potential, E=-grad V; in one-dimensional or radial symmetry this reduces to the negative derivative along the varying coordinate.

MixedJEE Physics · Original

Electric Field of an Ideal Dipole

For r much greater than dipole size, E = [3(p dot r_hat)r_hat-p]/(4 pi epsilon_0 r^3). Axial and equatorial forms follow from this expression.

MixedJEE Physics · Original

Electric Field of Discrete Charges

For point sources, E(r) = sum_i k q_i (r-r_i)/|r-r_i|^3. The field direction is the force direction on a positive test charge.

MixedJEE Physics · Original

Electric Field-Line Reasoning

Field lines are an oriented visualization of a vector field. A charged particle follows Newton's law, so inertia generally prevents its path from coinciding with a curved field line.

MixedJEE Physics · Original

Electric Flux Through an Open Surface

Electric flux is Phi = integral E dot dA. For a uniform field over a flat surface, Phi = E dot A = EA cos theta.

MixedJEE Physics · Original

Electric Force versus Gravity and a Medium

For two particles, F_e/F_g = k|q_1q_2|/(Gm_1m_2). In a dielectric, k becomes k_0/K; immersed bodies have effective weight (rho_body-rho_fluid)Vg.

MixedJEE Physics · Original

Electric-Dipole Potential

For r much larger than charge separation, an ideal dipole of moment p directed from negative to positive charge has potential k p dot rhat/r^2.

MixedJEE Physics · Original

Electrostatic Conductors and Spherical Shells

Within conductor material E = 0. A Gaussian surface there encloses zero net charge, fixing induced surface charge; total charge conservation then fixes outer-surface charge.

MixedJEE Physics · Original

Electrostatic Equilibrium and Stability

At equilibrium F(x_0) = 0. For a small displacement xi, stable one-dimensional equilibrium has F approximately -k_eff xi with k_eff > 0.

MixedJEE Physics · Original

Electrostatic Equilibrium and Stability

For one coordinate, equilibrium occurs where dU/dx=0; it is stable when d^2U/dx^2>0 and unstable when d^2U/dx^2<0, subject to the allowed constraints.

MixedJEE Physics · Original

Electrostatic Field Energy

In a linear isotropic medium, electrostatic energy density is one half E dot D; in vacuum D=epsilon_0 E. Integrate over the field region for total energy when appropriate.

MixedJEE Physics · Original

Electrostatic Force and Pressure on Plates

For ideal large plates, electrostatic pressure is one half epsilon E squared; force is pressure times overlap area, with edge effects neglected.

MixedJEE Physics · Original

Energy Density and Electric Field

In a linear isotropic electrostatic medium, u=(1/2)E dot D=(1/2)epsilon E^2; uniform-field total energy is u times field volume.

MixedJEE Physics · Original

Energy Loss During Redistribution

For two capacitors joined after source removal, compute U_i from each initial state and U_f from the common final voltage; Delta U=U_i-U_f is dissipated or radiated.

MixedJEE Physics · Original

Equipotential Surfaces

An equipotential surface has constant V. Therefore dV=0 for tangential displacement and E=-grad V has no tangential component in electrostatics.

MixedJEE Physics · Original

Field at the Centre of Charged Arcs

For a line charge on a circular arc, dE = k lambda dl/R^2 and dl = R dtheta; integrate vector components over the occupied angle.

MixedJEE Physics · Original

Field of a Uniform Spherical Shell

For a thin uniform spherical shell, E = 0 for r < R and E = kQ/r^2 for r > R. The ideal surface field is discontinuous by sigma/epsilon_0.

MixedJEE Physics · Original

Field of an Infinite Line Charge

For line density lambda in a homogeneous medium, E = lambda/(2 pi epsilon r), radial outward for positive lambda and inward for negative lambda.

MixedJEE Physics · Original

Field Scaling and Graphs

Characteristic magnitudes scale as plane r^0, infinite line r^-1, point charge r^-2, and ideal dipole r^-3 in their valid regimes.

MixedJEE Physics · Original

Fields in Cavities by Superposition

For uniformly charged bodies, write E_remaining = E_full + E_negative-cavity using correctly displaced centres. This method applies to charge-filled matter, not conductor shielding by itself.

MixedJEE Physics · Original

Fields of Infinite Charged Sheets

A nonconducting infinite sheet in vacuum gives magnitude |sigma|/(2 epsilon_0) on either side. Superpose sheets vectorially; a conductor surface has the appropriate boundary-field relation.

MixedJEE Physics · Original

Floating Nodes and Charge Neutrality

For each isolated node in an ideal capacitor network, the algebraic sum of plate charges attached to it equals its conserved initial node charge, often zero.

MixedJEE Physics · Original

Flux for Charges on Boundaries

Join identical solids around the boundary location, apply total flux q/epsilon_0 to the assembled closed surface, then divide by symmetry among copies and requested faces.

MixedJEE Physics · Original

Flux Through Cubes by Symmetry

For a central point charge, each cube face has Phi = q/(6 epsilon_0). Equivalent-face division is valid only when the full charge-surface geometry has the required symmetry.

MixedJEE Physics · Original

Force on a Movable Dielectric

For a quasistatic coordinate x, the electrical force follows the derivative of the appropriate total energy: at fixed Q, F=-d[Q^2/(2C)]/dx; with an ideal voltage source, F=(1/2)V^2 dC/dx after source work is included.

MixedJEE Physics · Original

Gauss's Law and Symmetry

For every closed surface, integral_closed E dot dA = Q_enclosed/epsilon_0. Replacing the integral by EA requires a justified constant magnitude and angle.

MixedJEE Physics · Original

Grounding and Electrostatic Induction

An ideal ground imposes V=0 relative to Earth; induced surface charges rearrange to satisfy conductor boundary conditions, and order matters when ground and external charges are removed.

MixedJEE Physics · Original

Infinite Charge Series

For collinear point charges, sum signed terms kq/r_n^2 along the axis. A valid infinite answer requires the resulting series to converge.

MixedJEE Physics · Original

Infinite Lines, Sheets, and Potential Difference

Infinite-source models have non-decaying or slowly decaying fields; a finite potential reference must be chosen and only potential differences are physically used.

MixedJEE Physics · Original

Layered Dielectrics by Thickness

For planar layers normal to the field with negligible fringing, each layer is a series capacitor of thickness d_i and permittivity epsilon_i.

MixedJEE Physics · Original

Mechanical Equilibrium of Charged Bodies

Static equilibrium requires vector sum F = 0. Resolve tension along convenient axes and calculate electric force from the actual field or other charges.

MixedJEE Physics · Original

Nonuniform Spherical Charge Density

Q_enclosed(r) = integral_0^r rho(r')4 pi r'^2 dr', and E(r) = Q_enclosed(r)/(4 pi epsilon_0 r^2). Normalize rho using the stated total charge when needed.

MixedJEE Physics · Original

Opposite-Polarity Charge Sharing

After batteries are removed, choose one final node polarity and conserve its algebraic charge; the capacitors in final parallel share the resulting voltage.

MixedJEE Physics · Original

Orbital Motion in Electrostatic Fields

For a circular orbit, mv^2/r = |q|E(r). In a point-charge field U = kQq/r, and with a central force angular momentum is conserved.

MixedJEE Physics · Original

Partial Dielectric by Area

Neglecting fringing, regions occupying disjoint plate areas with the same separation and common plate potentials form parallel capacitances.

MixedJEE Physics · Original

Planar Gauss-Law Boundary Relations

The normal field jump satisfies E_2n-E_1n = sigma/epsilon_0. An isolated nonconducting infinite sheet has equal opposite fields sigma/(2 epsilon_0); just outside a conductor E_n = sigma/epsilon_0.

MixedJEE Physics · Original

Point-Charge Potential and Superposition

With zero potential chosen at infinity, the potential of stationary point charges is the algebraic sum of kq_i/r_i at a field point not occupied by a point charge.

MixedJEE Physics · Original

Potential Difference and Conservative Work

For a time-independent electrostatic field, the work done by the field from A to B equals q(V_A-V_B); external quasistatic work is the opposite, with no kinetic-energy change.

MixedJEE Physics · Original

Potential Difference from Electric Field

For an electrostatic field, V_B-V_A=-integral_A^B E dot dl; path independence follows from curl E=0 in the region.

MixedJEE Physics · Original

Potential Energy of Charge Configurations

For stationary point charges assembled quasistatically from infinity, total electrostatic potential energy is the sum k q_i q_j/r_ij over i<j.

MixedJEE Physics · Original

Potential Graphs, Poisson Equation, and Charge Density

In a homogeneous linear medium, electrostatic potential satisfies div(grad V)=-rho/epsilon; charge-free regions satisfy Laplace equation.

MixedJEE Physics · Original

Potential of Continuous Charge Distributions

For a specified charge density and reference V(infinity)=0 where valid, integrate the point-charge contribution over the actual distribution without vector resolution.

MixedJEE Physics · Original

Potential of Spherical Shells and Spheres

For spherical symmetry and V(infinity)=0, use Gauss-law field outside and integrate inward; a conductor is equipotential, while a uniform volume charge has a nonzero interior field.

MixedJEE Physics · Original

Self-Force, Tension, and Electrostatic Pressure

Calculate force on an element from the field produced by all other charge, excluding its singular self-field. For conductor surfaces, electrostatic pressure is sigma^2/(2 epsilon_0).

MixedJEE Physics · Original

Shielding and Conducting Cavities

Electrostatic equilibrium enforces E = 0 in conductor material and constant conductor potential. An empty closed cavity has no field from external sources; a charged cavity induces net -q on its inner surface.

MixedJEE Physics · Original

Spatially Varying Dielectric

For planar plates with epsilon(x) varying only along the field and negligible fringing, electric displacement is uniform and the total voltage is the integral of D/epsilon(x).

MixedJEE Physics · Original

Spherical and Cylindrical Capacitors

With vacuum or uniform linear dielectric and negligible end effects, concentric spherical or long coaxial conductors have capacitance determined by their radii and geometry.

MixedJEE Physics · Original

Surface Charge and Electrostatic Pressure

Just outside a conductor E_n = sigma/epsilon_0, and the mechanical pressure from the field is P = sigma^2/(2 epsilon_0) in vacuum.

MixedJEE Physics · Original

Switching and Capacitor Network Redistribution

Ideal capacitor switching is treated by initial and final electrostatic states, with battery voltages, isolated-node charge conservation, and Q=C Delta V applied to each topology.

MixedJEE Physics · Original

Symmetry and Bridge Capacitor Networks

In a linear capacitor network, node charge balance and geometric/electrical symmetry determine node potentials; a capacitor between proven equipotential nodes is inactive.

MixedJEE Physics · Original

Symmetry and Mixed Potential Reasoning

Electrostatic potential is a scalar field obeying superposition; symmetry may set equal contributions or cancellations, but only after distances and charge signs are verified.

MixedJEE Physics · Original

Torque on an Electric Dipole

For a rigid dipole in a uniform field, tau = p x E and |tau| = pE sin theta, where theta is the angle from p to E.

MixedJEE Physics · Original

Uniformly Charged Solid Sphere

For volume density rho, E_inside = rho r/(3 epsilon_0). For r >= R, E = kQ/r^2. These results require spherical symmetry and uniform density.

MixedJEE Physics · Original

Vector Superposition of Forces

For discrete charges, F_net = sum_i F_i. Each F_i must be assigned its own direction before components are combined.

MixedJEE Physics · Original

Work-Energy in an Electric Field

For electrostatic fields, W_field = -Delta U = q(V_i-V_f) and Delta K = W_field when no other force does work.

MixedJEE Physics · Original

Zero-Field and Zero-Force Points

For a test charge q_0 != 0, zero force is equivalent to E_net = 0. Solve the vector equation with region and direction checks.

MixedJEE Physics · Original

Resistivity, Conductivity, and Material Properties

Learn how resistivity and conductivity depend on material, length, area, and how resistance changes with dimensions and temperature.

Exam level7 questions

Electrical Energy, Power, and Heating Effect

Calculate electrical energy, power dissipation, and understand the heating effect of current in various circuit configurations.

Exam level5 questions

EMF, Internal Resistance, and Terminal Voltage

Understand the concepts of EMF, internal resistance of cells, and how terminal voltage is affected in real circuits.

Foundation4 questions

Drift Velocity and Microscopic View of Current

Explore the microscopic origin of current, including drift velocity, relaxation time, current density, and the effect of electric field in conductors.

Challenging3 questions

Series and Parallel Combinations of Resistors

Master the calculation of equivalent resistance, voltage division, and current distribution in series and parallel resistor networks.

Exam level3 questions

Kirchhoff's Laws and Circuit Analysis

Apply Kirchhoff's current and voltage laws to analyze complex circuits and solve for unknowns.

Challenging2 questions

Ohm's Law and Basic Electrical Quantities

Understand Ohm's law, current, voltage, resistance, conductance, and how they relate in electrical circuits.

Exam level1 question

Wheatstone Bridge and Meter Bridge

Learn the principle and application of Wheatstone and meter bridges for measuring unknown resistances.

Exam level1 question

Appliance Ratings and Bulb Circuits

For a resistive appliance at rating, R_rated = V_rated squared/P_rated and I_rated = P_rated/V_rated. Actual power follows the actual circuit voltage or current; incandescent cold resistance can differ greatly.

MixedJEE Physics · Original

Bridge and Delta-Star Network Reduction

A delta-star transformation preserves resistance seen between all three external terminals. Each star arm equals the product of the two adjacent delta arms divided by the sum of all delta arms.

MixedJEE Physics · Original

Capacitors in DC Steady State

At DC steady state, replace an ideal capacitor by an open circuit, solve the resistive network for its terminal voltage, then use Q = CV and U = one-half CV squared.

MixedJEE Physics · Original

Cells in Parallel and Unequal Sources

For n identical parallel cells, E_eq = E and r_eq = r/n. For unequal parallel cells, use Kirchhoff equations or conductance-weighted Thevenin equivalence with consistent polarity.

MixedJEE Physics · Original

Cells in Series

For series cells, equivalent emf is the signed sum of emfs and equivalent internal resistance is the sum of internal resistances. Identical aiding cells give nE and nr.

MixedJEE Physics · Original

Current as Charge Flow

Conventional current is positive-charge flow. For a specified orientation, instantaneous current is dQ/dt and transferred charge is the time integral of current.

MixedJEE Physics · Original

Current Density and Continuity

Current density is current per normal area for uniform flow. Charge conservation gives equal steady current through successive sections; for fixed carrier density, Av_d remains constant.

MixedJEE Physics · Original

Drift Speed and Carrier Density

For one carrier species in a uniform conductor, current magnitude equals number density times carrier charge magnitude, cross-sectional area, and drift-speed magnitude.

MixedJEE Physics · Original

Effective Resistivity of Composite Conductors

For equal-length parallel wires, conductivities are area-weighted. For equal-area series layers, resistivities are length-weighted. Other geometries must be reduced from R = rho L/A.

MixedJEE Physics · Original

Electric Power in DC Circuits

Using the passive sign convention, positive VI means an element absorbs power and negative VI means it supplies power. For a resistor, absorbed power is nonnegative.

MixedJEE Physics · Original

Electrical Heating with Calorimetry

With efficiency eta, eta Pt equals mc Delta T plus mL for the included thermal stages. For an insulated thermodynamic system receiving heat and doing work, apply the first law with a declared sign convention.

MixedJEE Physics · Original

Electrical Work and Energy

For charge Q crossing a constant potential difference V, work is QV. For time-varying conditions, energy is the integral of VI over time; kilowatt-hour is an energy unit.

MixedJEE Physics · Original

EMF, Terminal Voltage, and Internal Resistance

For discharge current leaving the positive terminal, V_terminal = E - Ir. For charging current entering the positive terminal, V_terminal = E + Ir, with signs always checked from the chosen traversal.

MixedJEE Physics · Original

Finding Cell EMF and Internal Resistance from Loads

For a discharging cell connected to load R, E = I(R+r) and terminal voltage is IR. Two distinct loads produce simultaneous linear equations for E and r.

MixedJEE Physics · Original

Galvanometer Conversion and Half-Deflection

For galvanometer resistance G, ammeter range I needs shunt S such that IgG = (I-Ig)S. Voltmeter range V needs series resistance R = V/Ig - G. Half-deflection methods require the actual circuit, not an automatic S = G shortcut.

MixedJEE Physics · Original

Joule Heating

For an ohmic resistor with constant resistance, heat is I squared Rt, equivalently V squared t/R or VIt. Choose the form matching what is held fixed.

MixedJEE Physics · Original

Kirchhoff Junction Rule

Kirchhoff's current law is sum I = 0 at a node, using one consistent sign convention. It applies instantaneously when displacement or capacitor currents are represented appropriately.

MixedJEE Physics · Original

Kirchhoff Loop and Nodal Analysis

Kirchhoff's voltage law is the algebraic sum of potential changes around any lumped-circuit loop. Combine independent loop equations with junction equations to solve multi-source networks.

MixedJEE Physics · Original

Loaded Voltage Dividers

Replace the divider arm and load by their parallel equivalent before applying voltage division. An ideal voltmeter has infinite resistance and causes no loading.

MixedJEE Physics · Original

Maximum Power Transfer and Source Efficiency

For a linear source represented by Thevenin voltage Vth and resistance Rth, load power is Vth squared times R divided by (R+Rth) squared and is maximal at R = Rth. Efficiency there is 50 percent.

MixedJEE Physics · Original

Metal Resistance and Temperature Coefficient

Near a reference temperature T0 and over the stated linear range, R(T) = R0[1 + alpha(T-T0)]. Alpha belongs to the chosen reference convention and is not universally constant over large ranges.

MixedJEE Physics · Original

Meter Bridge and End Corrections

With gap resistances X and R and balance length l measured from X's end on a 100 cm ideal wire, X/R = l/(100-l). End corrections replace geometric lengths by their effective lengths.

MixedJEE Physics · Original

Meter Loading and Measurement Error

Model an ammeter by resistance RA in series at its insertion point and a voltmeter by RV in parallel with the measured element. Recompute the circuit before comparing measured and true values.

MixedJEE Physics · Original

Microscopic Ohm Law

For linear, homogeneous, isotropic conduction at fixed physical conditions, J equals sigma E. Combining E = V/L and I = JA for a uniform wire gives V = IR.

MixedJEE Physics · Original

Mobility, Relaxation Time, and Drude Conductivity

In the classical Drude model with field-independent relaxation time, mobility is drift speed per field and conductivity is nq times mobility; the electron form is ne squared tau over m.

MixedJEE Physics · Original

Ohmic and Non-Ohmic V-I Characteristics

An ohmic element obeys V = IR with constant R under fixed conditions. Graph slope depends on axes: slope of V versus I is R, while slope of I versus V is conductance.

MixedJEE Physics · Original

Parallel Resistors and Current Division

For ideal resistors connected between the same two nodes, conductances add. In two branches, current through one branch is inversely related to its own resistance.

MixedJEE Physics · Original

Potentiometer Comparison of EMFs

If the potential gradient k is unchanged, E = kl and E1/E2 = l1/l2. A valid null requires the test emf not exceed the available drop and correct polarity.

MixedJEE Physics · Original

Potentiometer Measurement of Internal Resistance

With balance length l1 for emf and l2 for terminal voltage across load R at unchanged gradient, E/V = l1/l2 and r = R(l1/l2 - 1).

MixedJEE Physics · Original

RC Charging and Discharging Transients

For one equivalent resistance seen by a capacitor, tau = Rth C. Charging from zero toward Vfinal gives Vc = Vfinal(1-e^-t/tau); natural discharge gives Vc = V0 e^-t/tau.

MixedJEE Physics · Original

Resistance from Resistivity and Geometry

For a homogeneous uniform conductor with constant cross-section and resistivity, resistance is rho L/A. Nonuniform conductors require summing differential elements dR = rho dl/A(l).

MixedJEE Physics · Original

Resistivity, Conductivity, and Material Choice

Bulk resistivity rho and conductivity sigma are reciprocals for an isotropic material. Their quoted values require a temperature and physical state; resistance follows only after geometry is included.

MixedJEE Physics · Original

Semiconductor Temperature Dependence

Semiconductors generally have a negative temperature coefficient over the elementary regime. Their resistance can vary nonlinearly, often with activated behavior, so the metal linear law is not automatically valid.

MixedJEE Physics · Original

Series Resistors and Voltage Division

For resistors connected end-to-end with no branching node, equivalent resistance is their sum. With an unloaded divider, each voltage drop is proportional to its resistance.

MixedJEE Physics · Original

Stretching, Recasting, and Cutting Wires

If material, temperature, and volume AL are unchanged, A is inversely proportional to L and R = rho L squared divided by volume. Cutting changes each piece resistance before any reconnection.

MixedJEE Physics · Original

Symmetry and Equipotential Nodes

Symmetry reduction is valid only when both network values and excitation respect the symmetry. Equipotential nodes may be joined; an element across them may be omitted for that operating condition.

MixedJEE Physics · Original

Systematic Reduction of Resistor Circuits

Ideal wires have zero voltage drop. Elements are parallel only if both endpoints match, and series only if their shared node has no other branch; after reduction, Ohm law recovers local quantities.

MixedJEE Physics · Original

Transmission Loss, Efficiency, and Fuse Rating

A line of resistance R loses I squared R. Efficiency is useful output divided by input. For loads on a common voltage, total current is total rated power divided by supply voltage, then a suitable standard fuse is selected above normal current.

MixedJEE Physics · Original

Uniform Wire Loops, Arcs, and Polygons

For a uniform wire of total resistance R, an arc occupying fraction f has resistance fR. Two complementary arcs between taps are parallel; polygon sides are handled by their two path lengths when symmetry permits.

MixedJEE Physics · Original

Wheatstone Bridge Balance and Imbalance

For arms P, Q in one branch and R, S in the other, with corresponding midpoints joined by the detector, balance requires P/Q = R/S. Off balance, solve the full network or use a Thevenin equivalent across the detector terminals.

MixedJEE Physics · Original

Magnetic field due to currents

Learn how electric currents produce magnetic fields, including the use of Biot-Savart law, Ampere's law, and the calculation of fields for various conductor shapes such as straight wires, circular loops, and solenoids.

Exam level9 questions

Magnetic properties of materials

Learn about diamagnetic, paramagnetic, and ferromagnetic materials, their magnetic susceptibility, temperature dependence, Curie temperature, and the role of domains.

Foundation6 questions

Magnetic moment and properties of magnets

Explore the concept of magnetic moment for current loops and bar magnets, properties of divided magnets, and effective length of magnets.

Exam level5 questions

Force and motion in magnetic fields

Understand the forces experienced by current-carrying conductors and moving charges in magnetic fields, including the Lorentz force, force between parallel conductors, and the motion of charged particles.

Exam level4 questions

Magnetic dipoles and torque

Study the behavior of magnetic dipoles in magnetic fields, including torque, potential energy, and the vector addition of dipole moments.

Exam level3 questions

Magnetic flux and units

Study the concept of magnetic flux through surfaces and the units used to measure magnetic field strength.

Foundation2 questions

Galvanometer conversion and measurement devices

Understand how to convert a galvanometer into an ammeter or voltmeter using shunt and series resistances, and the principles behind these measuring instruments.

Foundation1 question

Ampere's Circuital Law and Symmetry

For steady currents, the closed line integral of B dot dl equals mu_0 times the net current piercing the chosen surface. The shortcut requires enough symmetry to know B's direction and constancy.

MixedJEE Physics · Original

Biot-Savart Law

For a steady filamentary current, dB = (mu_0/4pi) I(dl cross r_hat)/r^2. The law assumes magnetostatics and superposition; conductor geometry sets the integration limits.

MixedJEE Physics · Original

Circular Radius from Momentum

For v perpendicular to uniform B, magnetic force supplies centripetal force, giving r = p/(|q|B) = mv/(|q|B) in the non-relativistic limit.

MixedJEE Physics · Original

Comparing Diamagnetic, Paramagnetic, and Ferromagnetic Response

Diamagnets have chi_m < 0 and mu_r < 1; paramagnets have small chi_m > 0 and mu_r slightly above 1; ferromagnets have large, nonlinear, history-dependent response below Curie temperature.

MixedJEE Physics · Original

Converting a Galvanometer into a Voltmeter

For galvanometer resistance G and full-scale current I_g converted to voltmeter range V, series resistance R satisfies V = I_g(G + R), so R = V/I_g - G.

MixedJEE Physics · Original

Converting a Galvanometer into an Ammeter

For galvanometer resistance G and full-scale current I_g converted to ammeter range I, the shunt S satisfies I_g G = (I - I_g)S, so S = I_g G/(I - I_g).

MixedJEE Physics · Original

Cyclotron Period and Frequency

For perpendicular non-relativistic motion in uniform B, angular frequency omega_c = |q|B/m, period T = 2pi m/(|q|B), and frequency f = |q|B/(2pi m).

MixedJEE Physics · Original

Diamagnetism

Diamagnets have small negative susceptibility, relative permeability slightly below one, and magnetization opposite to H. Ideal perfect diamagnetism has chi_m = -1 and B = 0 inside under the idealized bulk conditions.

MixedJEE Physics · Original

Dipoles in Non-Uniform Magnetic Fields

An ideal fixed dipole in a non-uniform field has force F = grad(m dot B) and torque m cross B. In a uniform field the gradient force vanishes, although torque may remain.

MixedJEE Physics · Original

Earth's Magnetic Field and Tangent Law

If Earth's total field makes dip angle delta below horizontal, B_H = B_E cos delta. With a perpendicular horizontal coil field B_c, equilibrium gives tan theta = B_c/B_H.

MixedJEE Physics · Original

Earth's Magnetic Field, Dip, and Compass Response

If total field B makes dip angle delta below the horizontal, B_H = B cos delta and B_V = B sin delta. A horizontal magnetic needle has T = 2pi sqrt(I_rot/(m B_H)) for small oscillations.

MixedJEE Physics · Original

Ferromagnetic Domains and Curie Temperature

A ferromagnetic domain is a macroscopic region with spontaneous near-saturation magnetization. Below Curie temperature domains exist; above it a ferromagnet loses spontaneous magnetization and becomes paramagnetic.

MixedJEE Physics · Original

Field at the Center of Arcs and Loops

A circular arc of radius R and angle phi radians gives B = mu_0 I phi/(4pi R) at its center. A full N-turn circle gives B = mu_0 N I/(2R).

MixedJEE Physics · Original

Field Inside a Current-Carrying Cylinder

For a long solid wire of radius R carrying uniformly distributed current I, B(r) = mu_0 I r/(2pi R^2) for r <= R and B(r) = mu_0 I/(2pi r) for r >= R.

MixedJEE Physics · Original

Field Inside a Long Solenoid

For an ideal long air-core solenoid with n turns per unit length, B = mu_0 n I inside and approximately zero outside. End effects are neglected.

MixedJEE Physics · Original

Field of a Short Magnetic Dipole

At distance r much larger than the magnet size, a dipole m produces B = (mu_0/4pi r^3)[3(m dot r_hat)r_hat - m]. The axial magnitude is twice the equatorial magnitude at the same r.

MixedJEE Physics · Original

Field of a Straight Current-Carrying Wire

At perpendicular distance a from a finite straight segment, B = (mu_0 I/(4pi a))(sin theta_1 + sin theta_2), with endpoint angles measured from the perpendicular. The infinite-wire limit is mu_0 I/(2pi a).

MixedJEE Physics · Original

Fields of Multiple Parallel Wires

Each ideal long wire contributes B_i = mu_0 I_i/(2pi r_i). The total field is the vector sum; in a common plane the contributions can be treated as signed perpendicular components.

MixedJEE Physics · Original

Finite Bar Magnets, Shape, and Magnetic Length

In the pole model, magnetic moment m = p_m times 2l_m, where p_m is pole strength and 2l_m is magnetic length. For a thin strip bent without changing pole strength, the new moment scales with the straight-line separation of its ends.

MixedJEE Physics · Original

Finite Magnetic Region and Arc Geometry

Within a uniform field region and for perpendicular motion, the trajectory is an arc of radius r = p/(|q|B). Entry tangent, circle center, chord, sagitta, and boundary width determine exit coordinates and deflection.

MixedJEE Physics · Original

Force Between Parallel Currents

For two long parallel wires separated by d in vacuum, F/L = mu_0 I_1 I_2/(2pi d) in magnitude. The sign or direction follows the current senses.

MixedJEE Physics · Original

Force on a Shaped Wire in Uniform Field

With constant current I and uniform B, F = I integral(dl cross B) = I(L_end-to-end cross B). For a closed loop in uniform B, the endpoint displacement is zero, so net force is zero.

MixedJEE Physics · Original

Force on a Straight Current-Carrying Conductor

For a straight segment carrying steady current I in uniform B, F = I(L cross B), where L points along conventional current and has magnitude equal to segment length.

MixedJEE Physics · Original

Galvanometer Figure of Merit and Calibration

If a current I produces n scale divisions, the galvanometer figure of merit is K = I/n in ampere per division. For a reading n, I = Kn; angular calibration similarly uses I/theta.

MixedJEE Physics · Original

Galvanometer Voltage Sensitivity

With galvanometer resistance G and current sensitivity theta/I = NAB/k, the voltage sensitivity is theta/V = NAB/(kG), using V = IG for the bare galvanometer.

MixedJEE Physics · Original

General Motion in Combined Electric and Magnetic Fields

The total Lorentz force is q(E + v cross B). The electric term can do work at rate qE dot v; the magnetic term has zero power. Special geometries should be solved from this vector equation rather than from a memorized path name.

MixedJEE Physics · Original

Hall Effect and Carrier Sign

At equilibrium qE_H + q(v_d cross B) = 0. In the simple one-carrier model, Hall coefficient R_H = 1/(nq), and Hall voltage magnitude is V_H = BI/(n|q|t) for sample thickness t.

MixedJEE Physics · Original

Helical Motion and Pitch

In a uniform magnetic field, v_perp produces circular motion of radius mv_perp/(|q|B), while v_parallel is unchanged. The helix pitch is the parallel distance traveled in one cyclotron period.

MixedJEE Physics · Original

Hysteresis, Coercivity, and Magnetic Materials

A hysteresis loop is the cyclic B-H or M-H response of a ferromagnet. Coercive field H_c is the magnitude of reverse magnetizing field required to reduce B or M to the specified zero crossing after saturation.

MixedJEE Physics · Original

Interaction Between Magnetic Dipoles

For separated small dipoles, dipole 1 produces B_1 proportional to m_1/r^3. Dipole 2 then has torque tau_2 = m_2 cross B_1 and force F = grad(m_2 dot B_1), giving F proportional to r^-4 for fixed orientations.

MixedJEE Physics · Original

Lorentz Magnetic Force and Direction

For a point charge q moving with velocity v in magnetic field B, the magnetic part of the Lorentz force is q(v cross B). Its magnitude is |q|vB sin theta and it is perpendicular to both v and B.

MixedJEE Physics · Original

Magnetic Dipole Moment of a Current Loop

For a planar N-turn loop carrying steady current I, the magnetic dipole moment is the axial vector m = NIA n-hat, where n-hat is fixed by curling the right-hand fingers with the current.

MixedJEE Physics · Original

Magnetic Field Lines and Absence of Monopoles

Gauss's law for magnetism states that net magnetic flux through every closed surface is zero: closed integral B dot dA = 0. Equivalently, div B = 0, so magnetic field lines have no sources or sinks.

MixedJEE Physics · Original

Magnetic Field on a Circular Loop Axis

For an N-turn circular loop of radius R carrying current I, the field at axial distance x is B = mu_0 N I R^2/[2(R^2 + x^2)^(3/2)], directed along the loop axis.

MixedJEE Physics · Original

Magnetic Flux and Gauss's Law for Magnetism

Magnetic flux is Phi_B = integral B dot dA. Gauss's law for magnetism states closed_integral B dot dA = 0, equivalently div B = 0, expressing the absence of magnetic monopoles in classical electromagnetism.

MixedJEE Physics · Original

Magnetic Force Does No Work

Because q(v cross B) is perpendicular to v, the instantaneous magnetic power F dot v is zero. In a purely magnetic field, kinetic energy and speed are constant although momentum direction may change.

MixedJEE Physics · Original

Magnetic Moment of Currents and Rotating Charge

A planar N-turn loop has m = NIA. A total charge q uniformly rotating around a circular loop with angular speed omega has I = q omega/(2pi) and m = q omega r^2/2 in magnitude.

MixedJEE Physics · Original

Magnetic Potential Energy and Work

For a fixed dipole in a uniform field, U = -m dot B = -mB cos theta. Slow external work from theta_1 to theta_2 equals Delta U = mB(cos theta_1 - cos theta_2).

MixedJEE Physics · Original

Magnetic Potential Energy, Equilibrium, and Small Oscillations

For a dipole in uniform B, U = -m dot B = -mB cos theta. Near theta = 0, tau is approximately -mB theta, so a body of rotational inertia J has period 2pi sqrt(J/(mB)).

MixedJEE Physics · Original

Magnetic Tension, Pressure, and Mechanical Balance

For a current element, dF = I dl cross B. In symmetric arc problems, local transverse load IB per unit length balances tension curvature T/R, giving T = IBR under the ideal flexible-wire assumptions.

MixedJEE Physics · Original

Magnetic Units, Dimensions, and Scaling Checks

Tesla satisfies 1 T = 1 N/(A m) = 1 Wb/m^2. Magnetic moment has unit A m^2, flux has unit Wb, and mu_0 has unit T m/A. Dimensional consistency is necessary but not sufficient for a correct formula.

MixedJEE Physics · Original

Magnetic-Field Graphs and Scaling

Scaling follows the applicable field law: an infinite wire gives B proportional to r^-1, a loop's far axial field gives x^-3, and a uniform-current wire gives B proportional to r inside. Boundary continuity must match the physical model.

MixedJEE Physics · Original

Magnetization and Magnetic Moment per Volume

For a uniformly magnetized sample of volume V and total magnetic moment m_total, M = m_total/V. In a linear isotropic material, M = chi_m H.

MixedJEE Physics · Original

Magnetizing Field of Solenoids and Toroids

For a long solenoid H = nI and B = mu H. For a toroid H = NI/(2pi r) locally. In a linear core, B_core/B_air = mu_r and the fractional increase is approximately chi_m.

MixedJEE Physics · Original

Mass Spectrometer and Charge-to-Mass Separation

After a velocity selector v = E/B_s, an analyzing field B_a gives r = mv/(|q|B_a). If ions are accelerated through V instead, r^2 = 2mV/(|q|B_a^2).

MixedJEE Physics · Original

Mixed Magnetic Problems and Consistency Tests

There is no single new law: use the relevant validated relations stage by stage. A reliable result must satisfy vector direction, SI dimensions, limiting cases, and the assumptions of every formula used.

MixedJEE Physics · Original

Moving-Coil Galvanometer Torque and Current Sensitivity

For N turns of area A in radial field B, equilibrium gives NABI = k theta. Current sensitivity is theta/I = NAB/k, assuming linear spring torque k theta.

MixedJEE Physics · Original

Multi-Range Meters and Shunt Networks

Each ammeter range obeys equal voltage across the galvanometer and its effective shunt; each voltmeter range obeys V_range = I_g times total series resistance. Ayrton networks must be reduced in the actual switch position.

MixedJEE Physics · Original

Net Force on a Loop in a Nonuniform Field

The exact force is I closed_integral(dl cross B). For a small magnetic dipole in a slowly varying field, F is approximately grad(m dot B). Uniform-field cancellation does not apply when B varies across the loop.

MixedJEE Physics · Original

Orbital Magnetic Moment and Angular Momentum

For a particle of charge q and mass m in orbital motion, mu_L = (q/2m)L. For an electron q = -e, so its orbital magnetic moment points opposite to its angular momentum.

MixedJEE Physics · Original

Orbital Magnetic Moment and Angular Momentum

For a point particle of charge q and mass m in orbital motion, mu_L = (q/2m)L. For an electron, q is negative, so magnetic moment and angular momentum point oppositely.

MixedJEE Physics · Original

Paramagnetism and Curie's Law

A linear paramagnet has small positive susceptibility and M parallel to H. For an ideal Curie paramagnet, chi_m = C/T, so at fixed material density M is proportional to H/T.

MixedJEE Physics · Original

Radius after Electric Acceleration

For a particle starting from rest and accelerated through potential difference V, K = |q|V. Non-relativistically, p = sqrt(2m|q|V), so perpendicular entry gives r = sqrt(2mV/(|q|B^2)).

MixedJEE Physics · Original

Relations Among B, H, M, Susceptibility, and Permeability

In SI for a linear isotropic medium, B = mu_0(H + M), M = chi_m H, and B = mu H. Therefore mu_r = mu/mu_0 = 1 + chi_m.

MixedJEE Physics · Original

Small Oscillations of a Magnetic Dipole

For small theta, I_rot theta_double_dot = -mB theta. Thus angular frequency is sqrt(mB/I_rot) and period is 2pi sqrt(I_rot/(mB)). The relevant field for a horizontal needle is Earth's horizontal component.

MixedJEE Physics · Original

Superconductors as Perfect Diamagnets

In the ideal Meissner state, bulk B = 0 and the effective susceptibility is -1 in SI for suitable geometry. The critical temperature T_c(B) generally decreases as applied field increases; beyond the critical boundary the material becomes normal.

MixedJEE Physics · Original

Superposition and Magnetic-Field Cancellation

In magnetostatics, B_total = sum B_i. Exact cancellation requires vector equality, not merely equal magnitudes; symmetric current paths can produce zero field even when individual contributions are nonzero.

MixedJEE Physics · Original

Torque on a Current Loop

A planar N-turn loop has moment m = NIA n_hat and torque tau = m cross B. The magnitude is mB sin theta, where theta is between the loop normal and B.

MixedJEE Physics · Original

Torque on a Magnetic Dipole

A magnetic dipole m in a uniform field B experiences torque tau = m cross B, of magnitude mB sin theta. The torque is zero when parallel or antiparallel and maximum at 90 degrees.

MixedJEE Physics · Original

Velocity Selector

For mutually perpendicular E, B, and v arranged so qE and q(v cross B) oppose, undeflected motion requires E = vB, hence v = E/B. The selected speed is independent of charge sign when geometry is reversed consistently.

MixedJEE Physics · Original

AC Circuit Analysis and LCR Circuits

AC circuits with resistors, capacitors, and inductors (LCR circuits) exhibit impedance, resonance, phase relationships, and power factor, all crucial for understanding circuit behavior.

Exam level11 questions

AC Generators and Transformers

AC generators convert mechanical energy to electrical energy using electromagnetic induction, while transformers transfer electrical energy between circuits, often changing voltage levels.

Foundation6 questions

Eddy Currents and Applications

Eddy currents are circulating currents induced in conductors by changing magnetic fields, leading to energy loss and effects like electromagnetic damping.

Exam level5 questions

Self and Mutual Inductance

Self-inductance is the property of a coil to oppose changes in its own current, while mutual inductance is the ability of one coil to induce EMF in another nearby coil.

Exam level5 questions

Faraday's Law and Lenz's Law

Faraday's law explains how a changing magnetic field induces an electromotive force (EMF), while Lenz's law determines the direction of the induced current to oppose the change causing it.

Exam level3 questions

Energy in Inductors and AC Circuits

Inductors store energy in their magnetic field, and energy considerations in AC circuits involve power dissipation and conservation.

Foundation2 questions

Motional EMF

Motional EMF is the voltage induced in a conductor moving through a magnetic field, depending on the speed, length, and orientation of the conductor.

Exam level2 questions

AC Waveform, Frequency, and RMS

For a sinusoid x = x0 sin(omega t + phi), omega = 2 pi f. The RMS value is peak divided by sqrt(2); this relation is not universal for non-sinusoidal waveforms.

MixedJEE Physics · Original

Eddy Currents

Eddy-current strength grows with changing flux and falls when the conductor is laminated or slotted because the current paths gain resistance and lose area.

MixedJEE Physics · Original

Faraday's Law

The induced emf equals the negative time rate of change of flux linkage. For N identical turns, flux linkage is N times the flux through one turn.

MixedJEE Physics · Original

Flux-Time Graphs and Average EMF

For one turn, emf is the negative slope of the Phi-t graph. For N turns, multiply the slope by N. Corners indicate sudden changes of emf in an idealized graph.

MixedJEE Physics · Original

Frequency Dependence of Reactance

XL is directly proportional to frequency; XC is inversely proportional. These opposite trends create filters and resonance.

MixedJEE Physics · Original

Induced Charge and Energy Conservation

With constant circuit resistance and negligible self-inductance, integrating I = emf/R gives the magnitude of total charge as N times flux change divided by R.

MixedJEE Physics · Original

LC Oscillations

In an ideal LC circuit total electromagnetic energy is constant and the oscillation frequency is 1/(2 pi sqrt(LC)).

MixedJEE Physics · Original

Lenz's Law and Current Direction

The minus sign in Faraday's law represents energy-conserving opposition to the change in magnetic flux. The right-hand grip rule converts the required induced field into current direction.

MixedJEE Physics · Original

Magnetic Flux

For a uniform field and a flat surface, magnetic flux is the dot product of the magnetic field and area vector. The area vector is perpendicular to the surface.

MixedJEE Physics · Original

Motional EMF in a Moving Conductor

For a straight rod of length l moving with velocity v in a uniform field B, the simple Blv form applies only when B, l, and v are mutually perpendicular.

MixedJEE Physics · Original

Moving Loops and Changing Overlap Area

For perpendicular uniform B, Phi = B times overlap area. The emf is B times the rate at which that overlap area changes.

MixedJEE Physics · Original

Mutual Inductance

Mutual inductance M relates the current change in one circuit to the induced emf in the other. For passive reciprocal coils, M12 equals M21.

MixedJEE Physics · Original

Phase Difference and Average Power

For sinusoidal steady state, average real power is Vrms Irms cos(phi). The factor cos(phi) is the power factor.

MixedJEE Physics · Original

Pure R, L, and C Circuits

For sinusoidal steady state, resistance is R, inductive reactance is omega L, and capacitive reactance is 1/(omega C). Ideal L and C consume zero average power.

MixedJEE Physics · Original

RL and RC Transients with Stored Energy

The RL time constant is L/R and the RC time constant is RC. After one time constant, a growing quantity reaches about 63 percent and a decaying quantity retains about 37 percent.

MixedJEE Physics · Original

Rotating Coil and AC Generator

For a coil of N turns and area A rotating at angular speed omega in uniform B, the peak emf is N B A omega.

MixedJEE Physics · Original

Series LCR Impedance and Phasors

The impedance magnitude is the hypotenuse formed by R and XL-XC. The phase angle follows tan(phi) = (XL-XC)/R.

MixedJEE Physics · Original

Series Resonance, Quality Factor, and Bandwidth

The resonant angular frequency is 1/sqrt(LC). For a series RLC circuit, Q measures sharpness and bandwidth in angular frequency is R/L.

MixedJEE Physics · Original

Solenoid Field and Self-Inductance

For a long ideal solenoid B = mu n I. Self-inductance L is flux linkage per current for a linear magnetic system, and its induced emf is proportional to current's rate of change.

MixedJEE Physics · Original

Transformer

For an ideal transformer, voltage ratio equals turns ratio and input power equals output power. A real transformer's efficiency is output power divided by input power.

MixedJEE Physics · Original

Mathematical relations in electromagnetic waves

Key mathematical relationships in electromagnetic waves include the connection between electric and magnetic field amplitudes, wave speed, wavelength, frequency, and the wave equation parameters.

Foundation8 questions

Nature and properties of electromagnetic waves

Electromagnetic waves are transverse waves consisting of mutually perpendicular oscillating electric and magnetic fields, both perpendicular to the direction of propagation, and exhibit properties such as speed, polarization, and ability to travel through vacuum.

Exam level5 questions

Energy, momentum, and radiation pressure of electromagnetic waves

Electromagnetic waves transport energy and momentum, exerting radiation pressure and force on surfaces, with quantifiable energy density and intensity.

Exam level4 questions

Production and propagation of electromagnetic waves

Electromagnetic waves are produced by accelerating charges and propagate through space, carrying energy and momentum, with their direction determined by the orientation of the electric and magnetic fields.

Exam level4 questions

Electromagnetic spectrum

The electromagnetic spectrum encompasses all types of electromagnetic waves, classified by wavelength or frequency, with each region having characteristic properties and applications.

Foundation3 questions

Conduction Current versus Displacement Current

For an ohmic medium, conduction-current density is sigma E. For a sinusoidal electric field of angular frequency omega, displacement-current density has amplitude omega epsilon E_0, so their amplitude ratio is sigma/(omega epsilon).

MixedJEE Physics · Original

Displacement Current

Displacement current is defined as epsilon_0 times the rate of change of electric flux in vacuum. For a capacitor carrying a charging current, the total displacement current through the full gap equals the conduction current in the wires.

MixedJEE Physics · Original

Displacement Current in a Capacitor

For a linear capacitor with constant capacitance C, charge is q = CV and the charging current equals C dV/dt. In an ideal capacitor this is also the displacement current through the complete inter-plate surface.

MixedJEE Physics · Original

Electric and Magnetic Energy Density

The instantaneous electromagnetic energy density is the sum of electric and magnetic terms. For a plane wave, the field-amplitude relation makes these terms equal, and sinusoidal time averaging contributes the usual factor of one-half.

MixedJEE Physics · Original

Electric and Magnetic Field Amplitudes

For a plane electromagnetic wave in a homogeneous isotropic medium, the instantaneous field magnitudes satisfy E/B = v when the fields are in phase. In vacuum the ratio is c. This relation also compares electric and magnetic forces on a moving charge.

MixedJEE Physics · Original

Electromagnetic Spectrum Order and Photon Energy

In vacuum, c = f lambda and photon energy is hf. Thus wavelength decreases as frequency and photon energy increase, giving the order radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray.

MixedJEE Physics · Original

Frequency, Wavelength, and Wavenumber

For a monochromatic wave, frequency f is oscillations per second, wavelength lambda is the spatial period, angular frequency is 2 pi f, and wavenumber is 2 pi/lambda.

MixedJEE Physics · Original

How Electromagnetic Waves Are Produced

Electromagnetic radiation is generated by accelerated charges or time-varying currents. In a source-free region, coupled time-varying electric and magnetic fields propagate without requiring a material medium.

MixedJEE Physics · Original

How Spectrum Bands Are Produced

Typical production mechanisms include accelerated charges in antennas for radio waves, microwave oscillators such as magnetrons, molecular rotational-vibrational transitions for infrared, electronic transitions for visible and ultraviolet, inner-shell transitions or rapid electron deceleration for X-rays, and nuclear transitions for gamma rays.

MixedJEE Physics · Original

Intensity and the Poynting Vector

The instantaneous Poynting vector is S = (1/mu)(E cross B). For a sinusoidal plane wave, intensity is the cycle average of its magnitude.

MixedJEE Physics · Original

Maxwell Equations as a Unified Picture

The integral Maxwell equations are Gauss's electric law, Gauss's magnetic law, Faraday's induction law, and the Ampere-Maxwell law. Together in a charge-free, current-free region they imply electromagnetic wave equations.

MixedJEE Physics · Original

Photon Momentum and Absorbing Surfaces

Electromagnetic energy U in vacuum carries momentum magnitude U/c. For normal incidence on a perfectly absorbing surface, momentum transfer per unit time and area gives radiation pressure I/c.

MixedJEE Physics · Original

Point Sources and the Inverse-Square Law

For an isotropic point source of radiated power P with negligible absorption, intensity at distance r equals P divided by the area 4 pi r squared. Points at the same radius have the same intensity.

MixedJEE Physics · Original

Radiation Pressure on Reflecting Surfaces

For normal incidence on a perfect reflector, momentum transfer is 2U/c and radiation pressure is 2I/c. The factor of two applies only to complete reflection with reversal of the normal momentum component.

MixedJEE Physics · Original

Reading a Plane-Wave Expression

A harmonic plane wave can be written as A cos(k vector dot r - omega t + phi). Its wave vector points along propagation, its magnitude is the wavenumber, and omega is the angular frequency.

MixedJEE Physics · Original

Refractive Index and Dielectric Constant

The refractive index is n = c/v. For a homogeneous linear medium, n = sqrt(mu_r epsilon_r); when mu_r is approximately one, n is approximately sqrt(epsilon_r).

MixedJEE Physics · Original

Spectrum Applications and Safety

Applications follow how radiation is generated, transmitted, absorbed, scattered, and ionizes matter. Higher photon energy generally increases ionizing capability, so application matching must also respect exposure and shielding requirements.

MixedJEE Physics · Original

Superposition and Counter-Propagating Waves

In linear media, the total electromagnetic field is the vector sum of all component solutions. For each component wave, E cross B points along that component's propagation direction; superposition does not permit amplitudes to be added without phase and direction information.

MixedJEE Physics · Original

Transverse Geometry and Propagation Direction

For a plane wave in an isotropic medium, E is perpendicular to B, and both are perpendicular to the wave vector k. The propagation direction is parallel to E cross B.

MixedJEE Physics · Original

Wave Speed in a Material Medium

In a homogeneous linear medium with permittivity epsilon and permeability mu, the phase speed is 1/sqrt(mu epsilon). Relative material parameters compare this speed with c.

MixedJEE Physics · Original

Interference of light

Examine the principle of superposition, Young's double slit experiment, fringe width, intensity distribution, and the conditions for constructive and destructive interference.

Exam level9 questions

Diffraction of light

Understand the bending of light around obstacles, single slit diffraction patterns, their width, and the effect of wavelength on diffraction.

Foundation8 questions

Lenses and mirrors

Explore the image formation, ray diagrams, lens and mirror formulas, and the behavior of light with concave/convex lenses and mirrors, including combinations and virtual objects.

Exam level6 questions

Polarization of light

Learn about the polarization of light, Brewster's law, Malus' law, and the use and function of polaroids.

Exam level6 questions

Dispersion and rainbow formation

Study how light splits into its constituent colors through dispersion in prisms and natural phenomena like rainbows, including minimum deviation and dispersive power.

Foundation5 questions

Optical instruments

Understand the working principles, magnification, resolving power, and design of devices like microscopes and telescopes, including their adjustments and measurement techniques.

Exam level4 questions

Total internal reflection and critical angle

Learn the conditions for total internal reflection, the concept of critical angle, and their applications in optics.

Foundation4 questions

Reflection and refraction of light

Study how light rays behave at the interface between different media, including the laws of reflection and refraction, Snell's law, and related geometric considerations.

Challenging2 questions

Aperture Size, Wavelength, and Diffraction

Diffraction is the redistribution of wave amplitude caused by finite apertures or obstacles. Its characteristic angular spread scales approximately as lambda/a, so longer wavelength or smaller aperture gives greater spreading.

MixedJEE Physics · Original

Apparent Depth at Normal Viewing

For a plane interface, small angles and an observer in medium n1 viewing an object in medium n2, apparent depth/real depth = n1/n2.

MixedJEE Physics · Original

Astronomical and Reflecting Telescopes

For an astronomical telescope in normal adjustment, angular magnification magnitude is f_o/f_e and tube length is f_o+f_e. A Galilean telescope uses a diverging eyepiece and gives an erect image.

MixedJEE Physics · Original

Brewster Geometry and Perpendicular Reflected-Refracted Rays

For nonmagnetic dielectric media, tan(i_B) = n2/n1 when light is incident from medium n1; this is the Brewster angle condition.

MixedJEE Physics · Original

Brewster's Law and Polarization by Reflection

For incidence from medium n1 into n2, Brewster angle i_B satisfies tan(i_B)=n2/n1. At this angle i_B+r=90 degrees, and the reflected light is polarized perpendicular to the plane of incidence.

MixedJEE Physics · Original

Coherent and Incoherent Sources

Sources are coherent when they emit waves of the same frequency with a time-independent phase difference. Incoherent sources have a fluctuating phase relation, so their cross term averages to zero over the detector response time.

MixedJEE Physics · Original

Combined Mirrors and Autocollimation

For coaxial or facing mirrors, apply reflection at each surface with signed coordinates. In autocollimation, placing an object at a converging element's focus returns parallel rays that retrace after plane reflection.

MixedJEE Physics · Original

Compound Microscope and Resolution

For normal adjustment in the standard long-tube approximation, M = (L/f_o)(D/f_e) in magnitude. Resolution improves with shorter wavelength and larger numerical aperture.

MixedJEE Physics · Original

Concave-Mirror Ray Diagrams and Image Nature

For a concave spherical mirror under paraxial conditions, an object outside F gives a real inverted image, while an object between P and F gives a virtual erect enlarged image.

MixedJEE Physics · Original

Convex Mirrors and Field of View

For real objects and paraxial rays, a convex spherical mirror has a virtual focus and produces 0 < m < 1 under the Cartesian convention.

MixedJEE Physics · Original

Critical Angle

For n1 > n2, sin(C) = n2/n1. The critical angle is defined only for incidence from the optically denser to the rarer medium.

MixedJEE Physics · Original

Cut Lenses, Aperture and Image Brightness

For a thin lens cut parallel to its principal axis, each piece retains focal length while aperture and image intensity decrease; a cut perpendicular to the axis changes a surface and therefore power.

MixedJEE Physics · Original

Diffraction Envelope and Interference Fringes

For slit width a and centre separation d, interference maxima satisfy d sin(theta)=m lambda while the single-slit envelope has minima at a sin(theta)=p lambda. An interference order is missing when both conditions occur at the same angle.

MixedJEE Physics · Original

Dispersion and Angular Dispersion

For a thin prism, angular dispersion between wavelengths 1 and 2 is (n1-n2)A; dispersive power is angular dispersion divided by mean deviation.

MixedJEE Physics · Original

Escape Cone and Visibility Through a Surface

For a point at depth h in higher-index medium, the escape-cone half-angle is C and the surface-circle radius is h tan C for a plane boundary.

MixedJEE Physics · Original

Fringe Width and Scaling

In the paraxial Young experiment, successive bright or dark fringes are separated by beta = lambda D/d. In a medium of refractive index mu, wavelength becomes lambda_0/mu and the fringe width decreases by the same factor.

MixedJEE Physics · Original

Generalized Two-Source Geometry

The total phase difference is obtained from the incident phase difference plus the propagation path difference to the observation point. Fringes are loci of constant total optical-path difference.

MixedJEE Physics · Original

Laws of Reflection and Ray Geometry

At a smooth reflecting surface, incident ray, reflected ray and normal are coplanar, and the angles measured from the normal are equal.

MixedJEE Physics · Original

Layered Media and Optical Path

For normal incidence, optical path length is sum(n_j t_j), while paraxial apparent thickness viewed from index n0 is sum(n0 t_j/n_j).

MixedJEE Physics · Original

Lens Displacement and u-v Experiments

For object-screen distance D and separation d between conjugate lens positions, f = (D^2-d^2)/(4D), provided D > 4f and the lens is thin.

MixedJEE Physics · Original

Lens Immersion and Surrounding Medium

For a thin lens in medium n_m, P = (n_l/n_m-1)(1/R1-1/R2). If n_l = n_m, P = 0; if n_l < n_m, the sign reverses relative to air.

MixedJEE Physics · Original

Lens Magnification

For a paraxial thin lens, m = h_i/h_o = v/u. Negative m denotes an inverted image and positive m an erect image.

MixedJEE Physics · Original

Lens Maker's Formula

For a thin lens in air, 1/f = (n-1)(1/R1-1/R2). In a medium replace n by the relative index n_lens/n_medium.

MixedJEE Physics · Original

Lens Ray Diagrams and Image Nature

Under the thin-lens paraxial model, a convex lens gives a real image for an object outside its focus and a virtual enlarged image inside it; a concave lens gives a virtual diminished image for a real object.

MixedJEE Physics · Original

Lens-Mirror Systems and Retracing

Use sequential paraxial imaging with a fixed coordinate system. When an image lies at a spherical mirror's centre of curvature, rays retrace; a plane mirror returns normally incident rays.

MixedJEE Physics · Original

Malus' Law and Multiple Polarizers

Malus' law states that linearly polarized light of intensity I_in incident on an ideal analyser whose axis is at angle theta to the polarization emerges with intensity I_out = I_in cos^2(theta). Apply it sequentially through multiple sheets.

MixedJEE Physics · Original

Maxima, Minima, and Fringe Visibility

For two coherent beams, maximum intensity occurs at phase difference 2m pi and minimum at (2m+1) pi. Visibility compares the modulation depth with the mean intensity.

MixedJEE Physics · Original

Minimum Deviation and Prism Refractive Index

For a prism in air at minimum deviation, i=e, r1=r2=A/2 and n = sin[(A+delta_m)/2]/sin(A/2).

MixedJEE Physics · Original

Mirror Magnification and Object-Image Separation

For a paraxial spherical mirror, transverse magnification is h_i/h_o = -v/u; its sign gives orientation and its magnitude gives size ratio.

MixedJEE Physics · Original

Motion and Rotation of a Plane Mirror

For motion normal to a fixed object, image speed relative to the object is twice the mirror speed. Rotating a mirror by theta rotates a reflected ray by 2 theta when the incident ray is fixed.

MixedJEE Physics · Original

Multiple Plane Mirrors and Repeated Images

For two plane mirrors inclined at angle theta, the standard image-count rule depends on 360 degrees/theta and on object symmetry; direct construction is safest for exceptional cases.

MixedJEE Physics · Original

Multiple Wavelengths and Fringe Coincidence

At a common bright fringe, m1 lambda1 = m2 lambda2 for integer orders. A common dark fringe additionally requires odd-half-integer conditions for both wavelengths and may not exist for every wavelength ratio.

MixedJEE Physics · Original

Optical Path and Phase Difference

Optical path length is the line integral of refractive index along a path. For monochromatic vacuum wavelength lambda_0, phase difference equals 2 pi times the optical-path difference divided by lambda_0.

MixedJEE Physics · Original

Optical Power and Lenses in Contact

In air for thin coaxial lenses in contact, P_eq = sum(P_i), where P = 1/f in metres and is measured in dioptres.

MixedJEE Physics · Original

Parallel Slab and Lateral Displacement

For a slab of thickness t with parallel faces, incidence angle i and refraction angle r, lateral shift is t sin(i-r)/cos r; this assumes the same external medium on both sides.

MixedJEE Physics · Original

Plane-Mirror Image and Minimum Mirror Size

For a stationary plane mirror, image distance equals object distance in magnitude and linear magnification is +1; a person needs a mirror of half their height to see the full body.

MixedJEE Physics · Original

Plate, Film, and Wedge Fringe Shift

A normally traversed plate of thickness t and refractive index mu replacing air adds optical path (mu-1)t. In YDSE this shifts the central fringe by D(mu-1)t/d, or by (mu-1)t/lambda fringe widths.

MixedJEE Physics · Original

Polarization and the First Polarizer

Light is transverse because its electric field is perpendicular to propagation. Linear polarization confines the electric-field oscillation to one fixed transverse direction. An ideal polarizer transmits the component along its pass axis.

MixedJEE Physics · Original

Polarization States and Optical Rotation

The polarization state is specified by two orthogonal transverse electric-field components and their phase difference. An optically active medium rotates the plane of linear polarization without requiring a change in intensity before analysis.

MixedJEE Physics · Original

Prism Combinations and Achromatic/Direct-Vision Conditions

For thin oppositely oriented prisms, net mean deviation and net angular dispersion are signed sums. Achromatism requires omega1 delta1 + omega2 delta2 = 0.

MixedJEE Physics · Original

Prism Geometry and Deviation

For a prism in air, r1+r2=A and deviation delta=i+e-A, with all angles measured consistently from face normals.

MixedJEE Physics · Original

Rayleigh Criterion and Resolving Power

By Rayleigh's criterion, two point sources are just resolved when the principal maximum of one diffraction pattern coincides with the first minimum of the other. For a circular aperture of diameter D, the limiting angle is approximately 1.22 lambda/D.

MixedJEE Physics · Original

Rays in Spheres, Drops and Bubbles

At each spherical boundary, apply Snell's law using the radius through the incidence point as normal; subsequent incidence angles follow from the internal triangle.

MixedJEE Physics · Original

Refraction at a Spherical Surface

For paraxial rays at a spherical interface, n2/v - n1/u = (n2-n1)/R using one consistent Cartesian sign convention.

MixedJEE Physics · Original

Refractive Index, Speed, Frequency and Wavelength

For a monochromatic wave in a transparent nondispersive approximation, n = c/v and lambda_medium = lambda_0/n; frequency is continuous across the interface.

MixedJEE Physics · Original

Separated Thin Lenses and Equivalent Focal Length

For two thin lenses in air separated by d, P_eq = P1 + P2 - d P1 P2 with d in metres; direct sequential imaging also gives final image position.

MixedJEE Physics · Original

Simple Microscope and Angular Magnification

For a convex magnifier of focal length f, angular magnification is D/f for final image at infinity and 1+D/f for final image at the near point D.

MixedJEE Physics · Original

Single-Slit Diffraction Minima and Central Maximum

For Fraunhofer diffraction by a slit of width a, minima satisfy a sin(theta) = m lambda for nonzero integer m. The central maximum lies between the first minima and is twice the approximate width of a secondary maximum.

MixedJEE Physics · Original

Snell's Law at a Plane Interface

For isotropic media and angles measured from the normal, n1 sin(i) = n2 sin(r). The relation follows from phase matching at a stationary plane boundary.

MixedJEE Physics · Original

Spherical-Mirror Formula

With the Cartesian sign convention and paraxial rays, a spherical mirror satisfies 1/f = 1/v + 1/u.

MixedJEE Physics · Original

Spherical-Mirror Geometry and Focal Length

For paraxial rays of a spherical mirror, the principal focus lies midway between pole and centre of curvature; focal length is unaffected by the surrounding transparent medium.

MixedJEE Physics · Original

Thin-Film Interference and Reflection Phase

For a film of refractive index mu and thickness t at refraction angle r, the geometric round-trip optical path is 2 mu t cos(r). Reflection from a lower-index to higher-index boundary adds a pi phase shift; the number of such reversals determines the bright/dark condition.

MixedJEE Physics · Original

Thin-Lens Formula

For a paraxial thin lens under the Cartesian convention, 1/f = 1/v - 1/u.

MixedJEE Physics · Original

Thin-Prism Approximation

For a thin prism in air under the paraxial small-angle approximation, delta = (n-1)A, with A and delta in the same angular unit.

MixedJEE Physics · Original

Total Internal Reflection

Total internal reflection requires n1 > n2 and i > C. At i = C the transmitted ray grazes the interface; real interfaces can still have evanescent fields or losses.

MixedJEE Physics · Original

Total Internal Reflection and Paths in Prisms

At each prism face, determine the local incidence angle from geometry, then compare it with C where sin C = n_out/n_in; otherwise apply Snell's law.

MixedJEE Physics · Original

Two-Wave Superposition and Intensity

For two mutually coherent waves of intensities I1 and I2 with phase difference phi, the time-averaged resultant intensity contains an interference cross term. For equal intensities I0, it reduces to a cosine-squared law.

MixedJEE Physics · Original

Wave-Optics Measurement and Mixed Systems

For a derived quantity expressed as a product of powers, maximum fractional uncertainty is bounded by the sum of absolute powers times the corresponding fractional uncertainties. Optical instruments must separately track aperture, focal lengths, aberrations, and any interference stage.

MixedJEE Physics · Original

Wavefronts, Rays, and Huygens' Principle

A wavefront is a surface of constant phase. Huygens' principle states that every point on a wavefront acts as a source of secondary wavelets; the forward envelope after time t is the new wavefront. In an isotropic medium, rays are normal to wavefronts.

MixedJEE Physics · Original

YDSE Path Difference and Fringe Positions

For slit separation d and observation angle theta, the far-field path difference is d sin(theta). Bright fringes satisfy m lambda and dark fringes satisfy (m+1/2) lambda. Under the small-angle approximation, y is approximately D theta.

MixedJEE Physics · Original

Photoelectric Effect

The photoelectric effect describes the emission of electrons from a material when it is exposed to light of sufficient frequency, governed by concepts such as threshold frequency, work function, stopping potential, and the Einstein photoelectric equation.

Exam level12 questions

de Broglie Wavelength and Matter Waves

All matter exhibits wave-like properties, with the de Broglie wavelength inversely proportional to momentum and dependent on factors like velocity, temperature, and particle type.

Exam level6 questions

Photon Properties and Energy-Momentum Relations

Photons are massless particles of light characterized by their energy, frequency, momentum, and charge neutrality, and their interactions obey conservation laws.

Foundation5 questions

Characteristic X-rays and Moseley's Law

Characteristic lines are discrete X-ray frequencies produced by electronic transitions into inner-shell vacancies. Their frequencies depend strongly on atomic number and approximately follow Moseley's square-root frequency relation.

MixedJEE Physics · Original

Charged Particle Accelerated Through Potential

For a non-relativistic particle accelerated from rest through a potential difference V, K = absolute q times V. Combining this with the de Broglie relation yields lambda = h divided by the square root of 2m absolute q V.

MixedJEE Physics · Original

Comparing Matter Waves of Different Particles

For any two particles, lambda_1/lambda_2 = p_2/p_1. Non-relativistically, use p = mv, p = square root of 2mK, or p = square root of 2m absolute q V according to the data.

MixedJEE Physics · Original

Continuous X-ray Spectrum and Cutoff Wavelength

In an X-ray tube, electrons accelerated through voltage V reach kinetic energy eV. Energy conservation sets the maximum photon energy h nu_max = eV and hence the Duane-Hunt cutoff wavelength.

MixedJEE Physics · Original

de Broglie Wavelength

The de Broglie hypothesis assigns wavelength lambda = h/p to a particle of momentum p. For non-relativistic motion p = m v; the momentum form remains the safest starting point whenever direction or collisions matter.

MixedJEE Physics · Original

Einstein Photoelectric Equation

Energy conservation for one-photon photoemission gives h nu = phi + K_max. The stopping potential is the reverse potential whose electric potential energy e V_s equals K_max.

MixedJEE Physics · Original

Electron Diffraction and Davisson-Germer Experiment

In the Davisson-Germer experiment, accelerated electrons scatter from a crystal and produce intensity maxima described by Bragg's law. Agreement between the inferred wavelength and h/p confirms the de Broglie hypothesis.

MixedJEE Physics · Original

Matter Wavelength and Kinetic Energy

For a non-relativistic particle, K = p squared divided by 2m. Substitution into lambda = h/p gives lambda = h divided by the square root of 2mK.

MixedJEE Physics · Original

Matter Wavelength in Electric and Magnetic Fields

Under a force, momentum evolves according to dp vector by dt = F vector, and the instantaneous wavelength is h divided by the momentum magnitude. A purely magnetic Lorentz force does no work, whereas an electric field can change kinetic energy.

MixedJEE Physics · Original

Matter Waves in Collisions and Decays

Total vector momentum is conserved for an isolated collision or decay. After solving for each relevant momentum magnitude, the associated wavelength is h divided by that magnitude.

MixedJEE Physics · Original

Photoelectric Current and Collector Potential

The photocurrent-voltage curve rises toward saturation under accelerating voltage and falls to zero at the stopping potential under reverse voltage. At fixed suitable frequency, saturation current scales with incident intensity.

MixedJEE Physics · Original

Photoelectric Effect Observations

For a fixed emitter, emission occurs only when incident frequency reaches the threshold frequency. Above threshold, maximum kinetic energy depends on frequency, while saturation current is primarily proportional to intensity.

MixedJEE Physics · Original

Photon Energy and Momentum

For a photon in vacuum, energy is proportional to frequency and momentum is energy divided by c. Frequency and wavelength obey c = nu lambda, so shorter-wavelength photons have larger energy and momentum.

MixedJEE Physics · Original

Photon Rate, Power, and Energy Density

For monochromatic radiation, total energy is the number of photons times h nu. Power is energy per unit time, so the photon emission rate equals power divided by single-photon energy.

MixedJEE Physics · Original

Radiation Pressure and Photon Momentum Transfer

Radiation force is the rate of photon momentum transfer. For normal incidence on an ideal absorber the pressure is intensity divided by c; for an ideal reflector it is twice that value.

MixedJEE Physics · Original

Stopping-Potential Graphs

Combining Einstein's equation with K_max = e V_s gives V_s = (h/e) nu - phi/e. The frequency-axis intercept is the threshold frequency, while parallel lines represent materials measured with the same fundamental constants.

MixedJEE Physics · Original

Thermal de Broglie Wavelength

A stated thermal speed model must be used consistently with lambda = h/(mv). For example, using root-mean-square speed gives p_rms = square root of 3mkT and lambda_rms = h divided by that momentum.

MixedJEE Physics · Original

Wave-Particle Duality and Resolution

Quantum objects display particle-like localized interactions and wave-like interference or diffraction. A probe generally resolves structures no smaller than a scale comparable to its wavelength, so increasing electron momentum improves ideal wavelength-limited resolution.

MixedJEE Physics · Original

Work Function and Threshold

The work function phi is the minimum energy required to remove an electron from a material surface. At threshold the emitted electron has zero maximum kinetic energy, giving phi = h nu_0 = h c / lambda_0.

MixedJEE Physics · Original

X-ray Tube Controls and Spectrum

In an X-ray tube, electron rate is controlled primarily by tube current and electron energy by accelerating voltage. Target material sets characteristic lines; voltage sets the continuous cutoff and must exceed shell thresholds for characteristic emission.

MixedJEE Physics · Original

Hydrogen spectrum and spectral series

Understand the origin, calculation, and interpretation of spectral lines, series limits, wavelength and frequency relations, and transitions in hydrogen and hydrogen-like ions.

Exam level8 questions

Mass defect, binding energy, and mass-energy equivalence

Learn how mass defect leads to nuclear binding energy, the use of E=mc², and how to calculate binding energy per nucleon and related quantities.

Exam level6 questions

Nuclear fission, fusion, and energy production

Understand the processes of nuclear fission and fusion, their energy yields, the role of binding energy, and applications like nuclear reactors and solar energy.

Exam level6 questions

Atomic models and Bohr theory

Learn the historical development of atomic models, especially the Bohr model, and how it explains the structure, radii, and energy levels of hydrogen and hydrogen-like atoms.

Foundation5 questions

Nuclear reactions and conservation laws

Study the equations for nuclear reactions, including conservation of mass number, atomic number, and other physical quantities.

Exam level3 questions

Rutherford scattering and atomic nucleus

Study the principles of Rutherford scattering, distance of closest approach, Coulomb force, and how these experiments revealed the structure and size of the nucleus.

Exam level3 questions

Nuclear size, radius, and density

Explore how nuclear size is measured, the formula for nuclear radius, its dependence on mass number, and the concept of nuclear density.

Exam level2 questions

Activity, Sample Size, Dating, and Tracers

Activity A = lambda N is measured in becquerels, where 1 Bq = 1 s^-1; 1 Ci = 3.7 x 10^10 Bq. N can be related to sample mass through N = (m/M)N_A, subject to detector efficiency when measured counts are used.

MixedJEE Physics · Original

Alpha Decay

Alpha decay is ^A_ZX -> ^(A-4)_(Z-2)Y + ^4_2He. It occurs when the final rest mass is lower; quantum tunnelling through the Coulomb barrier permits emission even when classical escape is forbidden.

MixedJEE Physics · Original

Angular-Momentum Quantization

Bohr's quantization condition restricts orbital angular momentum magnitude to positive integer multiples of hbar. It is equivalent to fitting an integer number of de Broglie wavelengths around the circular orbit.

MixedJEE Physics · Original

Beta Decay and Neutrinos

At nucleon level, beta-minus is n -> p + e- + anti-nu and beta-plus is p -> n + e+ + nu. Nuclear A stays fixed while Z changes by +1 or -1. Electron capture is another proton-to-neutron channel.

MixedJEE Physics · Original

Binding Energy per Nucleon and Stability Curve

Average binding energy per nucleon is B/A. Its curve rises steeply for light nuclei, peaks near iron-nickel mass numbers, and decreases slowly for heavy nuclei; a larger value usually indicates greater resistance to separation, not immunity to every decay.

MixedJEE Physics · Original

Bohr Postulates and Scope

The Bohr model postulates stationary states with quantized angular momentum and photon exchange only during transitions. Its standard Coulomb formulas apply to hydrogen and hydrogen-like one-electron ions, not to general many-electron atoms.

MixedJEE Physics · Original

Bohr Radius and Orbit Scaling

For an infinitely heavy hydrogen-like nucleus, the radius of the nth Bohr orbit is a_0 n^2/Z. With finite nuclear mass, replace electron mass by reduced mass, producing a small isotope correction.

MixedJEE Physics · Original

Counting Spectral Lines

If atoms populate every level from 1 through N and all downward transitions occur, the maximum number of distinct lines is N choose 2. A single atom follows one cascade, whereas a sample can realize all allowed paths.

MixedJEE Physics · Original

Coupled and Mixed Atomic Processes

Coupled problems can link atomic transitions to photoelectric emission, collisions, thermal matter waves, or secondary ionization. Each stage must satisfy its own threshold, conservation law, and approximation; source uncertainty must not be repaired by assuming a familiar result.

MixedJEE Physics · Original

de Broglie Standing Waves in Bohr Orbits

Combining lambda = h/(mv) with 2 pi r_n = n lambda gives m v_n r_n = n hbar. In a Coulomb Bohr atom, the electron wavelength therefore scales as n/Z.

MixedJEE Physics · Original

Decay Chains, Branching, and Constant Production

Population balance is production minus removal. For constant production alpha, dN/dt = alpha - lambda N and N = (alpha/lambda)(1 - e^(-lambda t)) when N(0)=0. Successive decay uses coupled Bateman equations; branching decay uses total lambda equal to the sum of branch constants.

MixedJEE Physics · Original

Distance of Closest Approach

For a head-on approach to a heavy stationary nucleus, the alpha particle's kinetic energy at large separation equals the Coulomb potential energy at the classical turning point. Nuclear recoil is neglected unless the problem supplies finite target mass.

MixedJEE Physics · Original

Excitation, Binding, and Ionization

Excitation energy is E_f - E_i for two bound levels with E_f > E_i. Ionization energy from level n is 0 - E_n = |E_n|. In collision excitation, the projectile must supply at least the required gap, with excess energy shared consistently.

MixedJEE Physics · Original

Exponential Radioactive Decay

For a single radioactive species with constant decay constant lambda and no production, dN/dt = -lambda N and N(t) = N0 exp(-lambda t). Activity is the positive decay rate A = -dN/dt = lambda N.

MixedJEE Physics · Original

Gamma Decay and Nuclear Levels

Gamma decay is ^A_ZX* -> ^A_ZX + gamma. Photon energy is h nu and momentum is h nu/c; for a freely recoiling nucleus the level-energy change equals photon energy plus recoil energy.

MixedJEE Physics · Original

Half-Life, Mean Life, and Decay Probability

For exponential decay, T_1/2 = ln2/lambda and mean life tau = 1/lambda. The probability that a nucleus survives to time t is exp(-lambda t), and the probability it has decayed by then is 1 - exp(-lambda t).

MixedJEE Physics · Original

Hydrogen-Like Energy Levels

For a one-electron Coulomb system with a heavy nucleus, the total energy of level n is -13.6 Z^2/n^2 eV. The magnitude is the binding energy required to remove the electron from that level.

MixedJEE Physics · Original

Kinetic, Potential, and Total Energy

For a circular orbit in a 1/r attractive potential, the virial relation gives U = -2K and E = K + U = -K = U/2. As an electron falls to lower n, K rises while U and E decrease.

MixedJEE Physics · Original

Mass Defect and Binding Energy

For a nucleus with Z protons and N neutrons, mass defect is the free-nucleon mass sum minus nuclear mass, and binding energy is Delta m c^2. With atomic masses, hydrogen-atom masses must be used consistently so electron masses cancel.

MixedJEE Physics · Original

Mass-Energy Equivalence and Nuclear Units

For an isolated process, total relativistic energy is conserved. A rest-mass difference Delta m corresponds to rest-energy difference Delta E = Delta m c^2; 1 u c^2 is approximately 931.5 MeV. Unit conversion must not insert c^2 twice.

MixedJEE Physics · Original

Nuclear Composition and Nuclide Families

Nuclide notation is written as ^A_ZX, where Z is atomic number, A is mass number, and N = A - Z. Isotopes have equal Z, isobars equal A, and isotones equal N; none of these labels alone implies equal stability.

MixedJEE Physics · Original

Nuclear Fission, Chain Reactions, and Reactor Power

Fission conserves A, Z, energy, and momentum and typically releases about 200 MeV per event for uranium-235. Reactor power equals event rate times energy per fission; moderation slows neutrons and control systems regulate neutron population.

MixedJEE Physics · Original

Nuclear Force, Short Range, and Saturation

The residual strong interaction between nucleons is attractive over typical nuclear separations, strongly repulsive at very short separation, approximately charge independent, and effective over only a few femtometres. Its short range produces saturation of binding.

MixedJEE Physics · Original

Nuclear Fusion

Fusion combines light nuclei. Its Q value is obtained from mass difference or total-binding difference; a positive value is released as product kinetic energy or radiation. High temperature raises collision energies but quantum tunnelling also matters.

MixedJEE Physics · Original

Nuclear Radius and Geometric Scaling

The empirical radius of a roughly spherical nucleus is R = R0 A^(1/3), with R0 about 1.2 fm. Ratios are more reliable than inserting a rounded R0, and the model describes bulk size rather than detailed nuclear shape.

MixedJEE Physics · Original

Nuclear Reaction Balancing and Q Value

For a reaction a + A -> b + B, nucleon number and electric charge are conserved, as are total energy and vector momentum. The reaction value is Q = (sum initial rest masses - sum final rest masses)c^2.

MixedJEE Physics · Original

Nuclear Volume and Nearly Constant Density

Using M approximately A m_n and R = R0 A^(1/3), a spherical nucleus has volume (4/3)pi R0^3 A and density 3m_n/(4pi R0^3), independent of A within the liquid-drop approximation.

MixedJEE Physics · Original

Orbital Current, Magnetic Moment, and Field

For orbital period T, the current magnitude is e/T. The orbital magnetic moment magnitude is eL/(2m), so Bohr quantization gives mu_n = n mu_B. At the centre of a circular orbit, B = mu_0 I/(2r), which scales as Z^3/n^5 in the simple Bohr model.

MixedJEE Physics · Original

Orbital Speed, Acceleration, and Period

Combining Coulomb centripetal force with angular-momentum quantization gives v_n proportional to Z/n and r_n proportional to n^2/Z. Therefore period 2 pi r_n/v_n scales as n^3/Z^2 and acceleration v_n^2/r_n scales as Z^3/n^4.

MixedJEE Physics · Original

Photon Energy in Atomic Transitions

For a transition between stationary states, conservation of energy gives h nu = hc/lambda = |E_i - E_f|. Emission has E_i > E_f; absorption has E_f > E_i.

MixedJEE Physics · Original

Photon Momentum and Atomic Recoil

A photon of wavelength lambda carries momentum h/lambda. Neglecting the small recoil correction to the line energy, a stationary atom of mass M acquires equal opposite momentum and speed h/(M lambda); its recoil energy is p^2/(2M).

MixedJEE Physics · Original

Quantization Beyond the Coulomb Atom

For a circular orbit under a stated central force F(r), solve m v^2/r = F(r) together with mvr = n hbar. For a uniform magnetic field perpendicular to motion, qvB supplies the centripetal force and yields r_n proportional to sqrt(n).

MixedJEE Physics · Original

Radiation Properties, Shielding, and Safe Use

Alpha particles have charge +2e and mass about 4u, beta particles are electrons or positrons, and gamma rays are photons. Their hazards depend on energy, exposure geometry, time, distance, shielding, and whether radioactive material is internal or external.

MixedJEE Physics · Original

Reaction Threshold Energy

For a non-relativistic two-body reaction a + A -> products with target A initially at rest and Q < 0, the exact threshold follows invariant energy conservation. In the usual small-|Q| approximation, K_th approximately -Q(1 + m_a/m_A).

MixedJEE Physics · Original

Reduced Mass, Isotopes, and Exotic Bohr Atoms

For masses m_1 and m_2, relative motion uses reduced mass mu = m_1 m_2/(m_1 + m_2). Hydrogenic radii scale as 1/(mu Z) and energies and Rydberg constants scale as mu Z^2 for fixed charge magnitudes.

MixedJEE Physics · Original

Rutherford Scattering and the Nuclear Atom

Rutherford scattering treats the alpha particle and nucleus as positive point charges interacting through Coulomb repulsion. The observed predominance of small deflections and rarity of backward scattering imply that nearly all positive charge and mass occupy a region far smaller than the atom.

MixedJEE Physics · Original

Rydberg Formula and Spectral Series

For emission from n_i to n_f with n_i > n_f, the wavenumber is R Z^2(1/n_f^2 - 1/n_i^2), subject to the reduced-mass value of R for the species. Lyman, Balmer, Paschen, Brackett, and Pfund have n_f = 1, 2, 3, 4, and 5.

MixedJEE Physics · Original

Series Limits, Longest and Shortest Wavelengths

For a series ending at n_f, the longest wavelength uses n_i = n_f + 1, while the shortest wavelength uses n_i approaching infinity. The limit wavenumber is R Z^2/n_f^2.

MixedJEE Physics · Original

Transition Combinations and Ritz Principle

Because each photon frequency equals a level-energy difference divided by h, transition frequencies obey the Ritz combination principle. For ordered levels E_A > E_B > E_C, nu_AC = nu_AB + nu_BC.

MixedJEE Physics · Original

Two-Body Recoil and Energy Sharing

In a two-body decay of a parent initially at rest, p1 = -p2. Under non-relativistic product motion, K1/K2 = m2/m1 and K1 + K2 = Q, giving K1 = Q m2/(m1 + m2).

MixedJEE Physics · Original

p-n junction formation and characteristics

Study how p-n junctions are formed, the creation of the depletion region, barrier potential, and how these properties change with biasing and temperature.

Foundation12 questions

Logic gates and Boolean algebra

Master the basic logic gates, their truth tables, Boolean expressions, universal gates, and how logic circuits are implemented.

Foundation6 questions

Semiconductor basics and doping

Learn about intrinsic and extrinsic semiconductors, doping processes, charge carrier types, and how doping affects electrical properties.

Exam level6 questions

Diode types and applications

Explore the working, characteristics, and uses of different diodes such as Zener diodes, including their role in voltage regulation and breakdown phenomena.

Foundation5 questions

Rectifiers and filters

Understand the operation of half-wave and full-wave rectifiers, the use of filters, and how rectification affects output voltage and frequency.

Foundation4 questions

Basic Logic Gates and Truth Tables

Boolean variables take values 0 or 1. The basic operations are AND as multiplication, OR as Boolean addition, and NOT as complementation; a circuit output is evaluated gate by gate.

MixedJEE Physics · Original

Bipolar Junction Transistor Currents

For a BJT, emitter current equals collector plus base current. In common-base notation alpha = Ic/Ie, while common-emitter current gain beta = Ic/Ib under the stated operating conditions.

MixedJEE Physics · Original

Boolean Algebra and Universal Gates

NAND and NOR are universal gates because each can realize NOT, AND, and OR. De Morgan's laws relate complemented products and sums and support systematic circuit simplification.

MixedJEE Physics · Original

Common-Emitter Transistor as Amplifier or Switch

In the active region of an npn common-emitter amplifier, the base-emitter junction is forward biased and collector-base junction reverse biased. Cutoff approximates an open switch and saturation approximates the ON state.

MixedJEE Physics · Original

Diode Biasing and I-V Characteristic

A p-n diode is forward biased when its p-side is at higher potential than its n-side. It is reverse biased for the opposite polarity; reverse current remains small and nearly voltage-independent before breakdown.

MixedJEE Physics · Original

Diode Models and DC Circuit Analysis

Piecewise diode analysis assigns every diode an ON or OFF state, solves the resulting linear circuit, and verifies that the calculated diode voltage and current support each assumed state.

MixedJEE Physics · Original

Doping, Majority Carriers, and Mass Action

Pentavalent donor impurities create n-type material with electrons as majority carriers; trivalent acceptors create p-type material with holes as majority carriers. At thermal equilibrium, carrier concentrations obey the mass-action law.

MixedJEE Physics · Original

Energy Bands and Material Classification

The valence band is the highest normally occupied band and the conduction band contains mobile electron states. The forbidden energy gap Eg separates them; semiconductors have a modest Eg whose carrier population changes strongly with temperature.

MixedJEE Physics · Original

Half-Wave and Full-Wave Rectifiers

Rectification produces pulsating dc from ac. For an ideal sinusoidal input, half-wave ripple repeats at the supply frequency and full-wave ripple repeats at twice the supply frequency.

MixedJEE Physics · Original

Intrinsic Semiconductors and Temperature

A pure semiconductor is intrinsic and satisfies n = p = ni. Over the usual semiconductor range, increasing temperature raises ni strongly, so resistance falls even though carrier mobility may decrease.

MixedJEE Physics · Original

Light-Emitting Diode

An LED is a heavily doped forward-biased p-n junction made from a suitable direct-band-gap semiconductor. Its approximate emitted photon energy is set by Eg, while a series resistor limits current and power.

MixedJEE Physics · Original

Mobility, Conductivity, and Resistivity

Mobility is drift-speed magnitude per electric-field magnitude. For electron concentration n and hole concentration p, their conventional-current contributions add to give the total conductivity.

MixedJEE Physics · Original

Optoelectronic Device Identification

Optoelectronic devices exchange electrical and optical energy through carrier generation or recombination. Identification should use operating mode and characteristic curves, not device name alone.

MixedJEE Physics · Original

P-N Junction, Depletion Layer, and Barrier

A p-n junction reaches equilibrium when diffusion current is balanced by drift current in the built-in electric field. The depletion width extends farther into the more lightly doped side.

MixedJEE Physics · Original

Photodiode

A photodiode is normally operated in reverse bias, where incident photons with energy at least comparable to Eg generate carriers and increase reverse current approximately with optical intensity.

MixedJEE Physics · Original

Solar Cell and Photovoltaic Operation

A solar cell is a large-area p-n junction operated in photovoltaic mode. Its terminal power is VI, and conversion efficiency is electrical output power divided by incident optical power.

MixedJEE Physics · Original

Wave Shaping and Capacitor Diode Circuits

In piecewise-linear wave-shaping circuits, diode states determine the active resistance and capacitor charging path. In a fixed state, the time constant is the resistance seen by the capacitor multiplied by capacitance.

MixedJEE Physics · Original

XOR, XNOR, and Combinational Logic

For two inputs, XOR is the sum of the mutually exclusive products A bar B and A B bar. XNOR is its complement and represents binary equality.

MixedJEE Physics · Original

Zener Breakdown and Device Limits

A Zener diode is designed to operate in reverse breakdown around Vz. Regulation requires current above the knee but below the current allowed by the maximum power rating.

MixedJEE Physics · Original

Zener Shunt Voltage Regulator

For an ideal shunt regulator in breakdown, Vo = Vz, series current is (Vs - Vz)/Rs, and Kirchhoff's current law gives Is = IL + Iz. Valid operation also requires safe minimum and maximum Iz.

MixedJEE Physics · Original

Cell Terminal Voltage and Internal Resistance Graph

For a cell of emf E and approximately constant internal resistance r delivering current I, terminal voltage is V = E - Ir, so a plot of V against I is a straight line with intercept E and slope -r.

MixedJEE Physics · Original

Concave-Mirror Focal Length by Parallax

In the no-parallax method, the image position is located by making a movable needle coincide in depth with the real image; measured object and image distances then satisfy the concave-mirror relation with the object placed beyond the focus.

MixedJEE Physics · Original

Convex-Lens u-v Relation and Graph

For a thin convex lens forming a real image with consistent sign convention, the measured conjugate distances satisfy 1/f = 1/v - 1/u, or 1/f = 1/v + 1/|u| when positive magnitudes are used.

MixedJEE Physics · Original

Diode I-V Curve and Static or Dynamic Resistance

For a nonlinear diode characteristic, static resistance at an operating point is V/I, whereas dynamic resistance is the local differential ratio dV/dI, approximated by a small Delta V divided by Delta I around that point.

MixedJEE Physics · Original

Galvanometer Resistance by Half Deflection

In the half-deflection method, a shunt S is connected across a galvanometer of resistance G while the source and series resistance remain fixed; circuit current division gives G exactly, with G approximately S only under the usual large-series-resistance condition.

MixedJEE Physics · Original

Meter-Bridge Balance and Data Consistency

For a uniform meter-bridge wire at a true null point l measured from the end adjacent to resistance R, the gap-resistance ratio is R/S = l/(100-l), subject to the actual circuit orientation and negligible end corrections.

MixedJEE Physics · Original

Multimeter Polarity Test for Components

In resistance mode, a multimeter applies an internal test voltage; reversing probes reverses device bias, so a healthy diode shows strongly asymmetric indications while an ohmic metal wire or resistor is approximately polarity independent.

MixedJEE Physics · Original

Newtonian Lens Formula and x-y Plot

For a thin lens with Newtonian object and image distances x and y measured from the corresponding focal points under the stated real-conjugate convention, xy = f squared; a plot of y against x is therefore a rectangular hyperbola.

MixedJEE Physics · Original

Ohm-Law Circuit Connections and V-I Test

To test Ohm's law, vary the current through a resistor while measuring series current and parallel terminal voltage; an ohmic conductor at constant physical conditions gives a linear V-I relation whose slope is resistance.

MixedJEE Physics · Original

Optical-Bench Focal-Length Measurement and Uncertainty

In an optical-bench lens experiment, object and image distances are differences of measured positions referenced to the lens optical centre; focal length follows the lens relation, and its uncertainty is propagated from the position-derived u and v values.

MixedJEE Physics · Original

Resistivity and Wire-Geometry Scaling

For a uniform wire at fixed temperature, R = rho L/A; therefore wires of the same material obey R2/R1 = (L2/L1)(A1/A2), and circular area scales as diameter squared.

MixedJEE Physics · Original

Resolution and Propagation of Measurement Uncertainty

For independent small limiting uncertainties, the maximum fractional uncertainty of a product of powers Q = product(x_i raised to n_i) is the sum of |n_i| times Delta x_i/x_i; a directly read scale cannot justify resolution finer than its least count.

MixedJEE Physics · Original

Screw-Gauge Pitch and Least Count

The pitch is the axial distance moved by the spindle per complete rotation, and the screw-gauge least count is the pitch divided by the number of equal divisions on the circular scale.

MixedJEE Physics · Original

Screw-Gauge Reading and Zero Correction

The observed screw-gauge reading is the pitch-scale reading plus the circular-scale reading times the least count, and the corrected value is obtained by subtracting the signed zero error measured with the studs in contact.

MixedJEE Physics · Original

Stokes-Law Viscosity by Terminal Speed

For a small sphere moving slowly through an effectively unbounded Newtonian liquid, Stokes drag is 6 pi eta R v; balancing drag with weight minus buoyancy gives terminal speed proportional to R squared and inversely proportional to viscosity.

MixedJEE Physics · Original

Travelling Microscope and Apparent Depth

For near-normal viewing through a plane slab, refractive index is real depth divided by apparent depth; the required depths are differences of corrected travelling-microscope readings, each resolved by its vernier least count.

MixedJEE Physics · Original

Uncertainty in Resistance from Voltmeter and Ammeter

For R = V/I and small independent limiting reading uncertainties, the maximum fractional uncertainty is Delta R/R approximately Delta V/V plus Delta I/I, with each direct uncertainty taken consistently from the instrument specification.

MixedJEE Physics · Original

Vernier Least Count and Direct Reading

For a direct vernier, the least count is one main-scale division minus one vernier-scale division, and the observed reading is the main-scale reading plus the coinciding vernier division multiplied by that least count.

MixedJEE Physics · Original

Vernier Zero Error and Correction

Vernier zero error is the signed reading shown when the jaws are in true contact; the zero correction is its negative, so the corrected measurement equals the observed measurement minus the signed zero error.

MixedJEE Physics · Original

Zener Characteristic, Power Rating, and Protection Resistor

For a Zener regulator in breakdown, the device voltage is approximately Vz, maximum safe current is Pmax/Vz, and the series resistance must be chosen from the worst-case source voltage and load current so Zener power remains within rating.

MixedJEE Physics · Original