Class 12 Physics — Important Board Questions with Answers

Coulomb's law states that the force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, acting along the line joining them. Mathematically, F = k q1 q2 / r^2, where k = 1/(4 pi epsilon0) = 9 x 10^9 N m^2 C^-2 in free space.

F = k q1 q2 / r^2

Given q1 = 2 x 10^-6 C, q2 = 3 x 10^-6 C, r = 30 cm = 0.3 m. Using F = k q1 q2 / r^2 = (9 x 10^9)(2 x 10^-6)(3 x 10^-6)/(0.3)^2 = (9 x 10^9 x 6 x 10^-12)/0.09 = 0.054/0.09 = 0.6 N. The force is repulsive since both charges are positive.

F = k q1 q2 / r^2

The electric field intensity at a point is the force experienced by a unit positive test charge placed at that point. It is a vector quantity given by E = F/q0, where q0 is the small positive test charge. Its SI unit is newton per coulomb (N/C) or equivalently volt per metre (V/m).

E = F/q0

Given Q = 5 x 10^-6 C, r = 0.3 m. The field due to a point charge is E = kQ/r^2 = (9 x 10^9)(5 x 10^-6)/(0.3)^2 = (45000)/0.09 = 5 x 10^5 N/C. The field is directed radially outward since the charge is positive.

E = kQ/r^2

An electric dipole is a pair of equal and opposite charges (+q and -q) separated by a small distance 2a. The dipole moment p = q x 2a, directed from the negative to the positive charge. On the axial line at distance r from the centre, the field due to +q is k q/(r-a)^2 (away) and due to -q is k q/(r+a)^2 (towards). Net E = kq[1/(r-a)^2 - 1/(r+a)^2] = kq[(4ar)/((r^2-a^2)^2)]. For a short dipole (r much greater than a), E = k(2p)/r^3, directed along the dipole moment.

E(axial) = 2kp/r^3

When a dipole of moment p is placed in a uniform electric field E at an angle theta, the two charges experience equal and opposite forces qE, forming a couple. The magnitude of the torque = force x perpendicular distance = qE x (2a sin theta) = (q x 2a) E sin theta = pE sin theta. In vector form, torque = p x E. The torque tends to align the dipole with the field.

torque = pE sin theta = p x E

Gauss's law states that the total electric flux through any closed surface is equal to 1/epsilon0 times the total charge enclosed by that surface. Mathematically, the surface integral of E over the closed surface = q(enclosed)/epsilon0. The closed surface over which the flux is calculated is called a Gaussian surface. The law is useful for finding the electric field of symmetric charge distributions.

flux = q(enclosed)/epsilon0

Consider an infinitely long straight wire with linear charge density lambda. Choose a coaxial cylindrical Gaussian surface of radius r and length L. By symmetry, the field E is radial and uniform over the curved surface. The flux through the curved surface = E x (2 pi r L); the flat ends contribute no flux. Charge enclosed = lambda L. By Gauss's law, E x 2 pi r L = lambda L / epsilon0, so E = lambda/(2 pi epsilon0 r). The field is directed radially and varies as 1/r.

E = lambda/(2 pi epsilon0 r)

The electric potential at a point is the work done in bringing a unit positive charge from infinity to that point against the electric field. It is a scalar quantity and its SI unit is the volt (V), where 1 volt = 1 joule per coulomb. The potential due to a point charge Q at distance r is V = kQ/r = Q/(4 pi epsilon0 r).

V = kQ/r

Given Q = 4 x 10^-6 C, r = 9 cm = 0.09 m. The potential due to a point charge is V = kQ/r = (9 x 10^9)(4 x 10^-6)/0.09 = (36000)/0.09 = 4 x 10^5 V. Thus the potential at the point is 4 x 10^5 volt.

V = kQ/r

The electric field is the negative gradient of the electric potential: E = -dV/dr. This means the field points in the direction in which the potential decreases most rapidly. The negative sign indicates that the electric field is directed from a region of higher potential to a region of lower potential, that is, potential decreases along the direction of the field.

E = -dV/dr

The capacitance of a conductor is the ratio of the charge given to it to the resulting rise in its potential: C = Q/V. It is a measure of the ability of the conductor to store charge. Its SI unit is the farad (F), where 1 farad = 1 coulomb per volt. Capacitance depends on the size and shape of the conductor and the surrounding medium.

C = Q/V

Let a parallel plate capacitor have plate area A, separation d, and charge Q with surface charge density sigma = Q/A. The uniform field between the plates is E = sigma/epsilon0 = Q/(A epsilon0). The potential difference between the plates is V = E d = Q d/(A epsilon0). Hence capacitance C = Q/V = Q/(Q d/(A epsilon0)) = epsilon0 A/d. So the capacitance increases with plate area and decreases with plate separation.

C = epsilon0 A/d

In series: 1/Cs = 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1, so Cs = 1 microfarad. In parallel: Cp = 2 + 3 + 6 = 11 microfarad. Thus the series combination gives 1 microfarad and the parallel combination gives 11 microfarad.

series: 1/Cs = sum(1/Ci); parallel: Cp = sum(Ci)

The energy stored in a charged capacitor is the work done in charging it. It can be written in three equivalent forms: U = (1/2) C V^2 = (1/2) Q V = Q^2/(2C), where C is the capacitance, Q the charge and V the potential difference. This energy is stored in the electric field between the plates.

U = (1/2) C V^2 = (1/2) QV = Q^2/(2C)

When a dielectric of dielectric constant K is fully inserted between the plates, the capacitance increases K times: C = K C0, where C0 is the capacitance with air. This is because the dielectric gets polarised and sets up an internal field opposite to the applied field, reducing the net field and hence the potential difference for the same charge; since C = Q/V, a smaller V means a larger C.

C = K C0

Ohm's law states that, at constant temperature, the current flowing through a conductor is directly proportional to the potential difference across its ends, so V = IR, where R is a constant called the resistance. Resistance is the opposition offered by a conductor to the flow of current and is defined as R = V/I. Its SI unit is the ohm.

V = IR

Resistivity (specific resistance) is the resistance of a conductor of unit length and unit area of cross-section; it depends on the material and temperature but not on its dimensions. The resistance of a wire is R = rho L/A, so it is directly proportional to its length L and inversely proportional to its area of cross-section A. The SI unit of resistivity is the ohm metre.

R = rho L/A

Drift velocity is the small average velocity with which free electrons move in a conductor under an applied electric field, opposite to the field. Consider a conductor of area A with n free electrons per unit volume moving with drift velocity vd. In time t, electrons in a length vd t cross a section, so charge crossing = (n)(A vd t)(e). Current I = charge/time = n A vd e t / t = n A e vd. Hence I = n A e vd.

I = n A e vd

In parallel: 1/Rp = 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1, so Rp = 1 ohm. In series: Rs = 2 + 3 + 6 = 11 ohm. Thus the parallel combination gives 1 ohm (less than the smallest resistance) and the series combination gives 11 ohm.

parallel: 1/Rp = sum(1/Ri); series: Rs = sum(Ri)

Kirchhoff's junction (current) law states that the algebraic sum of currents meeting at a junction is zero, that is, the total current entering a junction equals the total current leaving it; it is based on conservation of charge. Kirchhoff's loop (voltage) law states that the algebraic sum of the changes in potential around any closed loop of a circuit is zero; it is based on conservation of energy.

sum(I) at junction = 0; sum(V) around loop = 0

A Wheatstone bridge has four resistances P, Q, R and S arranged in a quadrilateral, with a galvanometer between the junctions of P-Q and R-S, and a cell across the other diagonal. When the bridge is balanced, no current flows through the galvanometer, so the junction connected by the galvanometer are at the same potential. Applying Kirchhoff's laws, the potential drops give P/Q = R/S. This is the balance condition; if three resistances are known, the fourth can be found.

P/Q = R/S

Given emf E = 2 V, internal resistance r = 0.5 ohm, external resistance R = 4.5 ohm. Current I = E/(R + r) = 2/(4.5 + 0.5) = 2/5 = 0.4 A. Terminal potential difference V = I R = 0.4 x 4.5 = 1.8 V (equivalently V = E - I r = 2 - 0.4 x 0.5 = 1.8 V).

I = E/(R + r); V = E - I r

When n identical cells, each of emf E and internal resistance r, are connected in series (all in the same direction), the effective emf is n E and the total internal resistance is n r. The current through an external resistance R is I = n E/(R + n r). Series grouping is advantageous when the external resistance is much larger than the internal resistance.

I = nE/(R + nr)

The resistance of a metallic conductor increases with a rise in temperature, because increased thermal motion of the atoms causes more frequent collisions of electrons, increasing the resistance. The relation is R(t) = R0 (1 + alpha (delta T)), where R0 is the resistance at the reference temperature, alpha is the temperature coefficient of resistance and delta T is the rise in temperature.

R = R0 (1 + alpha delta T)

A charge q moving with velocity v in a magnetic field B experiences a magnetic force F = q v B sin theta, where theta is the angle between v and B; in vector form F = q(v x B). The force is maximum (F = qvB) when the charge moves perpendicular to the field (theta = 90 deg), and it is zero when the charge moves parallel or antiparallel to the field (theta = 0 or 180 deg). The force is always perpendicular to the velocity, so it does no work.

F = q v B sin theta

The force on a current-carrying conductor is F = B I L sin theta. Here B = 0.2 T, I = 4 A, L = 0.5 m and theta = 90 deg (so sin theta = 1). F = 0.2 x 4 x 0.5 x 1 = 0.4 N. The direction of the force is given by Fleming's left-hand rule.

F = B I L sin theta

The Biot-Savart law states that the magnetic field dB due to a small current element I dl at a point P at distance r is directly proportional to the current I, the length dl and the sine of the angle theta between the element and the line joining it to P, and inversely proportional to the square of the distance r. Mathematically, dB = (mu0/4 pi) (I dl sin theta)/r^2, and its direction is perpendicular to the plane containing dl and r.

dB = (mu0/4 pi)(I dl sin theta)/r^2

Consider a circular loop of radius R carrying current I. By the Biot-Savart law, each element I dl is perpendicular to the line joining it to the centre (theta = 90 deg), so dB = (mu0/4 pi)(I dl)/R^2. All elements produce fields in the same direction (perpendicular to the plane of the loop) at the centre, so they simply add. Integrating dl around the loop gives the circumference 2 pi R: B = (mu0/4 pi)(I/R^2)(2 pi R) = mu0 I/(2R). For N turns, B = mu0 N I/(2R).

B = mu0 I/(2R)

Ampere's circuital law states that the line integral of the magnetic field B around any closed loop is equal to mu0 times the total current enclosed by the loop: the integral of B.dl = mu0 I(enclosed). Applying it to a long solenoid with n turns per unit length carrying current I, the magnetic field inside is uniform and given by B = mu0 n I, directed along the axis; the field outside an ideal long solenoid is nearly zero.

B = mu0 n I

When a charged particle of charge q and mass m enters a magnetic field B perpendicularly with speed v, the magnetic force qvB provides the centripetal force m v^2/r. Equating: q v B = m v^2/r, so the radius r = m v/(q B). The time period T = 2 pi r/v = 2 pi m/(q B), which is independent of the speed and radius. This principle is used in the cyclotron.

r = mv/(qB); T = 2 pi m/(qB)

Two long parallel wires separated by distance d carrying currents I1 and I2 experience a force per unit length F/L = (mu0 I1 I2)/(2 pi d); the force is attractive if the currents are in the same direction and repulsive if opposite. The ampere is defined as that steady current which, when maintained in two infinitely long parallel wires of negligible cross-section placed 1 metre apart in vacuum, produces a force of 2 x 10^-7 newton per metre of length between them.

F/L = mu0 I1 I2/(2 pi d)

A moving coil galvanometer works on the principle that a current-carrying coil placed in a magnetic field experiences a torque. The deflection of the coil is directly proportional to the current passing through it (I is proportional to the deflection). To convert a galvanometer into an ammeter, a small resistance (shunt) is connected in parallel with it, so most of the current bypasses the galvanometer. To convert it into a voltmeter, a high resistance is connected in series with it, so that only a small current flows and it can measure potential difference.

A current loop behaves as a magnetic dipole. The magnetic dipole moment of a coil of N turns each of area A carrying current I is m = N I A, and it is directed perpendicular to the plane of the loop (given by the right-hand rule). Its SI unit is ampere metre squared (A m^2). The torque on it in a field B is m x B.

m = N I A

Diamagnetic substances are weakly repelled by a magnetic field and move from stronger to weaker regions; they have a small negative susceptibility (e.g. bismuth, copper). Paramagnetic substances are weakly attracted by a magnetic field and have a small positive susceptibility (e.g. aluminium, sodium). Ferromagnetic substances are strongly attracted and can be permanently magnetised; they have a large positive susceptibility (e.g. iron, cobalt, nickel).

(1) Magnetic field lines are continuous closed curves that pass from the south to the north pole inside the magnet and from the north to the south pole outside it. (2) The tangent drawn at any point on a field line gives the direction of the magnetic field at that point. (3) Two field lines never intersect each other (as the field can have only one direction at a point). (4) The lines are crowded where the field is strong and spread apart where it is weak.

The three elements of the earth's magnetism are: (1) Magnetic declination - the angle between the geographic meridian and the magnetic meridian at a place. (2) Magnetic dip or inclination - the angle made by the earth's total magnetic field with the horizontal at a place. (3) Horizontal component of the earth's field (BH) - the component of the earth's total magnetic field in the horizontal direction, given by BH = B cos(dip). These three completely specify the earth's field at a place.

BH = B cos(dip)

Magnetic susceptibility (chi) is the ratio of the intensity of magnetisation (M) produced in a material to the magnetising field (H): chi = M/H; it measures how easily a material can be magnetised. Relative permeability (mu_r) is the ratio of the permeability of the material to that of free space. The relation between them is mu_r = 1 + chi.

mu_r = 1 + chi

Faraday's first law states that whenever the magnetic flux linked with a closed circuit changes, an emf is induced in the circuit, and it lasts as long as the flux is changing. Faraday's second law states that the magnitude of the induced emf is equal to the rate of change of magnetic flux linked with the circuit: e = -N (d(flux)/dt), where N is the number of turns. The negative sign is due to Lenz's law.

e = -N d(flux)/dt

Lenz's law states that the direction of the induced current (or emf) is always such that it opposes the change in magnetic flux that produces it. For example, if a magnet is pushed towards a coil, the induced current opposes its approach. Lenz's law is a consequence of the law of conservation of energy, because work has to be done against the opposing force, and this work appears as electrical energy.

Consider a conducting rod of length l moving with velocity v perpendicular to a uniform magnetic field B. In time dt the rod sweeps an area dA = l (v dt), so the change in flux is d(flux) = B dA = B l v dt. By Faraday's law, the induced emf e = d(flux)/dt = B l v. Alternatively, the free electrons in the rod experience a force qvB that pushes them to one end, setting up an emf e = B v l across the rod.

e = B l v

Self-inductance is the property of a coil by virtue of which it opposes any change in the current flowing through it, by inducing an opposing emf (back emf). It is defined as the flux linkage per unit current (N flux = L I) or from the induced emf e = -L (dI/dt), where L is the self-inductance. Its SI unit is the henry (H).

e = -L dI/dt

Mutual inductance is the property by which a change of current in one coil (primary) induces an emf in a neighbouring coil (secondary) due to the change in flux linkage. It is defined by e2 = -M (dI1/dt), where M is the mutual inductance, whose SI unit is the henry. It depends on the number of turns of the coils, their geometry (area and length), the distance and orientation between them, and the permeability of the core material.

e2 = -M dI1/dt

Given N = 500, initial flux = 0.01 Wb, final flux = 0.05 Wb, so change in flux = 0.05 - 0.01 = 0.04 Wb, and time = 0.1 s. The induced emf (magnitude) = N (change in flux)/time = 500 x 0.04/0.1 = 500 x 0.4 = 200 V.

e = N (change in flux)/time

Eddy currents are the circulating induced currents produced in the body of a conductor when the magnetic flux linked with it changes. They flow in closed loops within the conductor and generally cause heating and energy loss. Applications: they are used in induction furnaces (to melt metals by the heat produced), in electromagnetic braking of trains, in electric (induction) meters, and in induction cooktops. Laminating the cores of transformers reduces energy loss due to eddy currents.

The RMS (root mean square) value of an alternating current is that value of steady direct current which produces the same heating effect in a given resistance in the same time as the alternating current does. For a sinusoidal current of peak value I0, the RMS value is Irms = I0/sqrt(2) = 0.707 I0. Similarly Vrms = V0/sqrt(2). AC meters read RMS values.

Irms = I0/sqrt(2)

Inductive reactance is the opposition offered by an inductor to the flow of alternating current, given by XL = omega L = 2 pi f L; it increases with frequency. Capacitive reactance is the opposition offered by a capacitor to alternating current, given by XC = 1/(omega C) = 1/(2 pi f C); it decreases with frequency. Both are measured in ohm. For direct current (f = 0), XL = 0 and XC is infinite.

XL = omega L; XC = 1/(omega C)

In a series LCR circuit connected to an AC source, the total opposition to current is called impedance Z, given by Z = sqrt(R^2 + (XL - XC)^2), where XL is the inductive reactance and XC is the capacitive reactance. The phase angle phi between the applied voltage and the current is given by tan(phi) = (XL - XC)/R. The current is I = V/Z.

Z = sqrt(R^2 + (XL - XC)^2)

Resonance in a series LCR circuit occurs when the inductive reactance equals the capacitive reactance (XL = XC). At resonance the impedance is minimum (Z = R), the current is maximum, and the circuit behaves as purely resistive. The condition XL = XC gives omega L = 1/(omega C), so omega^2 = 1/(LC), and the resonant frequency f = 1/(2 pi sqrt(LC)). Such a circuit is used for tuning radios and televisions.

f = 1/(2 pi sqrt(LC))

The average power in an AC circuit is P = Vrms Irms cos(phi), where cos(phi) is called the power factor and phi is the phase difference between voltage and current. Thus the power factor is the ratio of true power to apparent power (cos phi = R/Z). Wattless current is the component of the AC current (Irms sin phi) that is 90 degrees out of phase with the voltage; it consumes no average power, so it is called the idle or wattless current.

P = Vrms Irms cos(phi)

A transformer works on the principle of mutual induction: an alternating current in the primary coil produces a changing flux that induces an emf in the secondary coil. For an ideal transformer, Vs/Vp = Ns/Np = Ip/Is, where V is voltage, N is number of turns and I is current. A step-up transformer has more turns in the secondary (Ns > Np) and increases the voltage; a step-down transformer has fewer turns in the secondary (Ns < Np) and decreases the voltage.

Vs/Vp = Ns/Np = Ip/Is

For an ideal transformer, Ns/Np = Vs/Vp. Given Vp = 2200 V, Vs = 220 V, Np = 5000. So Ns = Np x (Vs/Vp) = 5000 x (220/2200) = 5000 x 0.1 = 500 turns. Since the secondary has fewer turns than the primary, it is a step-down transformer, as expected.

Ns/Np = Vs/Vp

Displacement current is the current that arises due to a changing electric field (or changing electric flux) between the plates of a capacitor, even though no charge actually flows across the gap. It is given by Id = epsilon0 (d(electric flux)/dt). Maxwell introduced it to make Ampere's law consistent while charging a capacitor: the total current (conduction current plus displacement current) is continuous, so the modified Ampere-Maxwell law is the integral of B.dl = mu0 (I + Id).

Id = epsilon0 d(electric flux)/dt

(1) Electromagnetic waves are transverse in nature, with the electric field E and magnetic field B oscillating perpendicular to each other and to the direction of propagation. (2) They do not require a material medium and can travel through vacuum. (3) They travel through vacuum with the speed of light, c = 3 x 10^8 m/s. (4) They carry energy and momentum, and the ratio of the amplitudes of E and B equals c (E0/B0 = c).

c = E0/B0

In order of increasing frequency (decreasing wavelength): (1) Radio waves - used in radio and television communication. (2) Microwaves - used in radar and microwave ovens. (3) Infrared - used in remote controls and thermal imaging. (4) Visible light - used for vision and photography. (5) Ultraviolet - used to sterilise water and in detecting forgery. (6) X-rays - used in medical imaging and detecting fractures. (7) Gamma rays - used in cancer treatment (radiotherapy).

The speed of electromagnetic waves in vacuum is given by c = 1/sqrt(mu0 epsilon0), where mu0 is the permeability and epsilon0 the permittivity of free space. Substituting mu0 = 4 pi x 10^-7 and epsilon0 = 8.85 x 10^-12 gives c = 3 x 10^8 m/s, which equals the measured speed of light, showing that light is an electromagnetic wave. In a medium the speed is v = 1/sqrt(mu epsilon).

c = 1/sqrt(mu0 epsilon0)

The mirror formula relates the object distance u, image distance v and focal length f of a spherical mirror: 1/v + 1/u = 1/f. The linear magnification is m = -v/u = height of image/height of object. The focal length f = R/2, where R is the radius of curvature. The New Cartesian sign convention is used for the distances.

1/v + 1/u = 1/f

Using the sign convention, u = -20 cm and f = -15 cm. From the mirror formula 1/v = 1/f - 1/u = 1/(-15) - 1/(-20) = -1/15 + 1/20 = (-4 + 3)/60 = -1/60, so v = -60 cm. The magnification m = -v/u = -(-60)/(-20) = -3. The image is real (v negative), inverted (m negative) and magnified three times, formed 60 cm in front of the mirror.

1/v + 1/u = 1/f

The laws of refraction are: (1) The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane. (2) For a given pair of media and a given colour of light, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, called the refractive index: sin i/sin r = n (Snell's law). Refraction occurs because light travels at different speeds in different media.

sin i/sin r = n

Total internal reflection is the phenomenon in which a ray of light travelling from a denser to a rarer medium is completely reflected back into the denser medium when the angle of incidence exceeds a certain angle called the critical angle. Conditions: (1) light must travel from a denser to a rarer medium; (2) the angle of incidence must be greater than the critical angle C, where sin C = 1/n. Applications: optical fibres, sparkling of diamonds, and mirages.

sin C = 1/n

The critical angle C is given by sin C = 1/n, where n is the refractive index of the denser medium (glass). So sin C = 1/1.5 = 0.667, giving C = sin inverse (0.667) = 41.8 degrees (approximately 42 degrees). For angles of incidence greater than this, total internal reflection occurs.

sin C = 1/n

The lens maker's formula is 1/f = (n - 1)(1/R1 - 1/R2), where f is the focal length of the lens, n is the refractive index of the lens material with respect to the surrounding medium, and R1 and R2 are the radii of curvature of the two surfaces of the lens (taken with proper sign convention). It shows that the focal length depends on the material and the shape (curvature) of the lens.

1/f = (n - 1)(1/R1 - 1/R2)

Using the sign convention, u = -30 cm and f = +20 cm. From the lens formula 1/v - 1/u = 1/f, we get 1/v = 1/f + 1/u = 1/20 + 1/(-30) = 1/20 - 1/30 = (3 - 2)/60 = 1/60, so v = +60 cm. The magnification m = v/u = 60/(-30) = -2. The image is real (v positive for a lens), inverted (m negative) and magnified twice, formed 60 cm on the other side of the lens.

1/v - 1/u = 1/f

The power of a lens is a measure of its ability to converge or diverge light and is defined as the reciprocal of its focal length in metres: P = 1/f (f in metres). Its SI unit is the dioptre (D). A converging (convex) lens has positive power and a diverging (concave) lens has negative power. For two thin lenses in contact, the total power is the sum: P = P1 + P2.

P = 1/f; P = P1 + P2

When a ray passes through a prism, it is deviated; the deviation is least at the angle of minimum deviation Dm, when the ray passes symmetrically through the prism. The refractive index of the prism material is given by n = sin((A + Dm)/2)/sin(A/2), where A is the refracting angle of the prism and Dm is the angle of minimum deviation. This formula is used to determine the refractive index of a transparent material.

n = sin((A + Dm)/2)/sin(A/2)

An astronomical telescope has an objective lens of large focal length fo and an eyepiece of small focal length fe. In normal adjustment (final image at infinity), the magnifying power is M = fo/fe, and the length of the telescope is fo + fe. For high magnification, the objective should have a large focal length (and large aperture) and the eyepiece a small focal length. The final image is inverted.

M = fo/fe

Huygens' principle states that: (1) every point on a given wavefront acts as a source of new disturbance called secondary wavelets, which spread out in all directions with the speed of the wave; and (2) the new wavefront at a later instant is the forward envelope (tangential surface) of all these secondary wavelets. This principle is used to explain the laws of reflection and refraction and the propagation of light as a wave.

In Young's double slit experiment, two coherent sources produce alternate bright and dark fringes on a screen. The fringe width (the distance between two consecutive bright or dark fringes) is beta = lambda D/d, where lambda is the wavelength of light used, D is the distance between the slits and the screen, and d is the separation between the two slits. All fringes are of equal width, and the width increases with wavelength and D but decreases as d increases.

beta = lambda D/d

Given lambda = 600 nm = 600 x 10^-9 m, d = 1 mm = 1 x 10^-3 m, D = 1 m. Fringe width beta = lambda D/d = (600 x 10^-9 x 1)/(1 x 10^-3) = 600 x 10^-6 = 6 x 10^-4 m = 0.6 mm.

beta = lambda D/d

For constructive interference (bright fringe), the path difference between the two interfering waves must be an integral multiple of the wavelength: path difference = n lambda, where n = 0, 1, 2, ... For destructive interference (dark fringe), the path difference must be an odd multiple of half the wavelength: path difference = (2n - 1) lambda/2. Equivalently, the phase difference is 2n pi for constructive and (2n - 1) pi for destructive interference.

constructive: n lambda; destructive: (2n-1) lambda/2

Coherent sources are two sources of light that emit waves of the same frequency (or wavelength) and have a constant phase difference between them. Conditions for sustained (steady) interference: (1) the two sources must be coherent; (2) they must have the same frequency and nearly equal amplitudes; and (3) they must be narrow and close together, and the light should preferably be monochromatic. In practice, coherent sources are obtained from a single source (for example, using two slits).

Diffraction is the bending of light around the edges of an obstacle or aperture and its spreading into the geometrical shadow region. In diffraction at a single slit of width a, a central bright maximum is flanked by alternate dark and bright fringes. The condition for the dark fringes (minima) is a sin theta = n lambda, where n = 1, 2, 3, ... and theta is the angle of diffraction. The central maximum is the brightest and widest.

a sin theta = n lambda

(1) Interference is due to the superposition of waves from two (or more) different coherent sources, whereas diffraction is due to the superposition of secondary wavelets coming from different parts of the same wavefront. (2) In interference all bright fringes are of equal intensity and equal width, whereas in diffraction the central maximum is the brightest and the intensity of the secondary maxima decreases rapidly on either side.

Polarisation is the phenomenon of restricting the vibrations of the electric field of a light wave to a single plane perpendicular to the direction of propagation; it shows that light is a transverse wave. Brewster's law states that when unpolarised light is incident on a transparent surface at a particular angle called the polarising angle (theta_p), the reflected light is completely plane-polarised, and the refractive index of the medium is n = tan(theta_p).

n = tan(theta_p)

The photoelectric effect is the phenomenon of emission of electrons (called photoelectrons) from the surface of a metal when light of suitable frequency (usually ultraviolet or visible for some metals) falls on it. The emitted electrons carry kinetic energy. The effect occurs only when the frequency of the incident light is greater than a certain minimum value called the threshold frequency, and it provided evidence for the particle (photon) nature of light.

Einstein's photoelectric equation is h f = W0 + KE(max), where h f is the energy of the incident photon (h is Planck's constant and f the frequency of light), W0 is the work function (the minimum energy needed to eject an electron from the metal surface), and KE(max) is the maximum kinetic energy of the emitted photoelectron. It expresses conservation of energy: the photon's energy is used partly to free the electron and the rest appears as its kinetic energy. Thus KE(max) = h f - W0.

h f = W0 + KE(max)

The work function (W0) of a metal is the minimum energy required to just remove an electron from the surface of the metal without giving it any kinetic energy. The threshold frequency (f0) is the minimum frequency of the incident light below which no photoelectric emission takes place, however intense the light may be. They are related by W0 = h f0, where h is Planck's constant.

W0 = h f0

The energy of the incident photon E = hc/lambda = 1240/400 = 3.1 eV. Using Einstein's equation, the maximum kinetic energy KE(max) = E - W0 = 3.1 - 2.0 = 1.1 eV. Since the photon energy (3.1 eV) is greater than the work function (2 eV), emission occurs and the photoelectrons have a maximum kinetic energy of 1.1 eV.

KE(max) = hc/lambda - W0

(1) For a given metal, photoelectric emission occurs only if the frequency of the incident light is greater than a certain minimum value (threshold frequency), whatever the intensity. (2) The maximum kinetic energy of the emitted photoelectrons depends on the frequency of the incident light and the nature of the metal, but is independent of the intensity of the light. (3) The number of photoelectrons emitted per second (photoelectric current) is directly proportional to the intensity of the incident light. (4) The emission is instantaneous, with no measurable time lag.

The de Broglie hypothesis states that every moving particle has a wave associated with it, called a matter wave. The de Broglie wavelength is lambda = h/p = h/(m v), where h is Planck's constant, p is the momentum, m is the mass and v the velocity of the particle. In terms of kinetic energy, lambda = h/sqrt(2 m KE). This shows the dual (wave-particle) nature of matter; the wavelength is significant only for very small particles like electrons.

lambda = h/(m v)

From the scattering of alpha particles by a thin gold foil, Rutherford concluded that: (1) most of the atom is empty space, since most alpha particles passed straight through; (2) the entire positive charge and almost all the mass of the atom are concentrated in a very small central region called the nucleus, since a few alpha particles were deflected through large angles; and (3) the electrons revolve around the nucleus, and the size of the nucleus is very small compared with the size of the atom.

Bohr's postulates are: (1) The electron revolves around the nucleus only in certain fixed circular orbits called stationary states, in which it does not radiate energy. (2) Only those orbits are allowed for which the angular momentum of the electron is an integral multiple of h/(2 pi), that is, m v r = n h/(2 pi) (quantisation of angular momentum). (3) Energy is emitted or absorbed only when the electron jumps from one orbit to another, the energy of the emitted or absorbed photon being h f = E2 - E1.

m v r = n h/(2 pi)

The energy of the electron in the nth orbit of hydrogen is En = -13.6/n^2 eV. For n = 2, E2 = -13.6/2^2 = -13.6/4 = -3.4 eV. The negative sign shows that the electron is bound to the nucleus, and the energy increases (becomes less negative) as n increases.

En = -13.6/n^2 eV

Energy of the electron in n = 3: E3 = -13.6/9 = -1.51 eV. Energy in n = 2: E2 = -13.6/4 = -3.4 eV. The energy of the emitted photon = E3 - E2 = -1.51 - (-3.4) = 1.89 eV. This corresponds to the H-alpha line of the Balmer series (visible red light).

E(photon) = E(higher) - E(lower)

(1) Bohr's model applies successfully only to hydrogen and hydrogen-like single-electron atoms; it fails to explain the spectra of atoms having more than one electron. (2) It could not explain the fine structure of spectral lines or the relative intensities of the lines, and it does not account for the splitting of spectral lines in electric and magnetic fields (the Stark and Zeeman effects). Also, it arbitrarily assumes quantisation without explaining it (later explained by de Broglie).

Einstein's mass-energy relation is E = m c^2, which states that mass and energy are interconvertible, where c is the speed of light. Using this relation, the energy equivalent of 1 atomic mass unit (1 u = 1.66 x 10^-27 kg) is about 931 MeV (mega electron volt). This relation explains the large amount of energy released in nuclear reactions such as fission and fusion.

E = m c^2; 1 u = 931 MeV

The mass defect is the difference between the sum of the masses of the individual protons and neutrons (nucleons) and the actual mass of the nucleus; the actual nuclear mass is always less than the sum. This missing mass (delta m) is converted into energy that binds the nucleons together. The binding energy is the energy equivalent of the mass defect, BE = (delta m) c^2; it is the energy required to break the nucleus into its constituent nucleons.

BE = (delta m) c^2

The binding energy is the energy equivalent of the mass defect. BE = (mass defect in u) x 931 MeV = 0.0304 x 931 = 28.3 MeV. Thus the binding energy of the helium nucleus is about 28.3 MeV, and the binding energy per nucleon = 28.3/4 = 7.1 MeV.

BE (MeV) = (delta m in u) x 931

The three types of radioactive radiations are: (1) alpha rays, which are helium nuclei (2 protons + 2 neutrons), positively charged and with low penetrating power; (2) beta rays, which are fast-moving electrons, negatively charged and with greater penetrating power than alpha rays; and (3) gamma rays, which are high-energy electromagnetic waves (photons), electrically neutral and with very high penetrating power. In a magnetic field, alpha and beta rays are deflected in opposite directions while gamma rays are undeflected.

The radioactive decay law states that the rate of decay of a radioactive sample is directly proportional to the number of undecayed nuclei present at that instant: N = N0 e^(-lambda t), where lambda is the decay constant. The half-life T is related to it by T = 0.693/lambda. After each half-life, half of the sample remains, so after 3 half-lives the fraction remaining = (1/2)^3 = 1/8 of the original sample.

N = N0 (1/2)^(t/T)

Nuclear fission is the process in which a heavy nucleus (such as uranium-235) splits into two lighter nuclei of comparable masses, with the release of a few neutrons and a large amount of energy. For example, when a uranium-235 nucleus captures a slow neutron, it splits into barium and krypton nuclei plus three neutrons and energy. The released neutrons can cause further fissions, leading to a chain reaction, which is used in nuclear reactors and atom bombs.

Nuclear fusion is the process in which two light nuclei (such as isotopes of hydrogen) combine to form a heavier nucleus, with the release of an enormous amount of energy. It is the source of energy of the sun and stars, where hydrogen nuclei fuse to form helium. It requires very high temperature (millions of degrees) because the positively charged nuclei must overcome their strong electrostatic repulsion to come close enough to fuse.

An intrinsic semiconductor is a pure semiconductor (such as pure silicon or germanium) with no added impurity; its conductivity is low and is due to the equal number of electrons and holes generated thermally. An extrinsic semiconductor is one to which a small amount of a suitable impurity has been added (doping); this greatly increases its conductivity. Extrinsic semiconductors are of two types, n-type and p-type.

An n-type semiconductor is formed by doping a pure semiconductor (silicon) with a pentavalent impurity (such as phosphorus or arsenic), which donates free electrons; the majority carriers are electrons and the minority carriers are holes. A p-type semiconductor is formed by doping with a trivalent impurity (such as boron or aluminium), which creates holes; the majority carriers are holes and the minority carriers are electrons. Both types are electrically neutral overall.

When a p-n junction is formed, electrons from the n-side diffuse into the p-side and holes from the p-side diffuse into the n-side, and they recombine near the junction. This leaves a region near the junction that has no free charge carriers but has immobile charged ions; this region is called the depletion region (or depletion layer). The immobile ions set up an internal electric field that opposes further diffusion; the potential difference developed across the depletion region is called the potential barrier.

In forward biasing, the p-side is connected to the positive terminal and the n-side to the negative terminal of the battery; this reduces the width of the depletion region and the potential barrier, so a large current flows and the diode conducts. In reverse biasing, the p-side is connected to the negative terminal and the n-side to the positive terminal; this increases the width of the depletion region and the potential barrier, so only a very small (negligible) current flows and the diode does not conduct.

A rectifier converts alternating current (AC) into direct current (DC). In a half-wave rectifier, a single diode is connected in series with the AC source and a load resistor. During the positive half-cycle of the AC input, the diode is forward biased and conducts, so current flows through the load. During the negative half-cycle, the diode is reverse biased and does not conduct, so no current flows. As a result, output is obtained only during one half of each cycle, giving a pulsating DC. Its efficiency is low because half the input is wasted.

A full-wave rectifier converts both halves of the AC input into DC. It uses two diodes with a centre-tapped transformer and a load resistor. During the positive half-cycle, one diode is forward biased and conducts while the other is reverse biased; during the negative half-cycle, the second diode conducts while the first does not. In both half-cycles, the current through the load flows in the same direction, so output is obtained during the whole cycle. This gives a smoother, more efficient pulsating DC than a half-wave rectifier.

A Zener diode is a special heavily doped p-n junction diode designed to operate in the reverse breakdown region without being damaged. In this region, the voltage across it remains almost constant (equal to its Zener voltage) even when the current through it changes over a wide range. Because of this property, its main use is as a voltage regulator, that is, to provide a constant output voltage to a load in spite of changes in the input voltage or load current.

A logic gate is a digital circuit that gives an output based on its inputs following a logical rule (using binary 0 and 1). OR gate (output Y = A + B): Y = 1 if any input is 1; for inputs (0,0),(0,1),(1,0),(1,1) the outputs are 0,1,1,1. AND gate (Y = A.B): Y = 1 only if both inputs are 1; outputs are 0,0,0,1. NOT gate (Y = not A): it has a single input and inverts it, so input 0 gives output 1 and input 1 gives output 0.

OR: Y = A + B; AND: Y = A.B; NOT: Y = not A