Elements of Earth's magnetism

“A compass needle points north (declination) while tilting down into the ground (dip), just like a tilted flashlight beam.”

Earth’s field is described by declination (angle between magnetic and geographic north), inclination or dip (angle the field makes with the horizontal), and the horizontal component B_H = B cos I. The trap is mixing dip with the angle from vertical; dip is measured from the horizontal, so B_H uses cosine, not sine.

Memory trick: Declination = direction; Dip = tilt; B_H = B·cos I.

RMS value of AC

“A kitchen mixer’s heating equals a DC current equal to its sinusoidal peak divided by √2.”

For a sinusoidal current i = I₀ sin ωt, the root-mean-square value is I_rms = I₀/√2, giving the same heating as a DC current of that magnitude. Trap: using the same formula for non-sinusoidal waveforms leads to incorrect power calculations.

I_rms = I0/sqrt(2)

Memory trick: RMS = I_peak/√2 (sinusoid only)

Brewster's angle

“Glare off a lake vanishes at a certain angle because the reflected light becomes perfectly polarized.”

At Brewster’s angle θB, reflected and refracted rays are orthogonal, giving tanθB = n2/n1. For air to glass (n1≈1) it reduces to tanθB = n. The reflected beam is linearly polarized perpendicular to the plane of incidence; using the wrong index ratio flips the answer.

tan(theta_B) = n

Memory trick: Brewster: reflected ⟂ refracted → tanθ = n₂/n₁

Penetrating power of radiation

“Alpha particles are like a slow-moving bowling ball stopped by a sheet of paper, while gamma rays need lead shielding.”

α particles lose energy in a few cm of air or a sheet of paper, β particles penetrate mm of Al, while γ rays require several cm of lead. Order of penetration: α < β < γ. Trap: mixing range in air with shielding ability; β has longer range in air but lower penetrating power in solids.

Memory trick: Paper-α, Al-β, Lead-γ.

E and B in an EM wave

“A laser pointer’s beam is a plane wave where the electric and magnetic fields oscillate together, like two dancers moving side-by-side.”

In a uniform plane wave in free space E, B and the propagation vector k are mutually perpendicular and the fields are in phase; their amplitudes satisfy E0 = c B0. This relation breaks down in the near-field or waveguide modes where phase shifts can occur. Trap: assuming E ⟂ B and in-phase for any electromagnetic field.

E0/B0 = c

Memory trick: E ⟂ B ⟂ k, in phase, E=c B

Angle of contact

“A water droplet beads on a waxed windshield, showing a large contact angle between liquid and solid.”

The angle of contact θ is measured inside the liquid where solid, liquid and vapor meet. Young’s equation γ_SV = γ_SL + γ_LV cosθ links surface tensions; θ<90° means the liquid spreads (wetting), θ>90° means it beads. Measuring the external angle instead of the internal one flips the wetting prediction.

Memory trick: θ<90° → wetting; θ>90° → beading.

Independent horizontal and vertical motion — common mistake

“Like thinking a skier’s total speed down a slope equals its speed across the flat.”

The projectile’s speed changes because the vertical component varies, but the horizontal component remains u cosθ throughout flight. Mixing the resultant speed √(v_x²+v_y²) with v_x leads to errors, especially when asked for range or time of flight.

Memory trick: Horizontal speed = u cosθ always; don’t use √(v_x²+v_y²) for v_x.

Thermal expansion — common mistake

“A metal washer heated uniformly sees its central hole grow in diameter just like the outer edge.”

A hole in a plate expands as if the missing material also expanded; its dimensions increase with the same coefficient as the surrounding solid. Mistake: assuming the hole shrinks when the plate is heated.

Memory trick: Hole expands – think of missing material expanding

Temperature and molecular kinetic energy — common mistake

“In a room, oxygen and helium molecules jiggle differently; they don’t all move at the same speed even at equal temperature.”

Equal temperature gives equal average translational kinetic energy, not equal speed. The Maxwell-Boltzmann distribution yields a spread of speeds that depends on mass. Trap: assuming a single speed for all molecules in pressure or diffusion problems.

Memory trick: Same T → same KE, not same v

Field-potential relationship — common mistake

“A hilltop is high potential; a ball rolls down, not up.”

A common error is to think the field points toward higher potential. Since a positive test charge moves spontaneously toward lower potential, the field direction is opposite to the increase of V. Remember the minus sign in E = -dV/dx.

Memory trick: Field = downhill, not uphill.

No isolated magnetic poles — common mistake

“If you think a broken magnet leaves a north-only half, you’re treating it like a broken battery, but magnetic poles always stay paired.”

When a magnet is sliced, each fragment automatically forms a complete north-south pair; the magnetic domains re-orient to preserve dipoles. Assuming one piece becomes only north (or only south) contradicts ∮ B·dA = 0 and leads to wrong answers in flux problems.

Memory trick: Magnet cut = two complete magnets.

Bohr's quantised orbits — common mistake

“Think of a train moving on a fixed track – it doesn’t whistle until it switches tracks.”

A common mistake is to picture an electron radiating continuously while staying in a Bohr orbit. The Bohr postulate forbids radiation in a stationary state; a photon appears only during a transition, conserving angular momentum quantisation. Forgetting this leads to wrong energy-loss calculations.

Memory trick: No light on track, only at switch.

Mass defect and binding energy — common mistake

“Weighing a packed suitcase versus the empty box shows the box alone is lighter, not heavier.”

The binding energy must be positive, so the subtraction order is crucial: Δm = (mass of separated nucleons) – (mass of nucleus). Reversing gives a negative B, a frequent sign error in exams.

Memory trick: B = (mass of parts) – (mass of whole).

Accuracy, precision and uncertainty — common mistake

“A GPS that shows many points tightly grouped off the road is precise but not accurate.”

Precision reflects tight clustering of readings; accuracy requires the cluster centre to coincide with the true value. Systematic offset stays even after many trials, so a precise but shifted set is not accurate.

Memory trick: Cluster ≠ correct centre.

Acceleration — common mistake

“A ball thrown up stops at its highest point, but a sensor still records -9.8 m/s².”

At the highest point of a projectile, velocity is zero but the acceleration remains –g (≈9.8 m/s² downward) because gravity continues to act. Assuming zero acceleration whenever v=0 ignores the net force principle.

Memory trick: v=0, a=–g always for free fall.

Work–energy theorem — common mistake

“A block sliding down a rough slope feels gravity’s positive work and friction’s negative work; only their sum changes its kinetic energy.”

When applying the work-energy theorem, include every force’s work, including friction which does negative work. Ignoring non-conservative forces gives an incorrect kinetic-energy change. The sign of each work term must be kept consistent with displacement.

Memory trick: All forces count → ΣW = ΔK

Archimedes' principle — common mistake

“A steel ship floats because its hull encloses air, making the ship’s overall density lower than water.”

Archimedes' principle states the buoyant force equals the weight of displaced fluid. An object floats when its average density (total mass ÷ total volume, including cavities) is less than the fluid’s density; it sinks otherwise. The classic trap is using the material density of the solid part (e.g., steel) instead of the overall density, which fails for hollow or porous bodies.

Memory trick: Overall ρ = m_total / V_total → compare with ρ_fluid

Polarisation — common mistake

“When hearing a violin, you can't wear polarised earplugs to silence it, because sound doesn't polarise.”

Only transverse waves can be polarised; sound is longitudinal, its particle motion aligns with propagation, so a polariser has no effect. The misconception arises from confusing intensity reduction with polarisation. Remember: polarisation needs perpendicular oscillations.

Memory trick: Sound = longitudinal → no polariser effect.

RMS value — common mistake

“When you set a hair-dryer to 220 V, the heating element feels the RMS voltage, not the 311 V peak that would scorch it.”

RMS (root-mean-square) of a sinusoid equals V_rms = V_peak/√2, the DC value that gives the same heating effect. Thus 220 V mains is RMS; its peak is ≈311 V. Average real power is P = V_rms·I_rms·cosφ, so forgetting the cosφ factor for non-purely resistive loads is a common trap.

Memory trick: Peak = √2·RMS; Power = V·I·cosφ, not just V·I.

Depletion region — common mistake

“Think of the depletion zone as a cleared runway—no planes (carriers) can land there.”

The depletion zone is cleared of mobile electrons and holes; only fixed ionised atoms remain, producing the internal electric field. Mistake: assuming some carriers still move there. Remember the field exists because the region is carrier-free.

Memory trick: Runway cleared → carriers absent, ions stay.

Error in a power

“Doubling a circle’s radius doubles the area’s percentage uncertainty because area ∝ r².”

If a measured quantity x with relative error δx is raised to n, the result’s relative error is |n|·δx. Thus squaring doubles the percentage error, cubing triples it, and taking a square root halves it. A frequent slip is to keep the original error unchanged after exponentiation.

Memory trick: Power n → n×% error

Rocket equation (variable mass)

“A satellite fires its thruster and sheds 30 % of its mass; its speed jumps by u·ln(1/0.7), like a sled sliding faster as weight drops.”

For a rocket in free space, momentum conservation gives m dv = –u dm (u = exhaust speed relative to rocket, dm<0). Integrating from m₀ to m yields Δv = u ln(m₀/m). The sign of dm and the ratio m₀/m are crucial; using ln(m/m₀) gives a negative Δv. Trap: forgetting the minus sign or treating u as ground-relative exhaust speed leads to wrong Δv.

v = u ln(m0/m)

Memory trick: Δv = u·ln(initial mass / final mass)

Moment of inertia depends on the axis

“A figure skater pulls her arms in, reducing I about the vertical axis and spinning faster.”

I = Σmr² is defined relative to a specific axis; changing the axis changes each r. Use the parallel-axis theorem I = I_cm + Md² when shifting. Always state the axis before quoting I.

Memory trick: I = Σmr² → note axis!

Binding energy of a satellite

“To free a satellite you must supply energy equal to the magnitude of its negative orbital energy, GMm/2r.”

Total orbital energy E = -GMm/2r; the binding energy is |E| = GMm/2r, the energy required to escape. The trap: confusing kinetic energy (GMm/2r) with binding energy or forgetting the ½ factor.

BE = GMm/2r

Memory trick: BE = |E| = GMm/2r