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Vedatom Physics
CUT-OUT CARDS · PHYSICS.VEDATOM.COM
150 cards · six Parts
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Cut along the solid lines, fold along the dashed one: the prompt is on one face and the answer on the other. Every card names its Part and chapter, because a cut card ends up loose in a pile. Nothing here was written for this sheet — each card is a bench question, a worked warm-up or a formula-sheet definition already on the site.

◈ Bench oral 62 | △ Warm-up 13 | ● Definition 75

IMechanics25 cards
I · Vectors & Maths◈ 001
A student writes the van der Waals equation (P + a/V2)(V − b) = RT and claims a and b must have the same dimensions since they sit inside the same bracket. Where is the trap?
I · Vectors & MathsA
The bracket rule is about ADDITION, not the whole equation. a/V2 must match P, so a = P·V2 = M L5 T−2; and b must match V, so b = L3. They are added to different things, so they need not — and do not — share dimensions.
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I · Vectors & Maths◈ 002
You are told v = a sin(bx − ct) describes a wave. Without any physics, what does the ratio c/b have to be?
I · Vectors & MathsA
A speed. Both bx and ct must be dimensionless, so b = L−1 and c = T−1; their ratio c/b = L T−1. Dimensional analysis alone hands you the wave speed — the argument-must-be-dimensionless rule doing real work.
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I · Vectors & Maths◈ 003
A dimensionally correct equation is guaranteed to be physically correct — true or false, and what is the sharper statement?
I · Vectors & MathsA
False. Dimensions are blind to pure numbers and to added terms of the same dimension: s = ut + at2 (missing ½) and even s = ut − at2 pass the check. The sharp statement is the contrapositive: a dimensionally WRONG equation is always physically wrong.
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I · Vectors & Maths◈ 004
The period of a small liquid drop oscillating under its own surface tension T depends on its radius r, density ρ and T. Why does dimensional analysis fix the powers uniquely here?
I · Vectors & MathsA
Three unknowns (the powers of r, ρ, T) and three base dimensions M, L, T give three equations — just enough. It yields period ∝ √(ρr3/T). What it can never supply is the leading pure number, which needs the full hydrodynamic derivation.
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I · Vectors & Maths◈ 005
In g = 4π2L/T2, the length L is measured to 0.5% and the time for one swing to 0.5%. A student reports the error in g as 1%. What did they miss?
I · Vectors & MathsA
The exponent on T. Since g ∝ L·T−2, Δg/g = ΔL/L + 2·ΔT/T = 0.5% + 2(0.5%) = 1.5%. Powers weight their term — the squared period contributes double, the single most common slip in this classic experiment.
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I · Vectors & Maths◈ 006
Why is it wrong to convert 1 N to dyne and 1 J to erg with the same factor of 105?
I · Vectors & MathsA
They have different dimensions. Force is M L T−2 → factor (103)(102) = 105, so 1 N = 105 dyne. Energy is M L2 T−2 → factor (103)(102)2 = 107, so 1 J = 107 erg. The extra power of length changes the factor by 102.
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I · Work · Energy◈ 007
A block is pushed up a rough incline, stops, and slides back to its starting point. Over the whole round trip, is the total work done by friction zero?
I · Work · EnergyA
No. Friction always opposes the motion, so it does negative work on both legs; the round-trip total is −2f·(distance), never zero. Only a conservative force gives zero work around a closed path — that is exactly what disqualifies friction.
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I · Work · Energy◈ 008
Does the work–energy theorem still hold in a non-inertial (accelerating) frame?
I · Work · EnergyA
Yes — provided you also count the work done by the pseudo-force. In an accelerating frame, (work of real forces + work of the pseudo-force) = ΔKE measured in that frame. Forget the pseudo-force term and the books will not balance.
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I · Work · Energy◈ 009
A force stays perpendicular to a particle’s velocity at every instant. What does that fix, and what stays free?
I · Work · EnergyA
The speed (and hence KE) is fixed, because the power F·v is zero, so no energy is delivered. The path is still free to curve — this is precisely uniform circular motion, or a charge in a magnetic field.
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I · Work · Energy◈ 010
Two balls of equal mass are launched with the same speed, one at 30° and one at 60°. Which has the greater kinetic energy at the top of its arc?
I · Work · EnergyA
The 30° ball. At the peak only the horizontal velocity u·cosθ survives, so KEtop = ½m u2cos2θ — larger for the smaller angle. (Total energy is equal; they just bank different fractions as PE.)
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I · Work · Energy◈ 011
As a spring is compressed and then allowed to spring back to its natural length, what is the sign of the work done by the spring force on the block?
I · Work · EnergyA
Negative while compressing (the spring force opposes the inward displacement) and positive while re-expanding. Over the full cycle back to natural length the net work is zero — the signature of a conservative force.
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I · Work · Energy◈ 012
Can a system have positive kinetic energy while its total momentum is exactly zero?
I · Work · EnergyA
Yes. Two equal masses moving in opposite directions have cancelling momenta but each carries positive KE — KE is a scalar sum of squares, momentum a vector sum. This is why an explosion at rest can fling out fast fragments with zero net momentum.
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I · Circular Motion◈ 013
A car circles a track at constant speed. Is it accelerating?
I · Circular MotionA
Yes. Speed is constant but the velocity vector keeps changing direction, and any change in velocity is acceleration. It points toward the centre (centripetal) with magnitude v2/r, even though the speed never changes.
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I · Vectors & Maths● 014
Dimensional formula
I · Vectors & MathsA
every mechanical quantity written as MᵃLᵇTᶜ. An equation must be dimensionally homogeneous: every added term carries the same dimensions.
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I · Vectors & Maths● 015
Errors
I · Vectors & MathsA
absolute error Δa; fractional errors add for products and quotients, and powers multiply them. Significant figures survive multiplication as the least count of the inputs.
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I · Vectors & Maths● 016
Vector
I · Vectors & MathsA
magnitude + direction + triangle law of addition.
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I · Vectors & Maths● 017
Dot product
I · Vectors & MathsA
gives a scalar (work); cross product a vector ⊥ to both (torque, L). Derivative = slope of a graph; integral = area under it.
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I · Kinematics● 018
Velocity
I · KinematicsA
rate of change of displacement (slope of x–t); acceleration — rate of change of velocity (slope of v–t; area under a–t gives Δv). The constant-a equations below hold only when a is constant.
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I · Kinematics● 019
Projectile
I · KinematicsA
independent horizontal (uniform) and vertical (free-fall) motions.
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I · Newton's Laws● 020
Four interactions
I · Newton's LawsA
gravitational, electromagnetic, strong, weak. All contact forces (normal, friction, tension) are electromagnetic in origin.
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I · Newton's Laws● 021
Normal
I · Newton's LawsA
pushes ⊥ from a surface; tension pulls along a rope; a spring restores toward natural length. Weight W = mg acts at the centre of gravity.
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I · Newton's Laws● 022
Pseudo-force
I · Newton's LawsA
in a frame accelerating at a0, add −ma0 to every body; only then does ΣF = ma hold inside that frame.
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I · Newton's Laws● 023
Static friction
I · Newton's LawsA
self-adjusting, anything from 0 up to μsN; it equals the applied force until slipping.
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I · Newton's Laws● 024
Angle of repose θr
I · Newton's LawsA
steepest slope a block rests on: tan θr = μs; independent of mass.
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I · Work · Energy● 025
Work–energy theorem
I · Work · EnergyA
the net work of all forces equals the change in kinetic energy.
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IIWaves25 cards
II · SHM◈ 026
A uniform disc and a thin ring of equal mass and radius are each pivoted at their rim and set swinging. Which has the longer period, and why?
II · SHMA
The ring. Both have d=R, but Irim(ring)=2mR2 > Irim(disc)=3/2·mR2. Since T=2π√(I/mgd), the larger moment of inertia gives the longer period.
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II · SHM◈ 027
As the pivot of a swinging uniform rod moves out from its centre toward the end, the period first falls to a minimum, then rises. Where is the minimum?
II · SHMA
At d = radius of gyration k = L/√12. T→∞ as d→0 (pivot at the CoM — no restoring torque) and rises again for large d, so a minimum sits at d=k.
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II · SHM◈ 028
A physical pendulum and a simple pendulum share the same period. How long is the simple pendulum, and what is special about that point on the body?
II · SHMA
L = I/(md), the length of the equivalent simple pendulum. Its bob sits at the centre of oscillation O — and pivot ↔ O are interchangeable (swap them and T is unchanged).
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II · SHM◈ 029
A positively charged bob hangs as a simple pendulum. A uniform electric field is switched on pointing vertically downward. Does the period increase or decrease?
II · SHMA
Decrease. Effective gravity geff = g + qE/m rises, so T=2π√(L/geff) falls. (An upward field, or a negative charge, would lengthen it.)
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II · SHM◈ 030
A pendulum clock keeps correct time in a cold room, then is moved to a hot room. Does it gain or lose time?
II · SHMA
Lose. The rod expands, L↑ → T↑, so each swing takes longer and the clock runs slow.
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II · Waves & Sound◈ 031
A cork floats on a pond as ripples pass. Does it drift across the pond with the waves, or stay put?
II · Waves & SoundA
It stays put — bobbing up and down (and slightly to-and-fro) about a fixed spot. The wave carries energy across the water; the water itself only oscillates. No net transport of matter.
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II · Waves & Sound◈ 032
A y-vs-x graph and a y-vs-t graph of the same sinusoidal wave look identical. What does the repeat distance mean on each?
II · Waves & SoundA
On y-vs-x (a snapshot of the whole string at one instant) the repeat length is the wavelength λ. On y-vs-t (the history of one particle) the repeat is the period T. Same shape, different axes and meanings.
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II · Waves & Sound◈ 033
Which way does y = A sin(ωt − kx) travel, and how can you tell at a glance?
II · Waves & SoundA
In +x. Write it as −A sin(kx − ωt): the x and t terms carry opposite signs, which always means motion toward +x. Same signs (kx + ωt) means −x.
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II · Waves & Sound◈ 034
Why can the particle velocity be larger than, smaller than, or equal to the wave velocity depending on the wave?
II · Waves & SoundA
They measure different things: wave speed v = ω/k is fixed by the medium, while max particle speed Aω depends on amplitude and frequency. Since vp = −v·(∂y/∂x), the particle matches the wave speed only where the slope is 1 — i.e. when A = λ/2π.
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II · Waves & Sound◈ 035
Two identical waves travel along a string, one toward +x and one toward −x. Do they carry the same power?
II · Waves & SoundA
Yes. Power ∝ μω2A2v depends on amplitude, frequency and speed magnitude — not on direction. Each carries energy at the same rate, just opposite ways.
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II · SHM● 036
SHM
II · SHMA
restoring acceleration proportional to displacement, a = −ω2x; the shadow of uniform circular motion. Period independent of amplitude. Velocity leads displacement by π/2.
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II · SHM● 037
Simple harmonic motion
II · SHMA
acceleration is proportional to displacement and always directed toward the mean position: a = −ω2x. The projection of uniform circular motion onto a diameter (the phasor / reference circle).
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II · SHM● 038
Phase constant φ
II · SHMA
where in the cycle the motion starts. Velocity leads displacement by π/2; acceleration by π.
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II · SHM● 039
Springs
II · SHMA
in parallel the effective constant adds (k = k1+k2, stiffer); in series the compliances add (1/k = 1/k1+1/k2, softer).
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II · SHM● 040
Resonance
II · SHMA
driving at the natural frequency ω0 gives the largest response.
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II · Waves & Sound● 041
Progressive wave
II · Waves & SoundA
a disturbance that carries energy and momentum, not matter.
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II · Waves & Sound● 042
Transverse
II · Waves & SoundA
particles move ⊥ to travel (waves on a string, light).
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II · Waves & Sound● 043
Particle velocity
II · Waves & SoundA
u = ∂y/∂t is the slope of the displacement–time graph; the wave velocity v is fixed by the medium. They are linked by u = −v·(∂y/∂x): a particle's speed is the wave speed times the local slope.
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II · Waves & Sound● 044
Power & intensity
II · Waves & SoundA
a string carries P = ½μω2A2v; intensity ∝ A2.
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II · Waves & Sound● 045
Reflection
II · Waves & SoundA
at a fixed end the pulse flips (π phase change); at a free end it does not.
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II · Waves & Sound● 046
Superposition
II · Waves & SoundA
where waves overlap, displacements add. Two coherent waves give interference: in phase (δ = 2nπ) build up, out of phase (δ = (2n−1)π) cancel.
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II · Waves & Sound● 047
Standing wave
II · Waves & SoundA
two equal waves travelling opposite ways: y = 2A sin(kx) cos(ωt). Fixed nodes (always still) and antinodes (largest swing); no net energy transport. Node-to-node = λ/2, node-to-antinode = λ/4.
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II · Waves & Sound● 048
Strings & pipes
II · Waves & SoundA
boundary conditions pick discrete frequencies (harmonics). An open pipe gives all harmonics; a closed pipe only odd ones.
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II · Waves & Sound● 049
Speed of sound
II · Waves & SoundA
Newton assumed isothermal (v = √P/ρ, too low); Laplace corrected it to adiabatic (v = √γP/ρ). It rises with temperature (v ∝ √T) and, in air, with humidity.
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II · Waves & Sound● 050
Displacement vs pressure wave
II · Waves & SoundA
the pressure wave leads the displacement wave by π/2: where displacement is zero (a node of s) the pressure swing is greatest. Amplitude p0 = Bk·s0 = ρvω·s0.
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IIIThermodynamics25 cards
III · Thermal Physics◈ 051
A steel scale reads correctly at 20°C. On a hot 40°C day, does it read a length too large or too small?
III · Thermal PhysicsA
Too small. The scale itself expands, so its marked centimetres are longer than 1 cm → the object spans fewer divisions → reading < true length.
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III · Thermal Physics◈ 052
A metal plate has a circular hole. On heating, does the hole get bigger or smaller?
III · Thermal PhysicsA
Bigger. A hole expands exactly as if it were filled with the same metal — every linear dimension scales by (1+αΔT).
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III · Thermal Physics◈ 053
Convert 300 K and −40°C to the other scales.
III · Thermal PhysicsA
300 K = 26.85°C ≈ 80.3°F. −40°C = −40°F (the scales cross here) = 233.15 K.
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III · Thermal Physics◈ 054
Two rods, same length, α in ratio 3:2. Same ΔT. Ratio of their expansions?
III · Thermal PhysicsA
ΔL = LαΔT, same L and ΔT → ratio of ΔL = ratio of α = 3 : 2.
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III · Thermal Physics◈ 055
Why is there a gap left between two rails on a track?
III · Thermal PhysicsA
To allow free thermal expansion. Without a gap the prevented expansion creates a compressive thermal stress YαΔT that can buckle the rails.
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III · The Laws◈ 056
100 cal of heat is given to a gas and it does 40 J of work expanding. By how much does its internal energy rise?
III · The LawsA
ΔQ = 100 cal = 418 J, ΔW = 40 J. ΔU = ΔQ − ΔW = 418 − 40 = 378 J. Watch the units — heat came in calories, work in joules.
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III · The Laws◈ 057
A gas is compressed isothermally. What are the signs of W, ΔU and Q?
III · The LawsA
Compression → W < 0 (work done ON the gas). Isothermal → ΔU = 0. First law: Q = ΔU + W = W < 0, so heat is expelled. The work done on it leaves entirely as heat.
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III · The Laws◈ 058
Why is the internal-energy change of an ideal gas the same for an isothermal and a free expansion between the same volumes?
III · The LawsA
For an ideal gas U depends on T alone. Both processes start and end at the same temperature (free expansion keeps T fixed too), so ΔU = 0 for both — even though Q and W differ completely.
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III · The Laws◈ 059
Around a complete cycle, 250 J of heat is absorbed net. What is the net work done, and the change in internal energy?
III · The LawsA
A cycle returns to the same state → ΔU = 0. So Qnet = Wnet = 250 J. The enclosed PV-area equals 250 J, traversed clockwise.
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III · The Laws◈ 060
A gas expands adiabatically. Does it warm or cool, and where does the work come from?
III · The LawsA
Q = 0, so ΔU = −W. Expansion means W > 0, hence ΔU < 0 — the gas cools. The work pushing the piston is paid entirely out of internal energy.
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III · Kinetic Theory◈ 061
Hydrogen and oxygen sit in the same room at the same temperature. Whose molecules are faster, and by how much?
III · Kinetic TheoryA
Same T means same average KE per molecule, so ½m⟨v2⟩ matches. Lighter H2 must move faster: vrms ∝ 1/√M → √(32/2) = 4× faster than O2.
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III · Kinetic Theory◈ 062
PV = nRT works beautifully at 1 atm but fails badly at 100 atm. Which two assumptions break?
III · Kinetic TheoryA
(1) Molecules are no longer negligible points — their own volume matters (the b correction). (2) They are close enough to attract each other, softening wall impacts (the a correction). That is van der Waals.
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III · Kinetic Theory◈ 063
Why is γ = 1.67 for argon but only 1.40 for nitrogen?
III · Kinetic TheoryA
γ = 1 + 2/f. Argon is monoatomic (f = 3); nitrogen is a dumbbell with two extra rotational degrees of freedom (f = 5). More ways to store energy → smaller γ.
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III · Thermal Physics● 064
Temperature
III · Thermal PhysicsA
the quantity that decides the direction of heat flow; measured on the Celsius, Fahrenheit or absolute (Kelvin) scale. Absolute zero (0 K) is the point of least molecular motion.
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III · Thermal Physics● 065
Thermal expansion
III · Thermal PhysicsA
most solids grow on heating: linear (α), area/superficial (β) and volume/cubical (γ) coefficients, related α : β : γ = 1 : 2 : 3.
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III · Thermal Physics● 066
Thermal stress
III · Thermal PhysicsA
a rod clamped so it cannot expand develops a compressive stress YαΔT, independent of length.
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III · Thermal Physics● 067
Anomalous expansion of water
III · Thermal PhysicsA
water is densest at 4 °C; between 0 and 4 °C it contracts on heating — why ponds freeze top-down and aquatic life survives.
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III · Thermal Physics● 068
Specific heat (c)
III · Thermal PhysicsA
heat to raise unit mass by one degree; molar specific heat per mole.
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III · Thermal Physics● 069
Water equivalent
III · Thermal PhysicsA
w = mc/cwater: the mass of water needing the same heat.
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III · Thermal Physics● 070
Latent heat (L)
III · Thermal PhysicsA
heat per unit mass absorbed or released at a phase change (fusion, vaporisation) at constant temperature — the plateaus on a heating curve.
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III · Thermal Physics● 071
Principle of calorimetry
III · Thermal PhysicsA
in an insulated mix, heat lost by the hot bodies = heat gained by the cold, until a common equilibrium temperature.
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III · Thermal Physics● 072
Phase (P–T) diagram
III · Thermal PhysicsA
maps solid/liquid/gas regions; the triple point is where all three coexist, the critical point ends the liquid–vapour line.
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III · Thermal Physics● 073
Conduction
III · Thermal PhysicsA
energy passed molecule to molecule through a solid (Fourier's law).
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III · Thermal Physics● 074
Thermal resistance
III · Thermal PhysicsA
R = x/KA adds like electrical resistance: series R = R1 + R2, parallel 1/R = 1/R1 + 1/R2.
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III · Thermal Physics● 075
Convection
III · Thermal PhysicsA
heat carried bodily by a moving fluid (natural, from density differences, or forced).
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IVOptics25 cards
IV · Ray Optics△ 076
You raise your right hand facing a plane mirror. The image raises:
IV · Ray OpticsA
the hand that appears to be its left — Lateral inversion is a front-back flip; the raised hand appears on the opposite side, i.e. what looks like the image’s left hand.
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IV · Ray Optics△ 077
Two plane mirrors are inclined at 60°. How many images of a point object placed off the bisector are formed?
IV · Ray OpticsA
5 — 360/60 = 6 (even) → n = 6 − 1 = 5 images, independent of position.
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IV · Ray Optics△ 078
A person 1.8 m tall wants to see their whole body. What is the minimum vertical height (in m) of a plane mirror needed?
IV · Ray OpticsA
0.9 — Independent of distance: the mirror need only be half your height, 1.8/2 = 0.9 m, hung at the right level.
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IV · Ray Optics△ 079
A plane mirror is rotated by 12° while the incident ray is held fixed. Through what angle (in degrees) does the reflected ray turn?
IV · Ray OpticsA
24 — The reflected ray turns through twice the mirror rotation: 2 × 12° = 24°.
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IV · Ray Optics△ 080
A plane mirror rotates at a constant 2.0 rad/s about a vertical axis lying in its plane. A fixed laser beam strikes it and the reflected spot falls on a straight wall 10 m from the mirror. At the instant the reflected beam is perpendicular to the wall, the speed of the spot on the wall is
IV · Ray OpticsA
40 m/s — The reflected beam turns at 2ω = 4.0 rad/s. The spot position is x = d·tanφ, so its speed is 2ωd·sec2φ; at normal incidence φ = 0 and v = 2ωd = 4.0 × 10 = 40 m/s. (Away from the normal it grows as sec2φ — a favourite follow-up.)
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IV · Ray Optics△ 081
A point object moves with velocity v = (3î + 4ĵ) m/s in front of a plane mirror lying in the y–z plane (x is the normal). The speed of its image relative to the object is
IV · Ray OpticsA
6 m/s — A mirror reverses only the normal component: the image moves at (−3î + 4ĵ) m/s. Relative velocity = (−3−3)î + (4−4)ĵ = −6î → 6 m/s. The tangential components always cancel in the relative motion.
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IV · Ray Optics△ 082
An object moves toward a plane mirror at 5 m/s along the normal while the mirror simultaneously moves toward the object at 2 m/s. The speed of the image relative to the object is
IV · Ray OpticsA
14 m/s — With positions on the normal, ximg = 2x_mirror − xobj, so vimg = 2vm − vo. Taking the object’s direction as +: vo = +5, vm = −2 → vimg = −4 − 5 = −9 m/s. Relative to the object: |−9 − 5| = 14 m/s.
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IV · Ray Optics△ 083
A man’s eye is 1 m in front of a vertical plane mirror. A wall 3 m tall stands 3 m behind the mirror’s plane on the man’s side (2 m behind him). The minimum vertical length of mirror in which he can see the entire wall is
IV · Ray OpticsA
0.75 m — The wall’s image is 3 m behind the mirror, i.e. 4 m from the eye. The mirror (1 m from the eye) is an aperture: by similar triangles L/3 = 1/4 → L = 0.75 m. Unlike seeing yourself (H/2), seeing objects behind you DOES depend on the distances.
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IV · Ray Optics△ 084
Two plane mirrors are inclined at angle θ. A ray travelling parallel to the second mirror strikes the first mirror and, after reflection, falls normally on the second mirror and retraces its whole path. Then θ equals
IV · Ray OpticsA
45° — Being parallel to the second mirror, the ray makes glancing angle θ with the first; the reflected ray leaves at θ too. The vertex, the two strike points form a triangle with angles θ (vertex), θ (first mirror) and 90° (normal incidence): θ + θ + 90° = 180° → θ = 45°.
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IV · Ray Optics△ 085
Two parallel plane mirrors A and B are 10 m apart, and a point object sits between them, 2 m from A. The distance from A of the SECOND image formed behind mirror A is
IV · Ray OpticsA
18 m — Behind A the images sit at 2 m (direct image of the object) and then 18 m — the image, in A, of the object’s first image in B (which lies 8 m behind B, i.e. 18 m from A). The series behind A runs 2, 18, 22, 38, … m.
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IV · Ray Optics△ 086
A ray is reflected once from each of two plane mirrors inclined at 60°. Measured as the total angle through which the ray is turned, the net deviation is
IV · Ray OpticsA
240° — Each mirror deviates the ray by 180° − 2i, and for two mirrors at angle θ the deviations sum to 360° − 2θ, independent of the incidence angle — here 360° − 120° = 240°. The θ-independence is the whole point.
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IV · Ray Optics△ 087
A boy is 1.5 m tall with his eyes 0.1 m below the top of his head. He stands before a vertical plane mirror hung on a wall. For him to just see his complete image, the bottom edge of the shortest possible mirror must be at a height of
IV · Ray OpticsA
0.70 m — Eye level = 1.4 m. To see his feet, the mirror’s bottom edge must bisect the eye-to-feet drop: 1.4/2 = 0.70 m. (Top edge at 1.4 + 0.1/2 = 1.45 m, mirror length 0.75 m — but the QUESTION asks for the bottom edge.)
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IV · Ray Optics△ 088
A plane mirror rotates with angular velocity ω about an axis lying in its plane, in front of a FIXED point object. The image of the object revolves about that axis with angular velocity
IV · Ray OpticsA
2ω — The image is the object reflected across the mirror plane. When the mirror turns by θ, the reflection plane turns by θ and the image position swings through 2θ — so the image circles the axis at 2ω, twice the mirror’s rate.
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IV · Ray Optics● 089
Laws of reflection
IV · Ray OpticsA
the incident ray, reflected ray and normal lie in one plane; angle of incidence = angle of reflection.
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IV · Ray Optics● 090
Centre of curvature (C) / radius (R)
IV · Ray OpticsA
centre and radius of the sphere the mirror is cut from.
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IV · Ray Optics● 091
Principal focus (F)
IV · Ray OpticsA
where paraxial rays parallel to the axis converge (concave) or appear to diverge from (convex).
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IV · Ray Optics● 092
Plane mirror
IV · Ray OpticsA
image is virtual, erect, same size, as far behind as the object is in front, laterally inverted.
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IV · Ray Optics● 093
Refractive index
IV · Ray OpticsA
n = c/v, a measure of optical density. Frequency is fixed at a boundary; speed and wavelength both fall by n. Relative index n21 = n2/n1 = v1/v2 = λ1/λ2.
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IV · Ray Optics● 094
Critical angle (θc)
IV · Ray OpticsA
the angle of incidence in the denser medium for which the refracted ray grazes the surface (r = 90°).
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IV · Ray Optics● 095
Total internal reflection
IV · Ray OpticsA
beyond θc, all light is reflected back — the basis of optical fibres and the sparkle of diamond.
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IV · Ray Optics● 096
Apparent depth
IV · Ray OpticsA
an object in a denser medium looks nearer the surface; lateral shift is the sideways offset of a ray leaving a parallel slab (it emerges parallel to its original path).
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IV · Ray Optics● 097
Angle of deviation (δ)
IV · Ray OpticsA
the turn between incident and emergent rays. As i increases, δ falls to a minimum δm then rises; at δm the passage is symmetric (i1 = i2, r1 = r2 = A/2) and the ray inside runs parallel to the base.
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IV · Ray Optics● 098
Dispersion
IV · Ray OpticsA
a prism spreads white light because n is larger for shorter wavelengths, so violet deviates most, red least.
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IV · Ray Optics● 099
Dispersive power (ω)
IV · Ray OpticsA
measures the spread relative to the mean deviation.
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IV · Ray Optics● 100
Spectra
IV · Ray OpticsA
continuous (hot solids), line (atoms), band (molecules), absorption (dark Fraunhofer lines); a spectrometer (collimator · prism table · telescope) reads A and δm to get n(λ).
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VElectromagnetism25 cards
V · Electrostatics◈ 101
Why can two electric field lines never cross?
V · ElectrostaticsA
Because the field has one definite direction at each point. A crossing would give two tangents — two directions for E at once — which is impossible.
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V · Electrostatics◈ 102
Two equal positive charges sit a distance d apart. Where is E zero, and where is V zero on the joining line?
V · ElectrostaticsA
E = 0 at the midpoint (the two fields cancel). V is never zero — two positive charges make V > 0 everywhere. Zero field and zero potential are different questions.
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V · Electrostatics◈ 103
How much work does the field do when a charge moves along an equipotential?
V · ElectrostaticsA
Zero. V is constant, so ΔPE = qΔV = 0 — no work, whatever the path taken on that surface.
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V · Electrostatics◈ 104
A test charge is released from rest in a non-uniform field. Does it follow the field line it starts on?
V · ElectrostaticsA
Not in general. It accelerates along E initially, but once moving its inertia carries it off the (curved) line — field lines show direction of force, not the trajectory.
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V · Electrostatics◈ 105
Inside a charged hollow metal shell, what are E and V?
V · ElectrostaticsA
E = 0 everywhere inside (no enclosed charge, and the conductor screens it), but V is not zero — it equals the constant surface value kQ/R. Flat potential, zero field.
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V · Current Electricity◈ 106
A copper wire is stretched so its length doubles (volume conserved). By what factor does its resistance change?
V · Current ElectricityA
By 4. Stretching keeps volume AL constant, so if L doubles, A halves. R = ρL/A scales as L/A = L/(V/L) = L2/V, so doubling L quadruples R. A common trap is answering "2" by forgetting the area shrinks too.
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V · Current Electricity◈ 107
Two bulbs rated 60 W and 100 W (both at 220 V) are connected in SERIES across 220 V. Which glows brighter?
V · Current ElectricityA
The 60 W bulb. Its rated resistance R = V2/P is larger (60 W → higher R). In series the current is common, so power I2R is greatest in the larger resistance — the "weaker" bulb dominates. In parallel the 100 W bulb would win.
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V · Current Electricity◈ 108
Why does a potentiometer measure emf more accurately than a voltmeter?
V · Current ElectricityA
At balance it draws zero current from the cell, so there is no Ir drop — it reads the true emf ε. A voltmeter always draws some current, so it reads the terminal voltage ε − Ir, which is slightly low. Drawing no current is the whole point.
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V · Current Electricity◈ 109
In a metre bridge the balance point is found at 20 cm. How should you change the known resistance R to make the reading more reliable?
V · Current ElectricityA
Decrease R until the balance sits near 50 cm. End-resistance and length-measurement errors are smallest near the middle of the wire; a balance at 20 cm (or 80 cm) is in the error-prone region. Since X/R = ℓ/(100−ℓ), lowering R shifts ℓ toward centre.
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V · Current Electricity◈ 110
A capacitor charges through a resistor. At what multiple of τ is it 99% charged, and how much charge has flowed at t = τ?
V · Current ElectricityA
99% at about t = 4.6τ (since e^(−4.6) ≈ 0.01). At t = τ it is 1 − e^(−1) = 63% charged. People often think "one time constant = fully charged" — it is only 63%; 5τ is the practical "done" mark.
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V · Current Electricity◈ 111
n identical cells are wrongly connected in series with m of them reversed. What is the net emf and internal resistance?
V · Current ElectricityA
Net emf = (n − 2m)ε — each reversed cell subtracts twice its emf. The internal resistance is unchanged at nr, because resistances add regardless of polarity. So reversing cells hurts the emf but never lowers the resistance.
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V · Moving Charges◈ 112
A magnetic field can bend a fast electron into a circle but can never change its speed. Why not?
V · Moving ChargesA
The magnetic force F = qv × B is always perpendicular to the velocity, so F·v = 0 and it does zero work. With no work done the kinetic energy — and therefore the speed — is fixed; only the direction of v turns.
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V · Moving Charges◈ 113
A proton and an alpha particle are shot into the same field with the same speed, perpendicular to B. Whose circle is bigger?
V · Moving ChargesA
The alpha particle's, by a factor of two. r = mv/(qB) ∝ m/q. The alpha has four times the mass but only twice the charge, so m/q is doubled and its radius is twice the proton's.
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V · Electrostatics● 114
Coulomb's law
V · ElectrostaticsA
the force between two point charges, along the line joining them, inverse-square in the separation.
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V · Electrostatics● 115
Dipole
V · ElectrostaticsA
charges ±q a distance d apart; moment p = qd points − → +. Axial field is twice the equatorial field at the same distance.
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V · Electrostatics● 116
Potential V
V · ElectrostaticsA
work per unit charge brought from infinity; a scalar. E points down the steepest fall of V and is everywhere ⊥ to equipotentials. Inside a conductor E = 0; the surface is one equipotential.
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V · Electrostatics● 117
Electric flux Φ
V · ElectrostaticsA
field lines threading a surface; counts E across area, weighted by the tilt cosθ.
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V · Electrostatics● 118
Gauss's law
V · ElectrostaticsA
the total flux through ANY closed surface equals the enclosed charge over ε0; charge outside contributes nothing. Choose surfaces that match the symmetry (sphere, cylinder, pillbox).
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V · Electrostatics● 119
Capacitance C = Q/V
V · ElectrostaticsA
charge stored per volt; set by geometry alone. A dielectric (constant K) multiplies C by K by polarising and weakening the internal field.
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V · Electrostatics● 120
Battery connected
V · ElectrostaticsA
V fixed, Q and U rise with K. Isolated: Q fixed, V and U fall. Energy lives in the field itself, density ½ε0E2.
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V · Current Electricity● 121
Drift
V · Current ElectricityA
electrons crawl at vd (~mm/s) though the signal moves near light speed.
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V · Current Electricity● 122
Ohm's law
V · Current ElectricityA
V ∝ I when ρ is constant; ρ rises with temperature in metals, falls in carbon/semiconductors.
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V · Current Electricity● 123
EMF ε
V · Current ElectricityA
work per charge by the source; terminal voltage V = ε − Ir droops under load. Max power transfer at R = r (efficiency then only 50%).
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V · Current Electricity● 124
Wheatstone bridge
V · Current ElectricityA
balances at P/Q = R/S — the galvanometer reads zero.
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V · Current Electricity● 125
Potentiometer
V · Current ElectricityA
compares EMFs by balancing lengths, drawing no current.
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VIModern Physics25 cards
VI · Dual Nature◈ 126
A red beam and a violet beam are tuned to deliver equal intensity (equal W/m2) at the cathode. Compare the stopping potentials they produce, and compare the saturation currents.
VI · Dual NatureA
Violet has the higher frequency → larger Kmax → larger stopping potential. But equal intensity means equal power, and each violet photon carries more energy, so violet delivers fewer photons per second and hence the smaller saturation current. Equal intensity is not equal photon rate.
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VI · Dual Nature◈ 127
A student doubles the intensity of the light and claims the maximum kinetic energy of the photoelectrons has also doubled. Identify the error, and state what actually doubles.
VI · Dual NatureA
Kmax = hν − φ is fixed by frequency alone, so it does not change when intensity changes. Doubling the intensity doubles the number of photons per second, hence doubles the saturation current — not Kmax.
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VI · Dual Nature◈ 128
An electron and a proton have equal kinetic energy. Which has the larger de Broglie wavelength — and does the answer flip if instead they share equal momentum?
VI · Dual NatureA
At equal K, λ = h/√(2mK): the lighter electron wins with the larger wavelength. At equal momentum, λ = h/p is identical for both, so the comparison collapses to equality. The stated condition decides everything.
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VI · Dual Nature◈ 129
Why must the electron be bound inside the metal for the photoelectric effect, rather than free in space?
VI · Dual NatureA
A free electron cannot absorb a whole photon — energy and momentum cannot both be conserved in a single event. Inside a metal the lattice (or a nucleus) absorbs the recoil momentum, allowing complete absorption and ejection.
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VI · Dual Nature◈ 130
A perfectly reflecting mirror and a perfectly absorbing black plate of equal area face the same normal beam. Which feels the larger force, and by what factor?
VI · Dual NatureA
The mirror. An absorber receives P/c of momentum per second; a reflector reverses each photon and receives 2P/c — twice the force. (For a sphere the curvature erases this factor, but for a flat plate it is exactly 2.)
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VI · Dual Nature◈ 131
For two different metals, the stopping potential V0 is plotted against 1/λ. What feature of the two lines is identical, and what differs?
VI · Dual NatureA
The slope is identical — it equals hc/e, a universal constant independent of the metal. The lines differ only in their intercepts, which encode each metal’s work function φ/e. Parallel lines, simply shifted.
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VI · Semiconductors◈ 132
You heat a copper wire and a silicon rod side by side. Which one’s resistance rises and which falls — and what physically drives each change?
VI · SemiconductorsA
Copper (metal) rises: its carrier count is fixed, so hotter lattice vibrations just scatter electrons more (mobility ↓). Silicon falls: heat creates far more carriers (ni ∝ T^{3/2}e^{−Eg/2kT}), and that flood swamps the mobility drop. Opposite behaviours, opposite dominant effect.
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VI · Semiconductors◈ 133
A silicon crystal is doped n-type with phosphorus. Is the crystal now negatively charged, and what happens to its hole density?
VI · SemiconductorsA
Not charged — each donor releases one electron but stays a fixed positive ion, so the crystal is electrically neutral. The holes are crushed: nh = ni2/ne drops far below the intrinsic value (mass-action law).
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VI · Semiconductors◈ 134
Forward vs reverse bias on a p–n junction: in which does the depletion layer widen, and in which does current flow freely? Quote the Si knee.
VI · SemiconductorsA
Reverse bias widens the depletion layer (barrier grows; only leakage flows). Forward bias narrows it, and once V clears the ≈0.7 V knee (Si; 0.3 V for Ge) the current climbs almost vertically.
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VI · Semiconductors◈ 135
A full-wave bridge runs off 50 Hz mains. State its ripple frequency and say why it beats a half-wave rectifier on two counts.
VI · SemiconductorsA
100 Hz (2f): both half-cycles are folded up, giving two humps per input cycle. That is (i) easier to smooth and (ii) far more efficient (~81% vs ~40%), since no half-cycle is thrown away.
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VI · Semiconductors◈ 136
A transistor has α = 0.98. Find β, and explain what a small change in base current does to the collector current.
VI · SemiconductorsA
β = α/(1−α) = 0.98/0.02 = 49. A base current 49× smaller than the collector current controls it; a small ΔIB yields a large ΔIC = βΔIB — exactly the amplification a common-emitter stage exploits.
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VI · Semiconductors◈ 137
Why is a NAND gate called “universal”, and how would you make a NOT gate from a single NAND?
VI · SemiconductorsA
Universal means every logic function (AND, OR, NOT, and hence any circuit) can be built from NAND gates alone. A NOT is just a NAND with its inputs tied together: A NAND A = NOT A.
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VI · Nuclei◈ 138
Compare a nucleus of A = 125 with one of A = 8: how do their radii and their densities differ?
VI · NucleiA
R ∝ A^(1/3), so the radius ratio is (125/8)^(1/3) = 5/2 = 2.5× larger. The densities are identical (~2.3×1017 kg/m3) — nuclear density is independent of A, because volume grows in step with mass.
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VI · Dual Nature● 139
Photon
VI · Dual NatureA
light arrives in quanta of energy E = hν = hc/λ and momentum p = h/λ.
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VI · Dual Nature● 140
Work function φ
VI · Dual NatureA
least energy to free an electron; emission needs ν ≥ ν0 regardless of intensity.
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VI · Dual Nature● 141
Einstein's equation
VI · Dual NatureA
gives the maximum kinetic energy; the stopping potential V0 measures it.
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VI · Dual Nature● 142
de Broglie
VI · Dual NatureA
every moving particle has a wavelength λ = h/p, so matter diffracts too.
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VI · Atoms● 143
Quantised orbits
VI · AtomsA
angular momentum comes in units of h/2π, so only certain radii and energies are allowed. Energy is negative (bound) and rises toward zero as n grows.
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VI · Atoms● 144
Spectral series
VI · AtomsA
a jump from n2 to n1 emits a photon set by the Rydberg formula; Lyman (n1=1, UV), Balmer (n1=2, visible), Paschen (n1=3, IR). Hydrogen-like ions scale with Z2.
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VI · Atoms● 145
Continuous spectrum
VI · AtomsA
Bremsstrahlung with a sharp cutoff λmin set only by the tube voltage (Duane–Hunt).
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VI · Atoms● 146
Characteristic lines
VI · AtomsA
Kα, Kβ — inner-shell transitions fixed by the target element.
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VI · Atoms● 147
Bragg reflection
VI · AtomsA
lets a crystal diffract X-rays; only λ ≤ 2d can satisfy it.
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VI · Semiconductors● 148
Bands
VI · SemiconductorsA
a small gap Eg separates the valence and conduction bands; heat or doping supplies carriers. n-type (donors, majority electrons) and p-type (acceptors, majority holes) obey the mass-action law, staying neutral. Conductivity rises with T.
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VI · Semiconductors● 149
p–n junction
VI · SemiconductorsA
a one-way diode: forward bias conducts past the knee Vb, reverse bias blocks.
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VI · Semiconductors● 150
Rectifiers
VI · SemiconductorsA
make DC (full-wave ripple = 2f); transistor gains α, β amplify.
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