Need for modulation

“A walkie-talkie’s 2 m antenna works at 150 MHz, but transmitting voice (≈1 kHz) directly would need a 150 km antenna.”

Modulation shifts the low-frequency audio onto a high-frequency carrier, letting the antenna be ≈λ/2 of the carrier (hand-sized). This yields a resonant antenna and usable gain, but Friis shows free-space loss ∝(4πd/λ)², so higher f (smaller λ) actually increases loss. The benefit is antenna practicality, not reduced propagation loss.

Memory trick: λ↓ ⇒ antenna ↓, but loss ↑ (∝f²).

Thermal stress

“A metal bridge that’s bolted at both ends buckles inward when the sun heats it, because it cannot expand.”

If a rod is rigidly clamped at both ends, uniform heating creates a thermal strain αΔT that cannot appear, so a uniform stress σ = Y α ΔT develops. The formula assumes uniform temperature; a gradient causes varying stress and bending.

Memory trick: σ = Y·α·ΔT (fixed ends)

Relative velocity in two dimensions — common mistake

“Crossing a river on a drifting boat, you can’t just add boat speed and river speed as numbers.”

When the observer moves, the object's apparent speed is the vector difference, not a simple sum of magnitudes. Resolve each velocity into x-y components, then subtract. The exam trap is adding the speeds as scalars, giving wrong direction and magnitude.

Memory trick: Add components, not magnitudes.

Phase change and latent heat — common mistake

“When water boils on a stove, its temperature stalls at 100 °C while steam forms.”

During boiling, the temperature remains at the boiling point while latent heat of vaporisation is absorbed. The amount of heat equals m·L_v, not m·c·ΔT. A common mistake is to multiply the total mass of water by cΔT, which ignores that only the phase-changing mass uses L_v.

Memory trick: Boiling plateau: Q = m·L_v, not mcΔT.

Degrees of freedom — common mistake

“Choosing a car’s gear count without looking at the model is like assuming every gas has the same degrees of freedom.”

The common mistake is to assign a fixed f (e.g., 5 for diatomics) at all temperatures. Degrees of freedom must be counted as active quadratic energy terms; vibrational modes are frozen at low T. Forgetting this leads to wrong Cₚ, Cᵥ and γ values.

Memory trick: Count quadratic terms, not just xyz directions

Dielectric polarisation

“A polarised dielectric is like a crowd holding umbrellas opposite to a wind, weakening the net breeze.”

In an electric field a dielectric’s molecules align, creating an opposite field that reduces the net field. For an isolated charged capacitor Q stays fixed, so V drops and C becomes K C₀. If the capacitor remains connected to a battery V is fixed and Q rises by K. Forgetting which quantity is fixed is a frequent error.

Memory trick: Isolated → Q fixed, V↓; Battery → V fixed, Q↑.

Dia-, para- and ferromagnetism — common mistake

“A compass needle barely wobbles over a copper plate, showing diamagnetism’s weak repulsion.”

Diamagnetic substances have χ < 0, giving a very weak repulsion to magnetic fields; paramagnets have 0 < χ ≪ 1, so they are weakly attracted; ferromagnets have χ ≫ 1 and form permanent domains, giving strong attraction and hysteresis. The exam trap is to label any attracted material as ferromagnetic – paramagnets also attract, though faintly, and ferromagnets lose their behavior above the Curie temperature.

Memory trick: D-negative, P-tiny-positive, F-huge-positive

Spectral series — common mistake

“Reading a book by its first page is wrong; the series is named by the last page where the electron lands.”

The spectral series is identified by the fixed lower level (the final state) of the transition, not by the starting level. Lyman, Balmer, Paschen correspond to n=1,2,3 respectively. A common mistake is to use the initial quantum number, which leads to wrong series identification.

Memory trick: Name = final n, not start.

Random radioactive decay — common mistake

“A sand timer empties half its sand each flip only on average, not exactly each time.”

Half-life (t½) is the time in which, on average, half of a huge ensemble of identical nuclei decay. It does not guarantee that exactly half of a small sample disappear each interval. Assuming exact halving leads to wrong activity calculations.

Memory trick: Half-life = average half, not exact half each period

Significant figures in results — common mistake

“Rounding a pizza slice after each cut wastes the crust, just like trimming digits at every calculation step wastes precision.”

A common mistake is to round off after every intermediate operation, discarding guard digits and accumulating extra error. Keep extra digits through the calculation and round only the final answer. The exam trap is losing precision early, giving a wrong SF count.

Memory trick: Guard digits stay; round only at the end.

Projectile motion — common mistake

“A basketball’s arc is mis-calculated if you add launch speed and falling speed as plain numbers.”

Treat v as a vector: horizontal component stays u cosθ (a=0), vertical component follows v_y = u sinθ − g t. Using v = u+at on the total speed mixes components and gives wrong results.

Memory trick: Split, don’t blend – x const, y uses a.

Power — common mistake

“A toaster heats toast in 30 s using the same energy as a kettle but twice the power because it finishes faster.”

Energy (J) is the total work done; power (W) tells how fast that energy is delivered: P = E/Δt. The exam trap is to equate higher power with more work—actually a high-power device can do less work if it runs briefly.

Memory trick: Energy = Power × Time

P + ½ρv² + ρgh = constant

“Air speeding over a plane’s wing top reduces pressure, lifting the aircraft—Bernoulli’s principle.”

Along a streamline of steady, non-viscous, incompressible flow P + ½ρv² + ρgh remains constant. The trap: it cannot be used across pumps, turbines or in compressible high-speed streams where density varies.

P + ½ρv² + ρgh = constant

Memory trick: P + ½ρv² + ρgh = constant

f = 1/(2π√LC)

“When you tune a radio, the dial aligns L and C so the station’s frequency f = 1/(2π√LC) lights up the speaker.”

At resonance of a series LCR, X_L = X_C, so ω₀ = 1/√LC and f₀ = 1/(2π√LC). The total impedance reduces to R, giving maximum current and possibly voltages across L or C larger than the source. Beware: this formula applies only to series resonance; parallel resonance follows a different condition.

f = 1/(2π√LC)

Memory trick: Series resonance: ω=1/√LC → f=ω/2π, think ‘LC under 2π gives f’.

Accuracy versus precision

“A darts player hitting the same off-center spot repeatedly is precise but not accurate.”

Accuracy = closeness to true value; precision = repeatability. Systematic (bias) error hurts accuracy even with high precision. Exam trap: many decimal places do not guarantee accuracy if a bias exists.

Memory trick: Accurate = true; Precise = repeat

Work done by a variable force

“Pulling a rope to lift a boat, the tension rises as the boat leaves water – the work equals the shaded area under the tension-distance curve.”

For a force that varies with displacement, W = ∫ F·dx, i.e., the signed area under the F-x graph. The sign follows the direction of displacement relative to the force. A trap is using average force blindly; if F(x) is non-linear the area is not (Fmax+Fmin)/2·Δx.

Memory trick: Integrate → area; remember ‘W = ∫Fdx, not just avg F’

Perpendicular axis theorem

“A flat pizza’s spin about a vertical stick equals sum of spins about two perpendicular cuts through its centre.”

For a planar lamina, I_z (perpendicular to the plane) = I_x + I_y (in-plane orthogonal axes through the centre of mass). It stems from adding contributions of orthogonal coordinates. Trap: only valid for thin laminae lying in a single plane.

Memory trick: Pizza rule: I_z = I_x + I_y

Escape versus orbital velocity

“A rocket needs just enough speed to swing around Earth, but to break free it must add a √2 boost, like a skier needing extra push to jump off a hill.”

Escape velocity follows from ½mv² = GMm/r → v_esc = √(2GM/r) = √2·v_circ. A frequent mistake is to forget the √2 factor and equate escape speed with orbital speed.

Memory trick: v_esc = √2·v_circ

Springs in series and parallel

“Two garden hoses joined end-to-end (series) feel softer than each alone, while side-by-side (parallel) feel stiffer.”

For springs, parallel arrangement gives k_eq = k₁ + k₂ + … (stiffer), whereas series gives 1/k_eq = 1/k₁ + 1/k₂ + … (softer). The trap is swapping these with resistor rules; springs behave opposite to resistors.

Memory trick: Springs ↔ resistors opposite: parallel k add, series 1/k add.

Superposition — common mistake

“When two laser beams intersect, you only see bright fringes where their electric fields are in phase, not everywhere.”

A common mistake is I_total = I1+I2. Correctly, the resultant field E = E1+E2, so I ∝ |E|² = I1+I2+2√(I1I2)cosΔφ. Ignoring the cross term loses interference effects. Remember to include phase before squaring.

Memory trick: I = I1+I2+2√(I1I2)cosΔφ

Van der Waals real-gas equation

“A balloon inflates slower at high pressure because gas molecules push each other and take up space.”

Van der Waals equation (P + a n²/V²)(V - nb) = nRT adds a for intermolecular attraction and b for finite molecular volume. At low pressure it reduces to PV=nRT. Trap: inserting +bV instead of subtracting nb, or using aV² term incorrectly.

Memory trick: P + a(n/V)², V - nb

Field of standard distributions

“Standing near a massive charged wall, the electric pull feels the same everywhere, like a steady breeze on a flat plain.”

For ideal infinite distributions: sheet → E = σ/(2ε0) (uniform, both sides); line → E = λ/(2π ε0 r) (∝1/r); point → E = kQ/r² (∝1/r²). Toppers remember the factor 2 for a sheet and that only the perpendicular component matters.

Memory trick: Sheet constant, line 1/r, point 1/r²

Drift velocity — common mistake

“Students often picture electrons racing down the wire like a sprint, but the actual drift is a snail’s pace while the electric wave dashes ahead.”

Electrons move slowly under the applied field, but the electromagnetic disturbance travels at ~c/√ε_r, giving near-light-speed signal transmission. Forgetting this leads to using the tiny drift speed for timing problems, which yields orders-of-magnitude error.

Memory trick: Signal speed ≠ drift speed.

Magnetic force on a moving charge — common mistake

“Imagine a train on a straight track under a uniform magnetic roof; the magnetic field can’t push it sideways because its motion is parallel.”

When v is parallel to B, sinθ=0 and the magnetic force vanishes. Many students plug qvB regardless of direction, giving a non-zero answer. Remember to include the sinθ term; if v∥B, F=0.

Memory trick: v∥B → sinθ=0 → F=0