Physics C: E&M math practice

Every skill in this set

Pressure at depthFluids

P = P₀ + ρgh

Pressure depends on depth, not on the shape or total volume of the container. A narrow tube and a wide tank at the same depth read the same pressure.

Ohm's law and powerElectric Circuits

V = IR · P = IV = I²R = V²/R

Three equivalent power expressions exist so you can use whichever two quantities you were given. Choosing the one that needs a quantity you must first calculate wastes time and invites error.

Coulomb's lawElectric Force, Field & Potential

F = kq₁q₂ / r² · k = 8.99 × 10⁹ N·m²/C²

An inverse-square law: tripling the separation cuts the force to a ninth. Questions frequently change only the distance and ask for the factor.

Snell's lawGeometric Optics

n₁ sin θ₁ = n₂ sin θ₂

Light entering a denser medium bends toward the normal. Angles are always measured from the normal, not from the surface — measuring from the surface is the standard setup error.

Photon energyModern Physics

E = hf = hc/λ · h = 6.63 × 10⁻³⁴ J·s

Shorter wavelength means higher energy. Photoelectric-effect questions turn on whether a single photon clears the work function, which is why intensity does not help below threshold.

Heat engine efficiency and the Carnot limitThermodynamics

e = W/Q_H = 1 − Q_C/Q_H · e_max = 1 − T_C/T_H (KELVIN)

Kelvin is not optional in the Carnot formula. Using Celsius produces a flattering nonsense answer, sometimes above 1. Real engines fall short of the ceiling because reaching it requires reversible, and therefore infinitely slow, operation.

Work from a constant-pressure processThermodynamics

W_by gas = PΔV; in general the area under the PV path

Expansion gives positive work BY the gas, compression negative. A constant-volume process does no work however much pressure and temperature change, because there is no area under a vertical line.

Electric field of a point chargeElectric Force, Field, and Potential

E = kq/r² with k = 8.99 × 10⁹ N·m²/C²

Use the magnitude of q and let the geometry set the direction — away from positive, toward negative. Putting the sign into the magnitude produces component errors that are hard to find later.

Speed of a charge accelerated through a voltageElectric Force, Field, and Potential

qΔV = ½mv² → v = √(2qΔV/m)

Use energy when the problem gives a potential difference and asks for a speed — no geometry is needed. Use kinematics only when it gives plate dimensions and asks for a deflection.

Equivalent resistance of a parallel pairElectric Circuits

1/R = 1/R₁ + 1/R₂

Remember the final reciprocal. The equivalent is always SMALLER than the smaller member — if yours came out larger, you added resistances instead of reciprocals.

Terminal voltage with internal resistanceElectric Circuits

I = ε/(R + r) · V_terminal = ε − Ir

Internal resistance is in series with everything, so add it to the total BEFORE computing current. Computing I from R alone and subtracting afterward gives a current that is too large.

Motional emf and the retarding forceMagnetism and Electromagnetism

ε = BLv · I = ε/R · F = BIL = B²L²v/R

Answer in the order emf → current → force → power. Each link feeds the next, they are separate rubric points, and the check Fv = ε²/R verifies the whole chain in one line.

Radius of a charged particle in a magnetic fieldMagnetism and Electromagnetism

qvB = mv²/r → r = mv/(qB)

The magnetic force is always perpendicular to the velocity, so it changes direction but never speed. That is why the path is a circle at constant speed and why this relation drives the mass spectrometer.

Thin lens image distance and magnificationGeometric Optics

1/f = 1/d_o + 1/d_i · m = −d_i/d_o

Remember the final reciprocal, and read the sign: a negative image distance always means a virtual image that cannot be projected. Negative magnification means inverted.

Critical angle for total internal reflectionGeometric Optics

sin θ_c = n₂/n₁, valid only when n₁ > n₂

Total internal reflection needs BOTH a higher-to-lower index transition and an angle above the critical angle. If your sine comes out above 1, the media are the wrong way round — and that impossibility is itself the correct physical answer.

Double-slit fringe positionWaves, Sound, and Physical Optics

y = mλL/d for small angles

Convert nanometers, millimeters and meters to one unit before substituting. Note that CLOSER slits spread the pattern out, since d sits in the denominator.

Standing wave harmonics in a pipeWaves, Sound, and Physical Optics

open both ends: f_n = nv/2L · closed one end: f_n = nv/4L, n ODD only

A pipe closed at one end has a quarter-wavelength fundamental and supports only odd harmonics, so it sounds an octave below an open pipe of the same length.

Photoelectric maximum kinetic energyModern Physics

E(eV) = 1240 / λ(nm) · KE_max = hf − φ

The 1240 eV·nm shortcut removes two unit conversions. Raising the intensity increases the NUMBER of photoelectrons, never their maximum energy — only a shorter wavelength does that.

Half-life and activityModern Physics

N = N₀(½)^(t/T); activity follows the same law

Decay removes a constant FRACTION, not a constant amount. Counting halvings is faster and self-checking whenever the elapsed time is a whole multiple of the half-life, which on the exam it usually is.

Common questions