Electromagnetic Induction and Maxwell's Equations: every key term you need (+ practice quiz)
36 flashcard terms for AP Physics C: E&M Unit 5, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 21-question quiz — free, no account needed.
Φ_B = ∫B·dA, the amount of magnetic field threading a surface; measured in webers (1 Wb = 1 T·m^2). Only the component of B normal to the surface counts.
Faraday's Law
The emf induced around a closed loop equals the negative rate of change of magnetic flux through it: ε = −dΦ_B/dt. For N turns, ε = −N dΦ_B/dt.
Lenz's Law
The induced current flows in the direction whose own magnetic field opposes the change in flux that produced it; this is the physical meaning of the minus sign in Faraday's law and a consequence of energy conservation.
Ways to Change Flux
Flux Φ = BA cosθ can change by varying the field magnitude B, the loop area A, or the angle θ between B and the area normal; any of these induces an emf.
Motional emf
A conductor of length L moving at speed v perpendicular to B develops emf ε = BLv, because the magnetic force qv × B pushes charge carriers along the rod until an electric field balances it.
Sliding Rod on Rails
A rod sliding on conducting rails in a field B closes a circuit; ε = BLv drives I = BLv/R, and the rod feels a retarding force F = BIL = B^2L^2v/R opposing its motion.
Terminal Velocity of a Falling Rod
When a rod falls on vertical rails in a horizontal B, it reaches terminal speed when mg = B^2L^2v/R, giving v_t = mgR/(B^2L^2).
Eddy Currents
Circulating currents induced inside a bulk conductor by a changing flux; they dissipate energy as heat and produce forces that oppose the motion (magnetic braking).
Induced Electric Field
A changing magnetic flux creates a non-conservative electric field: ∮E·dl = −dΦ_B/dt. This E has closed field lines and exists even without a physical wire.
Non-conservative Induced E
Unlike electrostatic fields, an induced E field has nonzero circulation, so no scalar potential describes it and the work per unit charge around a loop is the emf, not zero.
Generator
A coil of N turns and area A rotating at angular speed ω in a field B produces a sinusoidal emf ε = NBAω sin(ωt), the basis of AC power generation.
Self-Inductance
L = NΦ_B/I (or Φ_total/I): the flux linkage a circuit produces through itself per unit current. Unit: henry (1 H = 1 V·s/A).
Inductor emf
An inductor opposes changes in its own current with a back emf ε = −L dI/dt; a steady current produces no inductor voltage.
Solenoid Inductance
For a long solenoid with n turns per length, length ℓ, and area A: L = μ₀n^2Aℓ = μ₀N^2A/ℓ. Inductance depends only on geometry (and core material).
Energy Stored in an Inductor
U_L = (1/2)LI^2, obtained by integrating power P = LI(dI/dt) as the current builds from zero.
Magnetic Energy Density
u_B = B^2/(2μ₀), the energy per unit volume stored in a magnetic field; the magnetic counterpart of u_E = (1/2)ε₀E^2.
LR Circuit (Current Rise)
When a battery ε is connected to R and L in series, I(t) = (ε/R)(1 − e^(−t/τ)) with time constant τ = L/R; the current starts at zero and approaches ε/R.
LR Circuit (Current Decay)
When the source is removed and the LR loop is closed, I(t) = I₀e^(−t/τ) with τ = L/R; the inductor keeps current flowing momentarily.
Immediately after a switch closes, an inductor acts like an open circuit (current cannot jump), so all the source emf appears across it.
Inductor at Steady State
After a long time in a DC circuit, an inductor acts like an ideal wire (zero voltage drop), and current is set by the resistors alone.
LC Circuit
A charged capacitor connected to an inductor oscillates: Q(t) = Q₀cos(ωt) with ω = 1/√(LC). Energy sloshes between electric (capacitor) and magnetic (inductor) forms.
LC Oscillation Frequency
ω = 1/√(LC), f = 1/(2π√(LC)); derived from Kirchhoff's loop rule L d^2Q/dt^2 + Q/C = 0, the same form as a mass on a spring.
LC Energy Conservation
Q₀^2/(2C) = (1/2)LI_max^2: the maximum electric energy equals the maximum magnetic energy, so I_max = Q₀/√(LC) = ωQ₀.
Mutual Inductance
M = N₂Φ₂₁/I₁: the flux linkage in coil 2 per unit current in coil 1; the emf induced in coil 2 is ε₂ = −M dI₁/dt, the principle of transformers.
Transformer
Two coils sharing a changing flux (iron core): V₂/V₁ = N₂/N₁ while, for an ideal transformer, I₁V₁ = I₂V₂. Works only with AC because a steady flux induces nothing.
Displacement Current
I_d = ε₀ dΦ_E/dt, Maxwell's addition to Ampère's law; a changing electric flux (as between charging capacitor plates) produces a magnetic field just as a real current does.
Ampère–Maxwell Law
∮B·dl = μ₀(I_enc + ε₀ dΦ_E/dt): the circulation of B is set by both conduction current and displacement current through the surface bounded by the loop.
Gauss's Law for Electricity
∮E·dA = Q_enc/ε₀ (Maxwell equation 1): electric field lines begin and end on charges; net flux through a closed surface measures enclosed charge.
Gauss's Law for Magnetism
∮B·dA = 0 (Maxwell equation 2): there are no magnetic monopoles, so B field lines are always closed loops and net magnetic flux through any closed surface is zero.
Faraday's Law (Integral Form)
∮E·dl = −dΦ_B/dt (Maxwell equation 3): a changing magnetic flux produces a circulating electric field.
Maxwell's Equations
The four laws (Gauss for E, Gauss for B, Faraday, Ampère–Maxwell) that together with the Lorentz force F = q(E + v × B) fully describe classical electromagnetism.
Electromagnetic Waves
Self-sustaining oscillations of E and B predicted by Maxwell's equations: a changing E creates B, a changing B creates E, propagating at c = 1/√(μ₀ε₀) ≈ 3×10^8 m/s.
Weber
SI unit of magnetic flux, 1 Wb = 1 T·m^2 = 1 V·s; an emf of 1 V is induced when flux through a loop changes at 1 Wb/s.
Henry
SI unit of inductance: 1 H = 1 V·s/A = 1 Wb/A. A 1 H inductor develops 1 V when its current changes at 1 A/s.
Back emf in Motors
A spinning motor coil is also a generator; its induced emf opposes the applied voltage, so a motor draws the most current at startup and less as it speeds up.
Flux Through a Coil vs. Loop
For a coil of N identical turns, the total flux linkage is NΦ_B, so both induced emf and inductance scale with N (and L with N^2 since Φ_B itself is proportional to N).