Electromagnetic Induction

Six topics on what a changing magnetic flux does and the circuits built to exploit it. Magnetic flux as field through an oriented area, three levers and not one, electromagnetic induction, where the EMF reads the slope of the flux and Lenz's law aims it against the change, induced currents that always feel a force resisting the motion that made them, inductance as a geometric constant that opposes changing current and never current itself, LR circuits, the capacitor's mirror with tau = L/R, and LC circuits, where a coil and a capacitor trade energy a quarter period apart at omega = 1/sqrt(LC).

AP exam 10-20%6 topics

Topics

13.1

Magnetic Flux

Electromagnetic Induction

Induced Currents and Magnetic Forces

Inductance

Circuits with Resistors and Inductors (LR Circuits)

Circuits with Capacitors and Inductors (LC Circuits)

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Magnetic flux

Φ = ∫ B·dA = BA cosθ for uniform B, with θ measured to the surface NORMAL. Units: weber (T·m²)

Faraday's law

ε = −N dΦ/dt. The EMF reads the SLOPE of the flux, never its size: a huge steady flux induces nothing

Three levers

Φ = BA cosθ changes if B changes, if A changes, or if θ changes. Any one moving is enough; all three frozen gives zero

Lenz's law

the induced current opposes the CHANGE in flux, not the field. Flux shrinking: current adds to it. Flux growing: current fights it

Motional EMF

ε = BLv for a rod of length L moving at v perpendicular to B. Polarity from qv × B on the carriers: that end goes positive

EMF, then current

geometry sets ε = −N dΦ/dt; the circuit sets I = ε/R afterward. Resistance changes the current, never the EMF

Sliding bar on rails

I = BLv/R, drag F = BIL = B²L²v/R opposing the motion. Agent power Fv = I²R exactly

Induced electric field

∮ E·dℓ = −dΦ/dt. Induced E lines close on themselves and are NON-conservative: no potential, one lap gains energy

Rotating loop

Φ = BA cos(ωt), ε = NBAω sin(ωt). EMF peaks where flux crosses ZERO, a quarter turn from the flux peak

Magnet through a coil

one pass writes two OPPOSITE lobes: current one way approaching, zero when centered, reversed leaving

Inductance

L = NΦ/I, a GEOMETRIC constant. Doubling I doubles Φ and leaves L alone. Units: henry (V·s/A)

Solenoid

L = μ₀ n² A ℓ = μ₀ N² A / ℓ. Goes as N²: double the turns, four times the inductance

Inductor voltage

V = L dI/dt. Opposes CHANGE in current, in either direction; a steady current drops zero volts however large

Zero resistance, real voltage

the back-EMF lives in the changing flux, not in ohms. At LR switch-on it equals the full battery voltage while I is still zero

Mutual inductance

ε₂ = −M dI₁/dt. A STEADY primary current induces nothing next door; transformers need AC

Field energy

U = LI²/2, stored in the field. Energy density B²/(2μ₀). At shutdown it must go somewhere: resistor heat, a spark, or an LC ring

LR endpoints

t = 0: inductor holds its prior current (fresh circuit: OPEN branch). Long time: dI/dt = 0, inductor is a WIRE. The capacitor's mirror

LR time constant

τ = L/R, seconds. MORE resistance means a FASTER transient, the opposite of RC. Never LR

LR rise and decay

rise: I = (ε/R)(1 − e^(−t/τ)). decay: I = I₀ e^(−t/τ). Inductor current is CONTINUOUS across every switching instant

LC equation

L d²Q/dt² + Q/C = 0, the mass-spring equation: L plays mass, 1/C plays stiffness. Q = Q₀ cos(ωt)

LC frequency

ω = 1/√(LC), T = 2π√(LC). Bigger L or C rings SLOWER; quadrupling C doubles the period. Independent of Q₀

LC energy trade

Q²/2C + LI²/2 is constant. I = 0 when Q = ±Q₀; I max = ωQ₀ when Q = 0, a quarter period later. No R, no decay

Maxwell's equations

∮ E·dA = q/ε₀; ∮ B·dA = 0; ∮ E·dℓ = −dΦ(B)/dt; ∮ B·dℓ = μ₀I + μ₀ε₀ dΦ(E)/dt

Unit 13 tools

Challenge bank

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60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

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Worked solution

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Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 6 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 8 through 13, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 42-question section, your choice. Even coverage means this is a retention check rather than a score estimate.

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31topics

114codes covered

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