Electromagnetic Induction

What this unit covers

The topics below follow the published Physics C: E&M course framework for Unit 6. This unit is worth 14–20% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Faraday’s Law & Lenz’s Law14 min · 3 objectivesCompute magnetic flux and identify the three ways it can change · Apply Faraday’s law ε = −N dΦ/dt to time-varying fields and moving conductors · Use Lenz’s law to determine the direction of induced currents
• Self-Inductance & Energy in Inductors13 min · 3 objectivesDefine self-inductance through ε = −L di/dt and L = NΦ/i · Derive the inductance of an ideal solenoid from its field and flux linkage · Compute stored energy with U = ½Li² and the field energy density B²/(2μ₀)
• LR Circuits: Solving the Current ODE14 min · 3 objectivesSet up the LR circuit differential equation from Kirchhoff’s loop rule · Solve the ODE to obtain i(t) = (ε/R)(1 − e^(−Rt/L)) and the decay solution · Analyze inductor behavior at t = 0 and t → ∞ and contrast it with capacitors
• Maxwell’s Equations: The Synthesis13 min · 3 objectivesState the four Maxwell equations and the physical claim each one makes · Explain why the displacement current completes Ampère’s law · Describe how the equations predict electromagnetic waves traveling at c = 1/√(μ₀ε₀)

Formulas in Unit 6

Every term in Unit 6

All 21 terms we publish for Electromagnetic Induction, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Energy stored in an inductor

U = ½LI², stored in the magnetic field. Energy density u = B²/2μ₀.

Motional EMF

EMF = BLv for a rod of length L moving perpendicular to B. Equivalent to Faraday's law applied to the growing circuit area.

Magnetic flux

Φ = ∫B·dA. Changing the field strength, the area, or the orientation all change flux.

Faraday's law

EMF = −dΦ/dt. Only a changing flux induces EMF; a constant field through a stationary loop induces none.

Lenz's law

The induced current opposes the change that produced it. Required by energy conservation — otherwise induction would create energy.

Applying Lenz's law

Determine whether flux is increasing or decreasing, then choose the current direction whose own field opposes that change.

Induced electric field

A changing magnetic field creates a non-conservative electric field with ∮E·dl = −dΦ/dt, which is why induced fields form closed loops.

Inductance

L = NΦ/I, depending only on geometry. A solenoid has L = μ₀n²Al.

Self-inductance and back EMF

EMF = −L(dI/dt). An inductor opposes changes in current, which is why it behaves as a break at t = 0 and a wire at t = ∞.

RL circuit

Current rises as I = (ε/R)(1 − e^(−Rt/L)) with time constant τ = L/R — the inductor's analogue of RC.

LC oscillation

Energy shifts between capacitor and inductor at angular frequency ω = 1/√(LC), the electrical analogue of a mass on a spring.

Maxwell's equations qualitatively

Gauss for electricity and magnetism, Faraday, and Ampère-Maxwell. Together they predict self-propagating electromagnetic waves traveling at c.

Three ways to change flux

Change B, change the enclosed area, or rotate the loop. Each gives a different term when differentiating Φ = BA cos θ.

Rotating loop generator

Φ = BA cos ωt gives EMF = BAω sin ωt — a sinusoidal output, which is the origin of alternating current.

Determining induced current direction

Find whether flux is increasing or decreasing, then pick the current whose own field opposes that change. Do not guess from the geometry.

Force opposing motion

A loop pulled out of a field experiences a retarding force, so work must be done. That work becomes the electrical energy dissipated.

Why Lenz's law must hold

A reinforcing induced current would accelerate the motion producing it and generate energy from nothing.

Inductor behavior at t = 0 and t = ∞

Initially an inductor opposes any current and behaves as a break; at steady state it behaves as a plain wire.

Energy in an RL circuit

The source supplies energy that partly dissipates in R and partly accumulates as ½LI² in the inductor's field.

Mutual inductance

Changing current in one coil induces EMF in a nearby one. The basis of the transformer, which requires AC to function.

Displacement current

Maxwell's addition to Ampère's law: a changing electric field acts as a current source of magnetic field, which is what closes the loop between the two fields and permits electromagnetic waves.

What examiners penalize here

• Faraday FRQs almost always follow one script: (1) write Φ_B symbolically as B·A·cosθ or an integral, (2) identify which factor depends on t, (3) differentiate to get ε = −N dΦ/dt, (4) if asked for current, divide by resistance, and (5) give the direction from Lenz’s law with an explicit “opposes the increase/decrease of flux” sentence. Write the derivative before plugging in numbers.
• Know each Maxwell equation by *claim*, not just symbol: Gauss (E) — charges make diverging E fields; Gauss (B) — no monopoles, B lines close; Faraday — changing Φ_B makes circulating E; Ampère–Maxwell — currents and changing Φ_E make circulating B. Exam questions ask “which equation forbids magnetic monopoles?” or “which term did Maxwell add, and why?” far more often than they ask you to compute with them.

Practice Physics C: E&M

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 6 units of AP Physics C: E & M

1. Unit 1 · Electric Charges, Fields, and Gauss’s Law
2. Unit 2 · Electric Potential
3. Unit 3 · Conductors, Capacitors, and Dielectrics
4. Unit 4 · Electric Circuits
5. Unit 5 · Magnetic Fields
6. Unit 6 · Electromagnetic Induction