Unit 13: Electromagnetic Induction — Faraday's Law and RL Circuits
Master Unit 13 AP Physics C E&M with practice tests on Faraday's law, motional EMF, RL circuit differential equations, and energy in magnetic fields.
Electromagnetic Induction in AP Physics C E&M
Unit 13 is the culminating unit of AP Physics C: Electricity & Magnetism, unifying electric and magnetic phenomena through Faraday's law. Induced EMF is driven by the time rate of change of magnetic flux — a time derivative — and the resulting RL circuit dynamics are governed by a first-order differential equation analogous to the RC circuit of Unit 11. Calculus is not merely useful here; it is the only language in which Faraday's law can be precisely expressed.
Faraday's Law of Electromagnetic Induction
Magnetic Flux
Magnetic flux is defined as the surface integral of the magnetic field through a surface: Φ_B = ∫B·dA. For a uniform field through a flat surface of area A at angle θ to the field: Φ_B = BA cosθ. AP FRQs test flux calculations for changing area, changing field magnitude, and changing orientation.
Faraday's Law
The induced EMF in any closed loop equals the negative rate of change of magnetic flux through the loop: EMF = −dΦ_B/dt. This time derivative is the core calculus content of Unit 13. AP-style FRQs test three scenarios: changing B (∂B/∂t ≠ 0), changing area (moving conductor), and changing angle (rotating loop).
Lenz's Law
The negative sign in Faraday's law encodes Lenz's law: the induced current flows in the direction that opposes the change in flux. In AP FRQs, students must state the direction of induced current using Lenz's law and verify it is consistent with the negative sign of the EMF.
Motional EMF
When a conductor of length L moves with velocity v perpendicular to a magnetic field B, the motional EMF is EMF = BLv. This can be derived from Faraday's law (the area of the circuit swept increases at rate Lv, so dΦ/dt = BLv) or from the magnetic force on charge carriers (F = qvB along the length). AP FRQs may involve a sliding bar on rails, a rotating rod, or a loop entering and exiting a field region.
Self-Inductance
A coil carrying current I generates a magnetic field and hence a magnetic flux through itself: Φ_B = LI, where L is the self-inductance measured in henries (H). When current changes, the self-induced EMF is EMF = −L(dI/dt). For a solenoid of n turns per unit length, cross-sectional area A, and length ℓ: L = μ₀n²Aℓ. AP FRQs ask students to derive L for solenoids using this relationship.
RL Circuits and Differential Equations
Current Growth in an RL Circuit
Applying KVL to a series RL circuit with EMF source ε: ε − IR − L(dI/dt) = 0. Rearranging: L(dI/dt) + RI = ε. This first-order linear ODE has the solution I(t) = (ε/R)(1 − e^(−t/τ_L)), where τ_L = L/R is the inductive time constant. The derivation by separation of variables is a standard AP FRQ requirement.
Current Decay in an RL Circuit
When the EMF source is removed: L(dI/dt) + RI = 0. Separation of variables gives I(t) = I₀e^(−t/τ_L). The inductor drives a decaying current that resists the change in magnetic flux through it.
Energy Stored in a Magnetic Field
The energy stored in an inductor carrying current I is U = ½LI². This is derived by integrating the power delivered to the inductor: U = ∫P dt = ∫L·I·(dI/dt) dt = ∫LI dI = ½LI². The magnetic energy density in a solenoid is u = B²/(2μ₀), analogous to the electric energy density u = ε₀E²/2.
Key FRQ Skills for Unit 13
- Faraday's law application: Compute dΦ_B/dt for changing B, changing area, or changing orientation; state magnitude and direction of EMF.
- Lenz's law direction: Determine induced current direction and justify it as opposing the flux change.
- RL circuit ODE: Write the KVL differential equation, solve by separation of variables, and express I(t) with correct τ_L = L/R.
- Energy in inductors: Derive U = ½LI² from the integral definition; compare to capacitor energy storage.
Common Errors in Unit 13
- Omitting the negative sign in EMF = −dΦ_B/dt and then incorrectly applying Lenz's law independently
- Confusing τ_L = L/R (RL) with τ = RC (RC) — noting that larger L slows the circuit while larger R speeds it up compared to RC behaviour
- Treating L(dI/dt) as a resistance rather than a reactive element in the KVL equation
- Computing flux as Φ = B·A without accounting for the angle between B and the area normal vector