Faraday's Law and Electromagnetic Induction

Electromagnetic induction closes AP Physics C: Electricity and Magnetism. It ties together fields, circuits and calculus, which is why it appears on the free response section so often.

Magnetic flux

Φ_B = ∫ B · dA

For a uniform field through a flat loop, Φ_B = BA cos θ, where θ is the angle between the field and the normal to the loop.

Faraday's law

ε = −N dΦ_B/dt

An emf appears whenever the flux changes, and the flux can change in three ways: the field changes, the area changes, or the angle changes. With calculus the rule applies even when the change is not steady: differentiate the flux, then evaluate.

Lenz's law: the direction

The induced current flows so that its own magnetic field opposes the change in flux. It does not oppose the field itself.

• A north pole approaching a loop increases the flux, so the loop makes a field that pushes back: its near face becomes a north pole.
• A north pole moving away decreases the flux, so the induced field points the other way and pulls it back.

Use the right-hand rule to turn the needed field direction into a current direction.

The moving rod

A rod of length L slides at speed v on rails in a field B perpendicular to the circuit, which has resistance R.

1. Emf: ε = BLv.
2. Current: I = BLv / R.
3. Magnetic force on the rod: F = BIL = B²L²v / R, opposing the motion.

Worked numbers: L = 0.5 m, B = 0.4 T, v = 3 m/s, R = 2 Ω. Then ε = 0.6 V, I = 0.3 A, and F = 0.4 × 0.3 × 0.5 = 0.06 N. The power to keep the rod moving, Fv = 0.18 W, equals the power dissipated, I²R = 0.09 × 2 = 0.18 W: energy is conserved.

Inductors

ε_L = −L dI/dt, stored energy U = ½LI²

In an LR circuit with a battery, the current rises as

I(t) = (ε/R)(1 − e^(−t/τ)), with τ = L/R

Right after the switch closes the inductor blocks any sudden change, so the current starts at zero. After a long time it acts like a wire, and the current is ε/R.

The free response approach

Write the flux as a function of time first, then differentiate. For direction, state in words which change is being opposed before naming clockwise or counterclockwise. For a differential equation, show the loop rule that produces it.