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United StatesPhysics C: Electricity and MagnetismSyllabus dot point

How does a changing magnetic flux induce an EMF, and what sets its direction?

Topic 13.2 Electromagnetic Induction: apply Faraday's law and Lenz's law to find the magnitude and direction of an induced EMF.

A calculus-based answer to AP Physics C E&M Topic 13.2, covering Faraday's law of induction, the rate of change of flux, Lenz's law for direction, motional EMF, and induced EMF in rotating coils.

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  1. What this topic is asking
  2. Faraday's law
  3. Lenz's law: the direction
  4. Motional EMF and the rotating coil
  5. Try this

What this topic is asking

The College Board (Topic 13.2) wants you to apply Faraday's law, that a changing magnetic flux induces an EMF, to find the magnitude of the EMF, and Lenz's law to find its direction. This is the central result of the whole course: changing magnetism makes electricity.

Faraday's law

Because ΦB=BAcos⁡θ\Phi_B = BA\cos\theta, the EMF is the derivative of that product. If only one variable changes, the others come out of the derivative: a changing field gives ε=−NAdBdt\varepsilon = -NA\dfrac{dB}{dt}, a changing area gives ε=−NBdAdt\varepsilon = -NB\dfrac{dA}{dt}, and a changing angle gives a rotating-coil EMF.

Lenz's law: the direction

The minus sign encodes Lenz's law: the induced current flows so as to oppose the change that produced it. If the flux through a loop is increasing, the induced current creates a field opposing that increase; if decreasing, it tries to maintain the flux. Physically, this is conservation of energy: an induced current that aided the change would accelerate it indefinitely, creating energy from nothing.

To apply it: find whether the flux is rising or falling and in which direction, then choose the current direction whose own field opposes that change (use the right-hand rule for the loop's field).

Motional EMF and the rotating coil

A conducting rod of length LL sliding at speed vv perpendicular to a field BB has a flux through the circuit that changes as the enclosed area grows, giving a motional EMF:

ε=BLv\varepsilon = BLv

(equivalently, the magnetic force qvBqvB pushes charges along the rod). A flat coil of NN turns and area AA rotating at angular frequency ω\omega in a uniform field has Φ=BAcos⁡(ωt)\Phi = BA\cos(\omega t), so

ε=−NdΦdt=NBAωsin⁡(ωt)\varepsilon = -N\frac{d\Phi}{dt} = NBA\omega\sin(\omega t)

a sinusoidal output, the basis of every AC generator.

Try this

Q1. A loop's flux changes at 0.400.40 Wb/s. Find the induced EMF (single turn). [1 point]

  • Cue. ∣ε∣=∣dΦdt∣=0.40|\varepsilon| = \left|\dfrac{d\Phi}{dt}\right| = 0.40 V.

Q2. A rod of length 0.500.50 m moves at 4.04.0 m/s perpendicular to a 0.300.30 T field. Find the motional EMF. [2 points]

  • Cue. ε=BLv=(0.30)(0.50)(4.0)=0.60\varepsilon = BLv = (0.30)(0.50)(4.0) = 0.60 V.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2022 (style)1 marksSection I (multiple choice). A magnet's north pole is pushed toward a conducting loop. The induced current in the loop, viewed from the magnet's side, flows (A) clockwise (B) counterclockwise (C) there is no current (D) it depends on the magnet's strength only. Justify your reasoning.
Show worked answer →

A 1-point MCQ on Lenz's law. The answer is (B).

By Lenz's law the induced current opposes the increasing flux, so the loop's near face must become a north pole to repel the incoming magnet. Viewed from the magnet's side, the current that makes the near face a north pole flows counterclockwise. The trap is (A): that would produce a south near face, which would attract (aid) the magnet, violating energy conservation.

AP 2024 (style)6 marksSection II (FRQ, quantitative). A square loop of side 0.100.10 m and resistance 2.0 Ω2.0\,\Omega sits perpendicular to a magnetic field that increases uniformly from 0.200.20 T to 0.800.80 T in 0.300.30 s. (a) Calculate the rate of change of flux. (b) Calculate the induced EMF and current. (c) State and justify the direction of the induced current using Lenz's law.
Show worked answer →

A 6-point FRQ on Faraday's law with a uniformly changing field.

(a) Rate of change of flux (2 points): area A=(0.10)2=0.010A = (0.10)^2 = 0.010 m squared. dΦdt=AdBdt=(0.010)0.80−0.200.30=(0.010)(2.0)=0.020\dfrac{d\Phi}{dt} = A\dfrac{dB}{dt} = (0.010)\dfrac{0.80 - 0.20}{0.30} = (0.010)(2.0) = 0.020 Wb/s.
(b) EMF and current (3 points): ∣ε∣=∣dΦdt∣=0.020|\varepsilon| = \left|\dfrac{d\Phi}{dt}\right| = 0.020 V. I=εR=0.0202.0=0.010I = \dfrac{\varepsilon}{R} = \dfrac{0.020}{2.0} = 0.010 A.
(c) Direction (1 point): the flux into the loop is increasing, so the induced current flows to oppose it, creating a field out of the loop (counterclockwise viewed from the side the field points toward).

Markers reward A dB/dtA\,dB/dt, the EMF and current, and a Lenz's-law direction with justification.

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