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How does a changing magnetic field create a voltage, and which way does the induced current flow?

Topic 12.4 Electromagnetic Induction and Faraday's Law: apply Faraday's law and Lenz's law to find the emf and current induced by a changing magnetic flux.

A focused answer to AP Physics 2 Topic 12.4, covering magnetic flux, Faraday's law of induction, the induced emf from a changing flux, Lenz's law for the direction of the induced current, motional emf, and applications to generators and transformers, with full worked examples.

Generated by Claude Opus 4.812 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

Magnetic flux $\Phi = B A \cos\theta$ measures how much field passes through a loop (field times area times the cosine of the angle between the field and the loop's normal), in webers (Wb). Faraday's law says a changing flux induces an emf: $\mathcal{E} = -N\dfrac{\Delta \Phi}{\Delta t}$ , proportional to the rate of change of flux and the number of turns $N$ . The flux can change by varying the field, the area, or the angle. Lenz's law (the minus sign) gives the direction: the induced current flows so as to oppose the change in flux that produced it. A constant field induces nothing, because only a change in flux matters. Motional emf arises when a conductor moves through a field, sweeping out area. This single principle, a changing magnetic flux drives a current, is how generators turn motion into electricity and how transformers change voltages, completing the link between electricity and magnetism.

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What this topic is asking
Magnetic flux
Faraday's law
Lenz's law and applications
Try this

What this topic is asking

The College Board (Topic 12.4) wants you to apply Faraday's law of induction, an emf is induced by a changing magnetic flux, and Lenz's law, which gives the direction of the induced current, to find the emf and current in a loop.

Magnetic flux

Flux is the bookkeeping quantity for induction: it counts how many field lines thread the loop. It is largest when the field passes straight through the loop and zero when the field skims along the plane of the loop. Crucially, induction depends not on the flux itself but on how fast it changes, which can happen three ways: change the field strength, change the loop's area, or rotate the loop to change the angle.

Faraday's law

Faraday's law is the heart of the topic: it is the rate of flux change that drives the emf, not the size of the field. A magnet sitting still in a coil induces nothing; the same magnet thrust quickly into the coil induces a large emf. Multiplying by $N$ (the number of turns) is why generators and transformers use many-turn coils, each turn adds its own contribution. The standard calculation is $\mathcal{E} = N \dfrac{\Delta \Phi}{\Delta t}$ in magnitude, with the direction handled separately by Lenz's law.

Lenz's law and applications

The minus sign in Faraday's law is Lenz's law: the induced current flows in the direction that opposes the change in flux that created it. If the flux through a loop is increasing, the induced current makes its own field to oppose the increase; if decreasing, it acts to maintain the flux. Lenz's law is really conservation of energy in disguise, the induced current must oppose the change, or it would create energy from nothing. A moving conductor in a field experiences a motional emf as it sweeps out area, the same law seen from the moving charges' point of view. The strategic payoff is enormous: this one principle, a changing flux induces a current that opposes the change, is the basis of the electric generator (rotating a coil in a field changes the flux and induces an AC voltage), the transformer (a changing current in one coil induces a voltage in another), and electromagnetic braking. It completes the unit by showing that electricity and magnetism are two faces of one phenomenon: currents make fields (Topic 12.3), and changing fields make currents (here).

emf from a shrinking loop

A square loop of side $0.10$ m sits in a uniform $0.50$ T field perpendicular to the loop. The loop is collapsed to zero area in $0.020$ s. Find the magnitude of the induced emf.

step 1 Initial and final flux

Initial area $A = (0.10)^2 = 0.010$ m squared. Initial flux $\Phi_i = BA = (0.50)(0.010) = 5.0 \times 10^{-3}$ Wb. Final flux (zero area) $\Phi_f = 0$ .

step 2 Change in flux

$\Delta \Phi = \Phi_f - \Phi_i = 0 - 5.0 \times 10^{-3} = -5.0 \times 10^{-3}$ Wb (magnitude $5.0 \times 10^{-3}$ Wb).

step 3 Apply Faraday's law

$\mathcal{E} = \dfrac{|\Delta \Phi|}{\Delta t} = \dfrac{5.0 \times 10^{-3}}{0.020} = 0.25$ V (one turn).

step 4 Direction by Lenz's law

The flux through the loop is decreasing, so the induced current flows to maintain it, in the direction that creates a field in the same sense as the original through the loop.

Try this

Q1. State what must change for an emf to be induced in a coil. [1 point]

Cue. The magnetic flux through the coil (its rate of change drives the emf).

Q2. State what Lenz's law tells you about the direction of an induced current. [1 point]

Cue. It flows so as to oppose the change in flux that produced it.

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 2024 (style)7 marksSection II (long FRQ). A single circular loop of area

0.020

m squared sits in a uniform magnetic field perpendicular to the loop. The field increases steadily from

0.10

T to

0.40

T in

0.50

s. (a) Calculate the change in magnetic flux through the loop. (b) Calculate the magnitude of the induced emf. (c) State and justify the direction of the induced current relative to the increasing field, using Lenz's law.

Show worked answer →

A 7-point FRQ on Faraday's and Lenz's laws.

(a) Flux change (2 points): $\Delta \Phi = A\,\Delta B = (0.020)(0.40 - 0.10) = (0.020)(0.30) = 6.0 \times 10^{-3}$ Wb.
(b) Induced emf (3 points): $\mathcal{E} = \dfrac{\Delta \Phi}{\Delta t} = \dfrac{6.0 \times 10^{-3}}{0.50} = 1.2 \times 10^{-2}$ V.
(c) Direction (2 points): by Lenz's law the induced current opposes the increase, so it flows to create a magnetic field opposite to the increasing applied field (that is, to oppose the rising flux through the loop).

Markers reward the flux change, Faraday's law for the emf, and Lenz's law for the opposing direction.

AP 2023 (style)1 marksSection I (multiple choice). A bar magnet is held stationary inside a coil. What is the induced emf in the coil? (A) maximum (B) zero (C) equal to the magnet's field (D) it depends on the magnet's strength. Justify your reasoning.

Show worked answer →

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

An emf is induced only when the magnetic flux through the coil changes. A stationary magnet gives a constant flux, so $\Delta \Phi = 0$ and the induced emf is zero. The trap is (D): a strong but unchanging field still induces nothing; it is the rate of change of flux that matters.

Related dot points

Sources & how we know this

AP Physics 2: Algebra-Based Course and Exam Description — College Board (2024)