Unit 13 · Topic 13.2

13.2 Electromagnetic Induction

Faraday's law says a changing magnetic flux through a loop induces an emf around it, $\mathcal{E} = -\dfrac{d\Phi_B}{dt}$ , and Lenz's law gives its direction. This is the principle behind generators and transformers, and it shows that a changing magnetic field creates an electric field.

Key terms

Faraday’s law
induced emf
Lenz’s law
changing magnetic flux
Maxwell’s equations

Faraday's law

When the magnetic flux through a loop changes, an emf appears around the loop. For a coil of N turns, each with the same flux, $\mathcal{E} = -N\dfrac{d\Phi_B}{dt}$ . If the loop is a closed conductor with resistance R, a current $I = \dfrac{\lvert\mathcal{E}\rvert}{R}$ flows.

Only the change matters. A huge steady flux induces nothing; a small flux that changes quickly can induce a large emf.

Since $\Phi_B = BA\cos\theta$ , there are three ways to change it: change the field's strength, change the loop's area (or how much of it is in the field), or change the angle by rotating the loop. A coil spinning at angular speed ω in a uniform field has $\Phi_B = BA\cos\omega t$ , so $\mathcal{E} = NBA\omega\sin\omega t$ . That's how a generator makes alternating current.

Lenz's law: the direction

The minus sign in Faraday's law is Lenz's law: the induced current flows in the direction that makes its own magnetic field oppose the change in flux. It doesn't oppose the flux itself, only the change.

Work through these steps every time:

Which way does the external field point through the loop?
Is the flux increasing or decreasing?
The induced field points opposite the external field if the flux is increasing, and the same way if it's decreasing.
Use the right-hand rule for a loop: point your thumb along the induced field, and your fingers curl in the direction of the induced current.

Why Lenz's law must be true

Lenz's law is conservation of energy in disguise. If the induced current helped the change instead of opposing it, a tiny push on a magnet would build a current that pulled the magnet in faster, which would build more current, creating energy from nothing. Instead, you must do work to push a magnet toward a coil, and that work becomes the electrical energy in the circuit.

Induced electric fields

What actually pushes the charges? A changing magnetic field creates an electric field, even in empty space. Its field lines form closed loops, so it isn't a conservative field, and the loop rule from 11.6 doesn't apply to it. Faraday's law in full is $\oint \vec{E} \cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$ . The emf is the work this field does on each coulomb going once around the loop.

This is one of Maxwell's four equations, along with Gauss's law (8.6), Gauss's law for magnetism (12.1) and the Ampère–Maxwell law (12.4). Together they show that changing electric and magnetic fields can sustain each other and travel as electromagnetic waves, such as light. You won't be asked to derive the wave, just to know the connection.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
A field switched off
A 200-turn coil with area 0.010 m² and resistance 8.0 Ω sits in a uniform 0.50 T field that points straight out of the page, perpendicular to the coil. The field drops steadily to zero in 0.25 s. Find the induced emf, the current, and its direction as seen from the front of the page.
Show the solutionHide the solution
1. Step 1: Change in flux through each turn: $\Delta\Phi_B = (0 - 0.50)(0.010) = -0.0050\text{ Wb}$ .
2. Step 2: $\lvert\mathcal{E}\rvert = N\dfrac{\lvert\Delta\Phi_B\rvert}{\Delta t} = 200 \times \dfrac{0.0050}{0.25} = 4.0\text{ V}$ , and $I = \dfrac{4.0}{8.0} = 0.50\text{ A}$ .
3. Step 3: Direction: the outward flux is decreasing, so the induced current makes its own field out of the page to replace what's lost. By the right-hand rule, a current making an outward field inside the loop flows counterclockwise as you look at the page.
Answer: 4.0 V and 0.50 A, counterclockwise as seen from the front
Example 2Calculator allowed
A spinning coil generator
A 50-turn coil of area 0.020 m² spins at 60 rad/s in a uniform 0.30 T field, about an axis perpendicular to the field. Find the maximum emf and when it occurs.
Show the solutionHide the solution
1. Step 1: Flux through each turn: $\Phi_B = BA\cos\omega t$ . Then $\mathcal{E} = -N\dfrac{d\Phi_B}{dt} = NBA\omega\sin\omega t$ .
2. Step 2: Maximum: $NBA\omega = (50)(0.30)(0.020)(60) = 18\text{ V}$ .
3. Step 3: It occurs when sin ωt = ±1, which is when the coil's plane is parallel to the field. At that moment the flux is zero but changing fastest. The trap is thinking the emf peaks when the flux peaks.
Answer: 18 V, when the plane of the coil is parallel to the field (zero flux)
Example 3Calculator allowed
Induced electric field inside a solenoid
A long solenoid has a radius of 5.0 cm. The uniform field inside it, along the axis, is increasing at 0.50 T/s. Find the size of the induced electric field 2.0 cm from the axis.
Show the solutionHide the solution
1. Step 1: By symmetry, the induced field forms circles around the axis with the same size at a given distance. Take a circular path of radius r = 0.020 m. It lies inside the solenoid, so the changing field passes through its whole area.
2. Step 2: $\oint \vec{E} \cdot d\vec{\ell} = E(2\pi r)$ and $\dfrac{d\Phi_B}{dt} = \pi r^2\dfrac{dB}{dt}$ .
3. Step 3: Faraday's law in size: $E(2\pi r) = \pi r^2\dfrac{dB}{dt}$ , so $E = \dfrac{r}{2}\dfrac{dB}{dt} = \dfrac{0.020}{2}(0.50) = 5.0 \times 10^{-3}\text{ V/m}$ .
4. Step 4: This field exists whether or not a wire is there; a wire just gives charges a path to follow it.
Answer: 5.0 × 10⁻³ V/m, circling the axis

Common mistakes

Thinking a strong field induces an emf by itself. Only a changing flux does.
Saying the induced current opposes the field. It opposes the change in flux; when flux is decreasing, the induced field points the same way as the external one.
Forgetting the number of turns N.
Assuming the emf is largest when the flux is largest. It's largest when the flux changes fastest.

On the exam

Direction questions are everywhere. Write out the Lenz's law chain (external field, increasing or decreasing, induced field, current) to earn the reasoning points.
Graph questions often show $\Phi_B$ against t and ask for the emf graph: the emf is the negative of the slope, so flat parts give zero and straight ramps give constant emf.

Connected topics

Videos

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Check yourself

4 questions on 13.2 Electromagnetic Induction. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A 200-turn coil with an area of 0.010 m² is perpendicular to a uniform magnetic field. The field increases steadily from 0.10 T to 0.50 T in 0.20 s. What is the magnitude of the induced emf?

Question 2 of 4Calculator allowed

A single loop of area 0.050 m² is perpendicular to a uniform field that changes as $B(t) = 0.30 + 0.20t^2$ , with B in teslas and t in seconds. What is the magnitude of the induced emf at t = 3.0 s?

Question 3 of 4Calculator allowed

The north pole of a bar magnet is moved toward a flat wire loop, along the loop’s axis. As seen by someone looking at the loop from the magnet’s side, which way does the induced current flow, and why?

Question 4 of 4Calculator allowed

A generator coil has 50 turns, each with an area of 0.030 m². It spins at 60 revolutions per second in a uniform 0.20 T field. What is the peak emf?

0 of 4 answered