Faraday's law, explained with examples
Faraday's law says that a changing magnetic field induces a voltage. It is how almost all of the world's electricity is generated, and it is a key topic in AP Physics 2, AP Physics C: E&M and A level Physics.
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Magnetic flux
Magnetic flux Φ measures how much magnetic field passes through an area:
Φ = B A cosθ
B is the magnetic flux density in tesla (T), A is the area in m², and θ is the angle between the field and the normal (perpendicular) to the area. Flux is measured in webers (Wb). It is greatest when the field passes straight through the surface (θ = 0) and zero when the field runs parallel to it. For a coil of N turns, the flux linkage is NΦ.
Faraday's law
ε = −N dΦ/dt, or for average values, ε = −N ΔΦ/Δt
The induced emf equals the rate of change of flux linkage. Flux can change in three ways: the field strength B changes, the area A changes, or the angle θ changes as a coil rotates. A steady field through a stationary coil induces nothing; only change matters. Faster changes induce larger emfs, and more turns multiply the effect.
Lenz's law: the minus sign
The minus sign expresses Lenz's law: the induced current flows in the direction that opposes the change that caused it. Push the north pole of a magnet into a coil, and the induced current makes that end of the coil a north pole, which repels the magnet. Pull it out, and the coil end becomes a south pole, attracting it back.
Lenz's law is a consequence of energy conservation. If the induced current helped the change instead of opposing it, the magnet would accelerate into the coil by itself, creating energy from nothing. Because the current opposes the motion, you must do work to move the magnet, and that work becomes electrical energy. You can feel this effect: a magnet dropped through a copper tube falls noticeably slowly.
Worked examples
Example 1: flux through a loop
A square loop of side 0.10 m sits in a 0.50 T field. Find the flux when the field is perpendicular to the loop, and when it is at 60° to the normal.
- A = 0.010 m². Perpendicular to the loop means θ = 0: Φ = 0.50 × 0.010 = 5.0 × 10−3 Wb.
- At 60° to the normal: Φ = 5.0 × 10−3 × cos 60° = 2.5 × 10−3 Wb.
Example 2: a changing field
A coil of 200 turns and area 0.020 m² is perpendicular to a field that rises steadily from 0.10 T to 0.40 T in 0.30 s. Find the induced emf.
- ΔΦ = ΔB × A = 0.30 × 0.020 = 6.0 × 10−3 Wb.
- ε = NΔΦ/Δt = 200 × 6.0 × 10−3/0.30 = 4.0 V (in size).
Example 3: a moving rod
A 0.40 m metal rod slides at 5.0 m/s along rails, perpendicular to a 0.25 T field. Find the emf across the rod.
- The rod sweeps out area at a rate Lv, so the flux changes at a rate BLv.
- ε = BLv = 0.25 × 0.40 × 5.0 = 0.50 V.
Example 4: induced current and force
The rails in Example 3 are joined by a 2.0 Ω resistor. Find the current and the force needed to keep the rod moving at constant speed.
- I = ε/R = 0.50/2.0 = 0.25 A.
- The current in the field feels a force F = BIL = 0.25 × 0.25 × 0.40 = 0.025 N, opposing the motion (Lenz's law). An equal applied force keeps the speed constant.
- Power check: F × v = 0.025 × 5.0 = 0.125 W, and I²R = 0.0625 × 2.0 = 0.125 W. The work you do becomes electrical energy.
Generators
A generator rotates a coil in a magnetic field. As the coil turns, the angle between the field and the coil's normal changes continuously, so the flux linkage changes and an alternating emf is induced. For a coil of N turns and area A rotating at angular speed ω in a field B, the peak emf is ε0 = NBAω. The emf is largest when the plane of the coil is parallel to the field, because that is when the flux is changing fastest, even though the flux itself is zero at that instant.
Example 5: a generator
A generator coil has 500 turns, area 0.015 m², and rotates 50 times per second in a 0.20 T field. Find the peak emf.
- ω = 2π × 50 ≈ 314 rad/s.
- ε0 = 500 × 0.20 × 0.015 × 314 ≈ 471 V.
Transformers
A transformer uses Faraday's law with no moving parts. An alternating current in the primary coil creates a changing flux in an iron core, which induces an alternating emf in the secondary coil. For an ideal transformer, Vs/Vp = Ns/Np. Transformers only work with alternating current, because a steady current gives a steady flux and so no induced emf. They let power stations step voltages up for efficient transmission and step them down again for homes.
Common mistakes
- Measuring θ from the plane of the coil instead of from its normal.
- Forgetting to multiply by the number of turns N.
- Thinking a strong but steady field induces an emf; only a changing flux does.
- Getting the induced current's direction wrong; it always opposes the change.
- Saying the emf is greatest when the flux is greatest; it is greatest when the flux changes fastest.
Practice questions
Try each one before opening the answer.
Q1. Find the flux through a 0.05 m² area in a 0.8 T field perpendicular to it.
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Q2. Flux linkage in a coil changes by 0.6 Wb in 0.2 s. Find the average emf.
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Q3. A 0.5 m rod moves at 4 m/s through a 0.3 T field. Find the emf.
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Q4. A transformer has 100 primary turns and 2000 secondary turns, with 12 V input. Find the output.
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Frequently asked questions
Is Faraday's law on the AP Physics exams?
Yes. Electromagnetic induction is in AP Physics 2 and in AP Physics C: Electricity and Magnetism, where it is treated with calculus.
Is Faraday's law in GCSE or A level Physics?
GCSE separate Physics covers the generator effect and transformers qualitatively. A level Physics covers flux, flux linkage, Faraday's and Lenz's laws quantitatively.
Key takeaways
- Flux Φ = BA cosθ; flux linkage = NΦ.
- Faraday: ε = −N dΦ/dt; only changing flux induces an emf.
- Lenz: the induced current opposes the change, which conserves energy.
- Moving rod: ε = BLv.
- Generators rotate coils; transformers use changing current with Vs/Vp = Ns/Np.