Faraday’s Law
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
A changing magnetic flux induces an emf: ε = −N dΦB/dt. It doesn’t matter whether B changes, the area changes, or the loop turns. Behind it is an induced electric field that circles around the changing flux: ∮E · dl = −dΦB/dt.
1. Flux change makes emf
2. Reading it right
Rate, not amount: a huge steady flux makes no emf. On a Φ–t graph, emf is the slope; steepest slope means largest emf.
Current: if the loop has resistance R, I = |ε|/R.
Induced E is different: its field lines close on themselves, so it isn’t conservative. You can’t define a potential for it, and a charge going around once gains energy qε.
Use symmetry like Ampère’s law: inside a solenoid with changing B, E(2πr) = πr² dB/dt.
3. Example, then try it
A 200-turn coil of area 0.010 m² sits perpendicular to a field that drops steadily from 0.60 T to 0.10 T in 0.25 s. Find the emf. If the coil’s resistance is 5.0 Ω, what current flows?
ε = N ΔΦ/Δt = (200)(0.010)(0.50) ÷ 0.25 = 4.0 V.
I = 4.0 ÷ 5.0 = 0.80 A.
A long solenoid of radius 3.0 cm has a field inside it increasing at 0.50 T/s. Find the induced electric field at 2.0 cm and at 6.0 cm from the axis.
Path: a circle of radius r centered on the axis. By symmetry E is tangent and constant, so ∮E · dl = E(2πr).
Inside (r = 2.0 cm): flux = πr²B, so E(2πr) = πr² dB/dt → E = (r/2)(dB/dt) = (0.010)(0.50) = 5.0 × 10⁻³ V/m.
Outside (r = 6.0 cm): flux is only inside the solenoid, πR²B. E = R²(dB/dt)/(2r) = (0.0009)(0.50) ÷ 0.12 ≈ 3.8 × 10⁻³ V/m.
Notice: E isn’t zero outside, even though B is.
Answer: 5.0 × 10⁻³ V/m inside at 2.0 cm; about 3.8 × 10⁻³ V/m outside at 6.0 cm.
4. Common mistakes
At one instant, the flux through a spinning generator coil is at its maximum. What is the emf at that instant?
The flux is at a peak, so its rate of change is momentarily zero, and so is the emf.
Ready to practice?
5 questions on Faraday’s Law, each with a hint and an explanation for every wrong answer.