Short on time? This page is the recap. For the figures, the derivations and the worked examples, use the full study guide. Differentiating a time-dependent flux and the induced electric field belong to the calculus-based course.
Recap
- A changing magnetic flux through a loop induces an emf around it (Faraday’s law). Steady flux, however large, induces nothing.
- In a coil or long solenoid of N turns, the total emf is N times the emf of one turn.
- The flux Φ_B = BA cos θ can change through B, through A or through θ.
- Lenz’s law: the induced current creates a field that opposes the change in flux. Find the direction with the right-hand rule.
- The emf graph is −N times the gradient of the flux-per-turn graph.
- A changing magnetic field creates an induced electric field that circulates round it, even in empty space.
- Maxwell’s equations predict electromagnetic waves at c = 1/√(μ₀ε₀) = 3.00 × 10⁸ m/s (not derived in the course).
Key relationships
| Situation | Emf |
|---|---|
| General (N turns) | ℰ = −N dΦ_B/dt |
| Field changes, area fixed | size of ℰ = NA dB⊥/dt |
| Area changes, field fixed | size of ℰ = NB dA⊥/dt |
| Rod of length L sliding at speed v on rails | size of ℰ = BLv |
| Coil rotating at ω (θ = ωt) | ℰ = NBAω sin ωt; peak NBAω |
| Induced current in a loop of resistance R | I = (size of ℰ)/R |
| Field form (Maxwell’s third equation) | ∮E·dl = −dΦ_B/dt |
| Induced E, inside / outside a solenoid of radius R | E = (r/2) dB/dt / E = (R²/2r) dB/dt |
Lenz’s law in four steps
- Direction of the external field through the loop.
- Is that flux increasing or decreasing?
- Increasing: induced field opposite. Decreasing: induced field the same way.
- Right-hand rule: thumb along the induced field inside the loop, fingers give the current.
Mistakes to avoid
- “Big B means big emf.” Only dΦ_B/dt matters.
- “Induced field always opposes B.” It opposes the change.
- Forgetting N, or forgetting that +Φ to −Φ is a change of 2Φ.
- Maximum flux ⇒ maximum emf. For a rotating coil the emf peaks when the flux is zero.
- Treating the induced E as conservative. Its loop integral is the emf, not zero.
Quick self-check
- The flux through each turn of a 100-turn coil changes at 2.0 × 10⁻³ Wb/s. What is the size of the emf? (0.20 V)
- A rod 0.50 m long slides at 3.0 m/s on rails in a perpendicular 0.20 T field. What is the emf? (0.30 V)
- The flux out of the page through a loop is decreasing. Which way does the induced current flow, as you look at the page? (Anticlockwise, so that its field inside points out of the page)
Next: practice questions.