Short on time? This page is the recap. For explanations, figures and the worked examples, use the full study guide.
Recap
- Flux measures the part of B perpendicular to a surface, times the area. The area vector is perpendicular to the surface (outward for a closed surface).
- The sign of Φ shows whether B is along (+) or against (−) the area vector.
- Flux changes if B, A or θ changes.
- Faraday: a changing flux induces an emf, even in an open loop. A current flows only in a closed circuit: I = ε/R.
- Lenz, in four steps: (1) direction of external B through the loop; (2) flux increasing or decreasing? (3) induced field opposite to B if increasing, same as B if decreasing; (4) thumb along the induced field, fingers give the current.
- Lenz’s law is energy conservation: work against the induced effect becomes electrical energy.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| Φ = BA cos θ | Wb = T·m²; θ from the area vector | find the flux through a flat loop |
| ε = −ΔΦ/Δt | V; average over Δt | find the induced emf |
| |ε| = N|ΔΦ|/Δt | N turns, Φ through one turn | find the emf of a coil |
| ε = Bℓv | rod of length ℓ, speed v ⊥ B | find the emf of a moving rod |
| I = ε/R | Ω | find the induced current |
| Flux–time graph | Induced emf |
|---|---|
| straight rise | constant, one sign |
| flat | zero |
| straight fall | constant, opposite sign |
| steeper slope | larger emf |
Assumptions behind the numbers
- The field is uniform over the loop, so Φ = BA cos θ applies.
- ΔΦ/Δt gives the average emf over the interval.
- Rails and rod have negligible resistance unless stated.
Mistakes to avoid
- Emf from Φ instead of ΔΦ/Δt. A steady flux gives no emf.
- θ from the surface. Use the angle to the area vector.
- Opposing the field instead of the change.
- Forgetting N for a coil.
- Wrong force direction on the rod. The magnetic force always opposes the motion.
Quick self-check
- B = 0.20 T at 60° to the area vector of a 0.050 m² loop. Find Φ. (5.0 × 10⁻³ Wb)
- The flux through each turn of a 100-turn coil changes by 2.0 × 10⁻³ Wb in 0.050 s. Find |ε|. (4.0 V)
- A 0.20 m rod moves at 3.0 m/s at right angles to a 0.10 T field. Find ε. (0.060 V)
- The flux into the page through a loop is decreasing. Which way does the induced field point inside the loop? (Into the page)
Next: practice questions.