Playground
A magnet moves through a coil, inducing EMF. Watch the galvanometer deflect as flux changes.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Induced EMFoutput Electromotive force induced in the loop | V | M·L²·T⁻³·I⁻¹ | -100 – 100 | |
| Magnetic flux Total magnetic flux through the loop | Wb | M·L²·T⁻²·I⁻¹ | 0 – 1 | |
| Number of turns Number of loops in the coil | dimensionless | 1 | 1 – 1000 |
Deep dive
Derivation
From experimental observation: ε ∝ dΦ_B/dt. The minus sign (Lenz's law) follows from energy conservation. In differential form: ∇ × E = −∂B/∂t, obtained by applying Stokes' theorem to the integral form.
Experimental verification
Faraday's original ring experiment (1831): a current pulse in one coil induced a transient current in a nearby coil. Modern verification includes precision measurements of AC generator output and magnetic braking experiments.
Common misconceptions
- A static magnetic field does NOT induce a voltage — only changes in flux matter
- The induced EMF opposes the change in flux, not the flux itself (Lenz's law)
- Flux can change by changing B, the area, or the angle between B and the surface normal
Real-world applications
- Electric generators and alternators in power plants
- Transformers for voltage conversion in power grids
- Induction charging pads for smartphones
- Magnetic stripe card readers and RFID systems
Worked examples
EMF from a collapsing field
Given:
- R_loop:
- 0.1
- B_initial:
- 0.5
- B_final:
- 0
- dt:
- 0.02
Find: Induced EMF
Solution
|ε| = |ΔΦ_B/Δt| = |Δ(BA)/Δt| = |0 − 0.5 × π × 0.01| / 0.02 = 0.785 V
Generator EMF (rotating coil)
Given:
- N:
- 100
- A:
- 0.05
- B:
- 0.2
- omega:
- 377
Find: Peak EMF
Solution
ε_peak = NBAω = 100 × 0.2 × 0.05 × 377 = 377 V
Scenarios
What if…
- scenario:
- What if the loop has 50 turns?
- answer:
- EMF multiplies by 50: ε = 50 × 0.785 = 39.25 V. More turns = more flux linkage per unit flux change.
- scenario:
- What if the field changes 10× faster?
- answer:
- EMF increases 10×: ε = 7.85 V. Faster changes induce larger voltages — the basis of spark ignition in car engines.
- scenario:
- What if the loop area shrinks instead of B changing?
- answer:
- Same effect — flux still changes. A loop collapsing from A to 0 in the same time gives the same EMF. Faraday's law cares about flux change, not its cause.
Limiting cases
- condition:
- dΦ_B/dt = 0
- result:
- ε = 0
- explanation:
- A constant magnetic flux induces no voltage — you need change.
- condition:
- N → large
- result:
- ε scales linearly with N
- explanation:
- More turns multiply the EMF — the principle behind transformers.
- condition:
- Rapid flux change
- result:
- ε → large
- explanation:
- Faster changes produce larger voltages — used in spark ignition systems.
Context
Michael Faraday · 1831
Faraday discovered that a changing magnetic field induces an electric current by moving a magnet through a coil. He had no formal mathematics — Maxwell later expressed it as an equation.
Hook
How does shaking a magnet inside a flashlight make it light up without batteries?
A circular loop of radius 0.1 m is in a magnetic field that changes from 0.5 T to 0 T in 0.02 s. Find the induced EMF.
Dimensions: [ε] = [Φ_B]/[t] → Wb/s = (V·s)/s = V ✓
Validity: Universally valid in classical electromagnetism — one of Maxwell's four equations. The integral form applies to any closed loop, moving or stationary. For moving conductors, both motional and transformer EMF must be considered.