How a changing field makes a voltage
In 1831, Michael Faraday discovered that moving a magnet near a coil makes a current flow in it. The key is change: a steady field does nothing, but a changing magnetic flux through a circuit induces an electromotive force (EMF). Generators, transformers, induction cooktops, wireless chargers and guitar pickups all work on this principle.
The equations
- Magnetic flux through one turn: Φ = B × A × cos θ, in webers, where B is the field in teslas, A the area of the loop and θ the angle between the field and the axis of the coil
- Faraday’s law: ε = −N × ΔΦ ÷ Δt, for a coil of N turns
- Induced current: I = |ε| ÷ R, for a circuit of resistance R
- Charge that flows: Q = N × |ΔΦ| ÷ R, independent of how fast the change happens
The minus sign is Lenz’s law: the induced current creates its own magnetic field that opposes the change in flux.
Example
A 100-turn coil of 50 cm² sits in a field that rises from 0 to 0.5 T in 0.1 s, with the field along the axis of the coil:
- Change in flux per turn: 0.5 × 0.005 = 0.0025 Wb
- Induced EMF: −100 × 0.0025 ÷ 0.1 = −2.5 V
- Current in a 10 Ω circuit: 0.25 A
- Charge moved: 0.025 C
Making the change ten times faster gives 25 V, but the same charge.
Three ways to change the flux
- Change the field, as in a transformer, where alternating current in one coil creates a varying field through the other.
- Change the area, as with a sliding bar on rails, which gives the motional EMF ε = B × L × v.
- Change the angle, as in a generator, where a rotating coil sees a flux that varies sinusoidally, giving an alternating voltage with a peak of N × B × A × ω.
Units
One tesla is 10,000 gauss. Earth’s magnetic field is about 25 to 65 microtesla, a fridge magnet about 5 mT, and an MRI scanner 1.5 to 3 T.