Electromagnetic induction

Electromagnetic induction occurs when magnetic flux through a circuit changes. A magnetic field can be strong and still produce no induced emf if the relevant flux stays constant.

Magnetic flux: the quantity that must change

Flux counts the component of B through a surface. The angle is measured from B to the surface normal, not to the plane itself.

Φ = BA cos θ for a uniform field through a flat loop.

Flux can change because B changes, the loop area changes, the loop rotates, or the loop moves into a region where the field is different.

Faraday's law

emf = −N dΦ/dt

The magnitude grows with the number of turns and with the rate of flux change. The minus sign encodes the direction described by Lenz's law.

Flux-change explorer

For a coil whose uniform flux changes from Φ₁ to Φ₂, |emf| = N|ΔΦ|/Δt.

N ΔΦ per turn (mWb) Δt (s)

Relative motion matters because it changes flux. Faster change produces larger induced emf; a stationary magnet and stationary coil give no induction merely because a magnetic field is present.

Lenz's law and energy conservation

The induced current creates a magnetic effect that opposes the change in flux, not necessarily the original magnetic field. If flux is increasing in one direction, the induced field points against that increase; if flux is decreasing, the induced field tends to sustain it.

This opposition is why mechanical work is required when a magnet is pushed into a conducting coil that carries induced current.

Motional emf is the same physics

A conducting rod moving through a magnetic field can separate charges because moving charges experience magnetic force. For a rod of length L moving perpendicular to a uniform B with speed v, the ideal magnitude is emf = BLv.

Faraday's law and motional emf are consistent descriptions of induction: what matters is the change in magnetic flux linked with the circuit.

A reliable induction workflow

Start by defining the loop and its surface normal. Then determine the external flux direction and ask whether that signed flux is increasing or decreasing. Only after that should you use Lenz's law to choose the induced field direction and the corresponding current direction.

A common mistake is to oppose the external field itself. Lenz's law opposes the change: if an upward flux is decreasing, the induced field is upward to resist the decrease. Another mistake is to use Φ = BA without checking orientation; the cos θ factor matters whenever the field is not parallel to the surface normal.

Worked examples

1. Flux through a loop

A 0.020 m² loop is perpendicular to a uniform 0.30 T field. Find the magnetic flux.

Solution

The surface normal is parallel to B, so θ = 0 and Φ = BA = 0.30 × 0.020 = 6.0 × 10⁻³ Wb.

2. Faraday magnitude

A 200-turn coil experiences a flux-per-turn change of 3.0 mWb in 0.050 s. Find the average induced emf magnitude.

Solution

|emf| = N|ΔΦ|/Δt = 200(3.0 × 10⁻³)/0.050 = 12 V.

3. Lenz direction

A north pole approaches a coil along its axis. Which magnetic pole should the near face of the coil act like?

Solution

The incoming north pole increases flux in its direction. The induced field must oppose that increase, so the near face behaves like a north pole, repelling the approaching magnet.