Electromagnetic Induction

Electromagnetic induction (AQA specification 3.7.5) describes how a changing magnetic field produces an EMF (and hence a current in a closed circuit). This is the basis of generators, transformers, and much of modern electrical technology.

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Magnetic Flux

Key Definition: Magnetic flux (Φ) through a surface is the product of the magnetic flux density B and the area A of the surface, when the field is perpendicular to the surface.

Φ = BA cos θ

where Φ is the magnetic flux (Wb, webers), B is the magnetic flux density (T), A is the area (m²), and θ is the angle between the field direction and the normal to the surface.

When B is perpendicular to the surface (θ = 0°): Φ = BA

When B is parallel to the surface (θ = 90°): Φ = 0

The unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T m² = 1 V s.

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Flux Linkage

For a coil of N turns, the flux linkage is:

NΦ = NBA cos θ

Flux linkage has units of Wb turns (or simply Wb, since N is dimensionless).

If the coil rotates in a uniform field at angular velocity ω:

θ = ωt

NΦ = NBA cos(ωt)

This sinusoidal variation in flux linkage is the basis of the AC generator.

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Faraday's Law

Key Definition: Faraday's law states that the magnitude of the induced EMF is equal to the rate of change of flux linkage through the circuit.

ε = −N(dΦ/dt) = −d(NΦ)/dt

The negative sign indicates that the induced EMF opposes the change producing it (Lenz's law).

In practice, for calculations:

|ε| = N × |ΔΦ/Δt|

Derivation from a Moving Conductor

Consider a straight conductor of length L moving at velocity v perpendicular to a uniform magnetic field B.

In time Δt, the conductor sweeps out an area ΔA = LvΔt.

Change in flux: ΔΦ = BΔA = BLvΔt

Rate of change of flux: ΔΦ/Δt = BLv

Therefore: ε = BLv (for a single conductor moving perpendicular to the field)

Worked Example 1 — EMF from a Moving Conductor

Question: A metal rod of length 0.40 m moves at 5.0 m s⁻¹ perpendicular to a uniform magnetic field of 0.25 T. Calculate the EMF induced across the rod.

Solution:

ε = BLv = 0.25 × 0.40 × 5.0 = 0.50 V

Worked Example 2 — EMF from a Changing Flux

Question: A coil of 200 turns and area 8.0 × 10⁻³ m² is placed in a uniform magnetic field. The field increases uniformly from 0 to 0.50 T in 0.10 s. Calculate the magnitude of the induced EMF.

Solution:

ΔΦ = BΔA? No — the area is constant, the field changes.

ΔΦ = ΔB × A = (0.50 − 0) × 8.0 × 10⁻³ = 4.0 × 10⁻³ Wb

|ε| = N × |ΔΦ/Δt| = 200 × (4.0 × 10⁻³ / 0.10) = 200 × 0.040 = 8.0 V

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Lenz's Law

Key Definition: Lenz's law states that the direction of the induced EMF (and hence the induced current) is such as to oppose the change in flux that produces it.

This is a consequence of conservation of energy. If the induced current enhanced the flux change, it would create a runaway process that would generate energy from nothing — violating the first law of thermodynamics.

Application of Lenz's Law

Example: A bar magnet is pushed into a solenoid (north pole first). The increasing flux through the solenoid induces an EMF that drives a current. By Lenz's law, this current must create a magnetic field that opposes the increasing flux — so the solenoid's near end becomes a north pole (to repel the incoming magnet). You must do work pushing the magnet in, and this work is the source of the electrical energy.

Example: If the magnet is pulled away, the flux decreases, and the induced current creates a field to try to maintain the flux — the near end becomes a south pole (to attract the retreating magnet).

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EMF from a Rotating Coil