Magnetic Flux and Electromagnetic Induction

Magnetic Flux and Electromagnetic Induction

Electromagnetic induction is the process of generating an EMF (and hence a current in a complete circuit) by changing the magnetic flux through a conductor. It is the principle behind generators, transformers, and many other devices.

Magnetic Flux

Magnetic flux, Φ, through a surface is:

Φ = BA cos θ

where:

• Φ = magnetic flux (weber, Wb)
• B = magnetic flux density (tesla, T)
• A = area of the surface (m²)
• θ = angle between the field lines and the normal to the surface

When the field is perpendicular to the surface (θ = 0): Φ = BA (maximum)

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

Magnetic Flux Linkage

For a coil of N turns:

Flux linkage = NΦ = NBA cos θ (units: Wb turns, or simply Wb)

Faraday's Law of Electromagnetic Induction

The induced EMF in a circuit is equal to the rate of change of magnetic flux linkage through the circuit:

ε = −dNΦ/dt = −N dΦ/dt

The magnitude form: ε = NΔΦ/Δt (for uniform rates of change)

Lenz's Law

The direction of the induced EMF (and current) is such that it opposes the change that produces it. This is the physical reason for the negative sign in Faraday's law.

Lenz's law is a consequence of conservation of energy: if the induced current aided the change, it would increase the flux further, inducing more current — a perpetual motion machine.

Worked Example — Straight Conductor

A wire of length 0.30 m moves at 4.0 m s⁻¹ perpendicular to a magnetic field of 0.50 T. Calculate the induced EMF.

The flux swept per unit time: dΦ/dt = BLv

ε = BLv = 0.50 × 0.30 × 4.0 = 0.60 V

Worked Example — Coil in a Changing Field

A coil of 200 turns and area 5.0 × 10⁻³ m² is in a magnetic field that decreases uniformly from 0.40 T to 0 in 0.020 s. Find the induced EMF.

ΔΦ = ΔB × A = (0.40 − 0) × 5.0 × 10⁻³ = 2.0 × 10⁻³ Wb

ε = NΔΦ/Δt = 200 × 2.0 × 10⁻³ / 0.020 = 20 V

EMF in a Rotating Coil

A coil rotating at angular velocity ω in a uniform field B:

Flux linkage: NΦ = NBA cos(ωt)

Induced EMF: ε = NBAω sin(ωt) (differentiating)

The EMF varies sinusoidally — this is the basis of an AC generator (alternator).

Peak EMF: ε₀ = NBAω

The EMF is maximum when the coil is parallel to the field (flux linkage changing most rapidly) and zero when perpendicular (flux linkage at maximum but momentarily not changing).

Applications

The AC Generator:

• Coil rotates in a magnetic field
• Slip rings maintain continuous contact
• Output: sinusoidal AC voltage
• ε = ε₀ sin(ωt) where ε₀ = NBAω

The DC Generator:

• Same as AC but uses a split-ring commutator
• Output is pulsating DC (always positive)

The Transformer:

An AC voltage in the primary coil creates a changing magnetic flux in an iron core, which links to the secondary coil, inducing an EMF.

V_s/V_p = N_s/N_p (for an ideal transformer)

For 100% efficiency: V_p I_p = V_s I_s (power in = power out)

Step-up transformer: N_s > N_p → voltage increased, current decreased

Step-down transformer: N_s < N_p → voltage decreased, current increased

Transformer Efficiency

Real transformers have losses due to:

• Resistive heating (I²R losses in the coils) — minimised by using thick, low-resistance wire
• Eddy currents in the core — minimised by using a laminated core (thin insulated layers)
• Magnetic flux leakage — minimised by using a closed core design
• Hysteresis losses — energy lost in repeatedly magnetising and demagnetising the core — minimised using soft iron

Efficiency = P_output/P_input × 100%

Large power transformers can achieve >99% efficiency.

Eddy Currents

When a conductor moves through a magnetic field (or the field changes), currents are induced in the bulk of the conductor. These eddy currents:

• Circulate in closed loops within the conductor
• Produce heating (I²R losses)
• Create their own magnetic field that opposes the motion (by Lenz's law), producing a braking effect

Applications of eddy currents:

• Electromagnetic braking (trains, roller coasters)
• Induction hobs (heating pans)
• Metal detectors
• Damping in analogue meters

Self-Inductance

When the current in a coil changes, the changing flux linkage induces an EMF in the same coil that opposes the change:

ε = −L dI/dt

where L = self-inductance (henrys, H)

This is why you get a spark when switching off an inductive circuit — the rapid decrease in current induces a large EMF.

Energy Stored in an Inductor

E = ½LI²

Analogous to E = ½CV² for a capacitor.