10.1 Faraday's Law & Induced Voltage

10.1 Faraday's Law & Induced Voltage

Quick Answer: Changing magnetic flux linkage induces EMF: e = −N dΦ/dt. A straight conductor of length ℓ moving at speed v perpendicular to field B gives e = Bℓv. Bigger B, faster change (or motion), and more turns all increase induced voltage.

CAAS SAR-66 Module 3 topic 3.11 Inductance/Inductor sits directly on the magnetism chapter. Topic 3.10 gave you poles, flux, B, H, and eddy currents. Topic 3.11 asks what happens electrically when flux through a circuit changes — or when a conductor moves so that it cuts flux. That induced voltage is the root of generators, transformers, ignition coils, and the inductor’s opposition to changing current.

Faraday’s Law (Statement and Formula)

Faraday’s Law of electromagnetic induction: an electromotive force (EMF) is induced in a circuit whenever the magnetic flux linkage of that circuit changes with time.

For a coil of N turns linked by flux Φ (webers):

e = −N (dΦ / dt)

or, with average values over a finite interval:

E_avg = −N (ΔΦ / Δt)

The minus sign is Lenz’s Law (polarity opposes the change) — covered in depth in §10.2. For magnitude questions on Module 3 you often compute |e| = N |dΦ/dt| and then assign polarity separately.

Worked example 1. A coil of 200 turns has flux rising from 0 to 0.004 Wb in 0.02 s (uniform change).

|E_avg| = N |ΔΦ/Δt| = 200 × (0.004 / 0.02) = 200 × 0.2 = 40 V.

Worked example 2. Same coil, same ΔΦ, but Δt = 0.01 s → |E_avg| = 200 × 0.4 = 80 V. Halving the time doubles the induced EMF — rate of change matters as much as the flux swing.

Flux linkage language (exam wording)

• Flux Φ through one turn.
• Flux linkage = NΦ when the same Φ links all N turns.
• Induced EMF is proportional to rate of change of flux linkage: e = −d(NΦ)/dt when N is constant.

If turns are in series aiding and share the same Φ, more turns → more volts for the same dΦ/dt.

Inducing Voltage in a Conductor Moving in a Magnetic Field

Faraday’s Law also covers motional EMF: a conductor moving so that it cuts magnetic flux lines has charges pushed along its length, producing a potential difference between the ends.

For a straight conductor of active length ℓ, moving at velocity v in a uniform field B, with B, ℓ, and v mutually perpendicular:

e = B ℓ v

If the motion is not perpendicular, only the component of velocity that cuts flux counts: e = B ℓ v sinθ, where θ is the angle between v and B (maximum when θ = 90°).

Worked example 3. B = 0.5 T, ℓ = 0.2 m, v = 10 m/s, all perpendicular.

e = 0.5 × 0.2 × 10 = 1.0 V.

Worked example 4. Same B and ℓ, but v = 40 m/s → e = 4.0 V. Speed up the cutting of flux → more induced voltage.

Generator picture (preview of machines)

A simple AC generator rotates a coil in a magnetic field so that each side of the coil cuts flux with a sinusoidal component of velocity relative to B. Instantaneous EMF follows Faraday: flux through the coil is Φ = Φ_max cos(ωt) (or sin), so e ∝ dΦ/dt yields a sine wave. Module 3.12–3.17 develop machines and AC waveforms; here you only need the induction mechanism.

Fleming’s right-hand rule (generator / induced EMF)

For a conductor moving in a field (conventional current / EMF direction):

This is the generator rule (right hand). Fleming’s left-hand rule remains the motor rule (force on a current-carrying conductor). Do not swap them on the exam.

Factors Affecting Induced Voltage

Module 3 expects you to list and apply the controlling factors. Whether you write Faraday’s form or the motional form, the physics is the same: more flux cut per second → more EMF.

Aircraft / hangar mental model. Pull a permanent magnet slowly past a coil → small meter kick. Snap the magnet past the same coil → larger kick (same ΔΦ, smaller Δt). Wind more turns on the coil → still larger kick. That is Faraday’s Law as a technician feels it.

Coil vs single wire

A transformer primary changing current changes core flux; secondary EMF follows Faraday with its own N_s. An inductor with changing self-current has changing self-flux — that is self-induction (§10.2).

Relating Flux Change to Circuit Behaviour

Any of these produce induced voltage:

1. Move a magnet relative to a coil (or move the coil relative to a magnet).
2. Vary current in a neighbouring coil so mutual flux changes (mutual induction).
3. Vary current in the same coil so its own flux linkage changes (self-induction).
4. Rotate conductors in a field (generator armature).

Eddy currents in solid cores (§9.3) are Faraday EMFs driving loops inside the metal — unwanted heating in transformers, sometimes useful as damping. Laminations interrupt those paths; the useful winding EMF is collected in insulated copper turns instead.

Units and Sign Sanity Checks

Common exam trap. A constant DC current through a coil that has already reached steady flux does not keep inducing voltage by Faraday; induction needs change. (At switch-on/off there is change — back EMF / kick — §10.3.)

Section Synthesis Table

Master Faraday’s Law and the three magnitude factors — field strength, rate of change of flux, and number of turns — before §10.2 names self and mutual inductance and locks polarity with Lenz’s Law.