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Electromagnetic induction is the process by which a changing magnetic field produces (induces) an e.m.f. in a conductor. It was discovered by Michael Faraday in 1831, and it powers the entire modern world: every generator, every transformer, every wireless charger, every induction cooker, every metal detector, every RFID reader, every electric guitar pick-up, every eddy-current brake — all are applications of the two laws we meet in this lesson, Faraday's law and Lenz's law.
This lesson continues OCR H556 Module 6.3.
Before we can state Faraday's law we need one more quantity: magnetic flux, symbol Φ (Greek phi). Intuitively, magnetic flux is a measure of the total "amount" of magnetic field passing through a surface.
For a flat surface of area A in a uniform field B making an angle θ between the field and the normal to the surface,
Φ = BA cos θ
The unit is the weber (Wb): 1 Wb = 1 T m².
Two special cases to commit to memory:
θ = 0°): Φ = BA — the maximum.θ = 90°): Φ = 0 — no flux links through the loop.graph LR
A[Area A] -->|normal n| N((B at angle θ to n))
Watch the angle!
θis the angle between the field and the normal to the surface, not between the field and the surface itself. If the field lies in the plane of the loop,θ = 90°andΦ = 0. Mixing this up is the most common flux error at A-Level.
For a coil of N turns, each turn is linked by the same flux Φ. The total "flux linkage" is
Flux linkage = NΦ
Flux linkage also has units of weber (some textbooks write "weber-turns" for clarity). It is the quantity that appears in Faraday's law for a coil.
A circular coil of 50 turns has radius 8.0 cm. It is placed in a uniform magnetic field of 0.12 T. Calculate the flux linkage when (a) the field is perpendicular to the plane of the coil, (b) the field is parallel to the plane of the coil, (c) the field is at 30° to the plane of the coil.
(a) Field perpendicular to coil — this means the field is parallel to the normal, so θ = 0° and cos θ = 1.
A = π r² = π × (0.08)² = 2.011 × 10⁻² m²
Φ = BA = 0.12 × 2.011 × 10⁻² = 2.413 × 10⁻³ Wb
NΦ = 50 × 2.413 × 10⁻³ = 0.121 Wb
(b) Field parallel to coil — field in the plane of the coil means the angle between field and normal is 90°. cos 90° = 0, so NΦ = 0.
(c) Field at 30° to the plane of the coil — careful! "30° to the plane" means 60° to the normal, so θ = 60° in the formula.
NΦ = NBA cos θ = 50 × 0.12 × 2.011 × 10⁻² × cos 60°
= 0.121 × 0.5
= 0.060 Wb
Faraday observed that whenever the flux linking a coil changes, an e.m.f. is induced in the coil. Quantitatively, the induced e.m.f. is equal to the rate of change of flux linkage:
ε = −d(NΦ)/dt
The minus sign is Lenz's law (see next section). The magnitude is the rate of change of flux linkage in webers per second (= volts).
Any way of changing the flux linkage will induce an e.m.f.:
cos θ with time. (OCR does NOT require the derivation of rotating-coil EMF at A-Level, but does expect you to know that rotating a coil is one way to induce an e.m.f.)Faraday's law is the fundamental principle behind all generators and transformers.
Lenz's law fixes the direction of the induced current: the induced current flows in the direction that opposes the change in flux which caused it.
Lenz's law is a direct consequence of conservation of energy. If the induced current flowed in the same direction as the change, then the change would accelerate and we would get something for nothing — perpetual motion. Instead, the induced current creates a field that opposes the change, and an external agent must do work to maintain the change, keeping energy conservation intact.
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