1Magnetic Flux

Before talking about induction, we need the same idea used in Gauss’s Law, but for magnetic fields: how much field “passes through” a surface.

Magnetic Flux
Φ_B = B · A = BA cos θ
Unit: weber (Wb), where 1 Wb = 1 T·m². θ is the angle between B and the area vector (normal to the surface).

2Faraday’s Laws of Electromagnetic Induction

Faraday’s central discovery: a changing magnetic flux through a circuit induces an EMF in it — the field itself doesn’t need to touch the wire, and no battery is required. Move a magnet near a coil, or change the current in a nearby circuit, and a voltage appears.

Faraday’s Law
ε = − N dΦ_B / dt
N = number of turns in the coil. The induced EMF is proportional to the rate of change of flux — a steady, unchanging flux, however large, induces nothing at all.

3Lenz’s Law

The minus sign in Faraday’s Law isn’t just bookkeeping — it encodes an entire physical law of its own. Lenz’s Law states that the induced current always flows in a direction that opposes the change in flux that produced it.

Why It Must Be True
Lenz’s Law is really a statement of energy conservation. If the induced current instead aided the change in flux, it would reinforce itself, growing without bound and creating energy from nothing — clearly impossible. Opposition is what makes work necessary to sustain the change, and that work is exactly what becomes electrical energy.

4Motional EMF

A very common and intuitive case: slide a conducting rod along rails in a magnetic field, and an EMF appears because the free charges in the moving rod experience a magnetic force (qv×B), which pushes them along the rod exactly like a battery would.

Motional EMF (rod moving perpendicular to B and to itself)
ε = B l v
l = length of the rod, v = its speed. This is really just Faraday’s Law in disguise — as the rod moves, it sweeps out area, changing the enclosed flux at rate Blv.

5Energy Consideration: A Quantitative Study

If the moving rod is part of a closed circuit with resistance R, the induced EMF drives a current, and that current experiences a magnetic force opposing the rod’s motion (Lenz’s Law in action) — so an external agent must do work to keep the rod moving at constant speed.

Induced Current
I = Blv / R
Power Delivered by the External Agent
P = Fv = B²l²v² / R
This exactly equals I²R, the rate of heat dissipation in the resistor — confirming that mechanical work put in is fully converted to electrical energy, with nothing gained or lost. A clean, worked demonstration of energy conservation.

6Eddy Currents

Induction isn’t limited to wires — any bulk conductor moving through (or sitting in a changing) magnetic field develops swirling loops of induced current within its volume, called eddy currents.

• Useful applications: magnetic braking (eddy currents oppose motion, providing smooth, contact-free braking), induction furnaces (eddy currents heat the material directly), speedometers.
• Unwanted effects: eddy currents in transformer cores and motor armatures cause wasteful heating. This is minimized by laminating the core — using thin, insulated sheets instead of a solid block, which breaks up the current loops and raises resistance to eddy current flow.

7Inductance

Mutual Inductance

When current in one coil changes, it changes the flux through a second, nearby coil, inducing an EMF in the second coil — even though they’re not electrically connected.

Mutual Inductance
Φ₂ = M I₁, so ε₂ = −M dI₁/dt
M depends only on the geometry of the two coils and their relative positioning. Unit: henry (H). This is exactly how a transformer works.

Self-Inductance

A coil’s own changing current changes its own flux, inducing a “back EMF” in itself that opposes the change — a kind of electrical inertia.

Self-Inductance
Φ = L I, so ε = −L dI/dt
L for a long solenoid: L = μ₀N²A/l. Self-inductance is why current in an inductor can’t change instantaneously — it always resists sudden changes, the electrical analogue of mass resisting acceleration.
Energy Stored in an Inductor
U = ½ L I²
Stored in the magnetic field itself — the magnetic analogue of ½CV² for a capacitor’s electric field.

8AC Generator

An AC generator converts mechanical rotation into an alternating EMF by spinning a coil inside a magnetic field, continuously changing the flux through it.

Instantaneous EMF
ε = ε₀ sin(ωt) = N B A ω sin(ωt)
N = turns, B = field strength, A = coil area, ω = angular speed of rotation. Peak EMF ε₀ = NBAω occurs when the coil’s plane is parallel to B (flux changing fastest); EMF is zero when the coil’s plane is perpendicular to B (flux at its momentary maximum, but momentarily unchanging).

Formula Summary

Magnetic FluxΦ = BA cosθ
Faraday’s Lawε = −N dΦ/dt
Motional EMFε = Blv
Induced CurrentI = Blv/R
Power (Energy Check)P = B²l²v²/R
Mutual Inductanceε₂ = −M dI₁/dt
Self-Inductanceε = −L dI/dt
Solenoid LL = μ₀N²A/l
Energy StoredU = ½LI²
AC Generator EMFε = NBAω sinωt

Solved Examples

Example 1 · Motional EMF

A rod of length 0.5 m moves with a speed of 4 m/s, perpendicular to both its length and a uniform magnetic field of 0.2 T. Find the EMF induced across the rod.

Solution: ε = Blv = 0.2 × 0.5 × 4 = 0.4 V.

Example 2 · Faraday’s Law

A coil of 100 turns and area 0.02 m² is held with its plane perpendicular to a magnetic field that increases uniformly from 0.1 T to 0.5 T in 0.2 s. Find the magnitude of the induced EMF.

Solution: ε = N·A·(dB/dt) = 100 × 0.02 × [(0.5 − 0.1)/0.2] = 100 × 0.02 × 2 = 4 V.

Quick Check

1. Lenz’s Law is fundamentally a statement of the conservation of:
✓ Energy (correct)
Charge
Momentum
Magnetic flux
2. Laminating a transformer core is done primarily to reduce:
Mutual inductance
Self-inductance
✓ Eddy current losses (correct)
Magnetic flux
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