The full, readable lecture — what magnetic flux means, how a moving magnet makes electricity, Faraday's law and Lenz's law, the EMF of a sliding rod, self and mutual inductance behind the transformer, the AC generator, and the everyday machines all of this powers. As you scroll, the panel on the right plays out each idea with an everyday object you already know — rain through a hoop, a magnet pushed into a coil, a sliding rod, a bicycle dynamo.
1 — Magnetic flux
Magnetic flux measures how many magnetic field lines thread through a surface. Picture rain falling straight down onto a hoop: hold the hoop flat and the most drops pass through; tilt it and fewer come through; turn its rim edge-on to the rain and none pass at all. Magnetic field lines behave the same way through a coil.
Magnetic fluxΦ = B A cos θ
B = flux density (tesla, T) · A = area of the loop · θ = angle between B and the normal to the loop
- Magnetic flux (Φ) — the product of the field and the area it passes through perpendicularly. Unit: the weber (Wb), where 1 Wb = 1 T·m².
- θ = 0° — the loop faces the field square-on, cos θ = 1, flux is maximum (Φ = BA).
- θ = 90° — the loop lies along the field, cos θ = 0, flux is zero.
- Flux density (B) — flux per unit area, B = Φ/A; this is the strength of the field itself.
Exam point: flux is changed in three ways — change B, change the area A, or change the angle θ. Every kind of induction in this chapter is just one of these three at work.
2 — Electromagnetic induction
Electromagnetic induction is the production of an EMF (and a current) in a circuit by a changing magnetic flux. The classic demonstration: connect a coil to a sensitive galvanometer and push a bar magnet into it. The needle kicks to one side while the magnet moves in, returns to zero when it stops, and kicks the other way as you pull it out.
- A current flows only while the flux is changing — a moving magnet, not a still one.
- Move the magnet faster and the needle kicks further — the EMF is bigger.
- More turns on the coil, or a stronger magnet, also gives a bigger kick.
- It does not matter what moves — push the magnet or move the coil; only the relative change counts.
Michael Faraday (1831) discovered this. No battery is in the circuit — the moving magnet itself is the source of the electrical energy, converted from the mechanical work you do pushing it.
3 — Faraday's law of induction
Faraday's law puts a number on the kick: the induced EMF is proportional to the rate of change of flux through the coil, multiplied by the number of turns. It is the change that matters, not the flux itself — a strong but steady field induces nothing.
Faraday's lawε = −N (dΦ / dt)
N = number of turns · dΦ/dt = rate of change of flux (Wb/s) · ε in volts (V)
- Plunge the magnet in twice as fast → dΦ/dt doubles → the EMF doubles.
- Double the number of turns N → the EMF doubles.
- The minus sign comes from Lenz's law (next section) — it fixes the direction.
Faraday's law — quick numerical
A 200-turn coil has its flux change from 0 to 0.05 Wb in 0.1 s. Find the average EMF.
ε = N ΔΦ/Δt = 200 × (0.05 / 0.1) = 200 × 0.5 = 100 V
4 — Lenz's law & energy conservation
Lenz's law tells us the direction of the induced current: it always flows so as to oppose the change in flux that produced it. Push a magnet's north pole towards a coil and the near face of the coil becomes a north pole too, pushing back against the incoming magnet. Pull the magnet away and the coil turns into a south pole, trying to hold the magnet back.
Why the minus sign in Faraday's lawε = −N (dΦ / dt)
the minus sign is Lenz's law — the induced effect fights the change
- Lenz's law is a direct statement of the conservation of energy.
- Because the coil pushes back, you must do work to move the magnet — that work becomes electrical energy.
- If the induced current helped the change, you would get free energy from nothing — impossible.
Everyday proof: drop a magnet down a copper pipe and it falls in slow motion. The changing flux induces eddy currents in the pipe whose magnetic field opposes the fall — Lenz's law braking the magnet.
5 — Motional EMF
When a conducting rod slides along two rails inside a magnetic field, it sweeps across field lines and a steady EMF appears across its ends. The free electrons in the rod feel a magnetic force and pile up at one end, making that end negative — the rod becomes a little battery for as long as it keeps moving.
Motional EMFε = B L v
B = flux density (T) · L = length of the rod in the field (m) · v = speed of the rod (m/s)
- This is Faraday's law in disguise: the rod sweeps area A = L·(v·t), so dΦ/dt = B·L·v.
- The EMF is bigger with a stronger field, a longer rod, or a faster slide.
- If the rails complete a circuit, an induced current flows and (by Lenz's law) a force opposes the motion — you must keep pushing.
motional EMF — quick numerical
A 0.5 m rod slides at 4 m/s through a 0.3 T field. Find the EMF.
ε = B L v = 0.3 × 0.5 × 4 = 0.6 V
6 — Self & mutual inductance
A coil's own changing current makes a changing flux that induces an EMF back in itself — this is self-inductance, measured by L. Place a second coil alongside it and the changing flux links the neighbour too, inducing an EMF there as well — mutual inductance, measured by M. This is the heart of every transformer.
Self & mutual inductanceεself = −L (dI / dt) · εmutual = −M (dI₁ / dt)
L, M measured in henry (H) · 1 H = 1 V·s/A
- Self-inductance (L) — a coil opposes changes in its own current; it acts like electrical inertia.
- Mutual inductance (M) — a changing current in coil 1 induces an EMF in nearby coil 2 without any wire between them.
- Transformer — two coils on one iron core; the turns ratio sets the voltage: V₂/V₁ = N₂/N₁.
Note: a transformer only works on AC — it needs a continuously changing current to keep the flux changing. A steady DC current gives no mutual EMF once it has settled.
7 — The AC generator (dynamo)
Rotate a coil steadily inside a magnetic field and the flux through it varies as cos θ = cos ωt, so by Faraday's law the induced EMF varies as sin ωt — a smooth alternating voltage. A bicycle dynamo is exactly this: the wheel spins a small magnet past a coil and lights your lamp with home-made AC.
EMF of an AC generatorε = ε₀ sin ωt · ε₀ = N B A ω
ε₀ = peak EMF · ω = angular speed of rotation (rad/s)
- The EMF is zero when the coil is flat-on to the field (flux is maximum but not changing).
- The EMF is maximum when the coil is edge-on (flux is zero but changing fastest).
- Spin faster (bigger ω), use more turns, a stronger field, or a bigger coil → bigger peak EMF.
Generator vs motor: a generator turns motion into electricity; a motor runs the same machine backwards, turning electricity into motion. Power stations are just enormous generators spun by steam, water or wind.
8 — Applications & exam recap
Electromagnetic induction quietly powers modern life. The same changing-flux idea appears in every device below.
| Device | How induction does the work |
|---|
| Transformer | changing current in one coil induces a different voltage in another (mutual inductance) |
| Generator / power station | a coil spun in a field gives sinusoidal AC (Faraday's law) |
| Induction cooktop | an AC coil induces eddy currents in the pan, heating it directly |
| Electric guitar pickup | a vibrating steel string changes the flux through a coil, inducing the signal |
| Metal detector & transformer-meter | induced eddy currents reveal hidden metal |
- Magnetic flux: Φ = B A cos θ, measured in webers.
- Induction = EMF from a changing flux (move a magnet near a coil).
- Faraday's law: ε = −N dΦ/dt — faster change, bigger EMF.
- Lenz's law: the induced current opposes the change (energy conservation).
- Motional EMF: ε = B L v for a rod sweeping a field.
- Self-inductance L and mutual inductance M (henry) → the transformer.
- AC generator: ε = ε₀ sin ωt, with ε₀ = N B A ω.