Move a magnet, and a current is born from nothing. Step through it: the live panel on the right shows each idea as a real, everyday picture — rain through a hoop, a magnet diving into a coil, a transformer humming on a pole — while you read. Scroll down.
1 — Magnetic flux Φ = BA cosθ
Before induction, one idea: magnetic flux — how much magnetic field "pours through" a loop. Picture rain falling and a hoop held out in it. Face the hoop straight up at the shower and the most rain passes through; tilt it edge-on and almost none does.
Magnetic fluxΦ = B A cos θ
B = field strength (tesla) · A = loop area (m²) · θ = angle between B and the loop's normal
unit: weber (Wb) = T·m²
- θ = 0° (face-on) — cos 0 = 1, flux is maximum: field threads straight through.
- θ = 90° (edge-on) — cos 90 = 0, flux is zero: field skims past, none captured.
Why it matters: nothing is induced unless this Φ changes. Flux is the quantity every law in this chapter watches.
2 — Inducing a current
Faraday's famous experiment: a coil wired to a sensitive galvanometer, no battery anywhere. Push a bar magnet into the coil and the needle kicks. Pull it out — it kicks the other way. Hold the magnet still inside — nothing.
- Magnet moving in — flux through the coil rising → current one way.
- Magnet moving out — flux falling → current reverses.
- Magnet held still — flux constant → zero current, even though the field is strong.
The whole secret: it is the changing flux that drives the current, never the field by itself. Move the coil instead of the magnet and it works just the same.
3 — Faraday's law of induction
How big is the induced EMF? Faraday measured it: the faster you change the flux — and the more turns of wire the flux links — the larger the voltage. A lazy push barely moves the needle; a sharp plunge slams it across.
Faraday's lawEMF = − N (dΦ / dt)
N = number of turns · dΦ/dt = rate of change of flux
EMF in volts (V)
Two ways to get a bigger EMF: change the flux faster (a brisk movement, a stronger magnet) or add more turns N. The minus sign is Lenz's law, coming next.
worked — a 200-turn coil
Flux drops from 0.04 Wb to 0 in 0.1 s. EMF?
EMF = N·ΔΦ/Δt = 200 × 0.04 / 0.1 = 80 V
4 — Lenz's law & energy conservation
Which way does the induced current flow? Lenz's law: always the way that opposes the change that made it. Push a north pole toward the coil and the coil greets it with a north pole of its own — they repel, and you feel the magnet resist.
Lenz's law (the minus sign)EMF = − N (dΦ / dt)
induced current opposes the change in flux
This is simply energy conservation. If the coil helped the magnet in, you would get free energy forever. Instead you must do work against the push-back — and that work becomes the electrical energy of the induced current.
Everyday Lenz: drop a magnet down a copper pipe and it falls in slow motion — the induced currents fight its fall the whole way.
5 — Motional EMF
You don't even need a magnet to move. Slide a metal rod along two rails sitting in a magnetic field — like a window squeegee gliding across glass. The field pushes the rod's free electrons sideways; one end charges up +, the other −, and a voltage appears.
Motional EMFEMF = B L v
B = field (T) · L = rod length in the field (m) · v = speed (m/s)
Faster slide, longer rod, or stronger field → bigger EMF. Close the circuit and a current flows — this is the seed of every generator: it turns motion straight into voltage.
worked — a sliding rod
B = 0.5 T, L = 0.2 m, v = 4 m/s?
EMF = B·L·v = 0.5 × 0.2 × 4 = 0.4 V
6 — Self & mutual inductance
A coil reacts even to its own changing current. As its field grows or collapses, the flux through its own turns changes — so it induces a voltage in itself that opposes the change. That sluggishness is self-inductance, L.
Self & mutual inductanceself: EMF = − L (dI / dt)
mutual: EMF₂ = − M (dI₁ / dt)
L, M in henry (H)
- Self-inductance L — a coil opposes changes in its own current; it stores energy in its magnetic field.
- Mutual inductance M — a changing current in coil 1 induces an EMF in a nearby coil 2, with no wires between them.
This is the bridge: mutual inductance — energy crossing the air gap from one coil to another — is exactly how a transformer works.
7 — The transformer
Wrap two coils on the same iron ring. Send AC into the primary; its changing flux runs round the core and threads the secondary, inducing a voltage there. The turns ratio sets the trade.
Transformer equationVₛ / Vₚ = Nₛ / Nₚ
ideal: Vₚ Iₚ = Vₛ Iₛ (power in = power out)
- Step-up — more turns on the secondary (Nₛ > Nₚ) → higher voltage, lower current.
- Step-down — fewer secondary turns → lower voltage, higher current (your phone charger).
worked — pole transformer
Vₚ = 11000 V, Nₚ = 5000, Nₛ = 100?
Vₛ = Vₚ·Nₛ/Nₚ = 11000 × 100/5000 = 220 V
Why AC? Transformers need a changing flux, so they only work on AC — that is why the grid is alternating current.
8 — The AC generator
Run motional EMF in a circle. Spin a coil steadily between magnet poles and the flux through it rises, falls, reverses and rises again — so the induced EMF swings up and down as a sine wave. This is the generator (dynamo) behind every power station.
Generator EMFEMF = N B A ω sin(ωt)
peak EMF, EMF₀ = N B A ω · ω = angular speed (rad/s)
The EMF is largest when the coil's plane is along the field (cutting lines fastest) and zero when face-on (momentarily not cutting any). Whether driven by steam, falling water or wind, every generator turns rotation into this AC sine.
9 — Recap & applications
Every effect in this chapter springs from one sentence: a changing magnetic flux induces an EMF. Faraday gives its size, Lenz gives its direction, and energy is always conserved.
- Induction cooktop — a coil under the glass makes a changing flux that drives eddy currents in the steel pan; the pan's own resistance heats the food.
- Bicycle dynamo & power stations — a spinning magnet or coil = an AC generator lighting the lamp or the grid.
- Guitar pickups — a vibrating steel string changes the flux through a coil, inducing the tiny signal your amplifier hears.
The chapter in five linesΦ = B A cos θ
EMF = − N dΦ/dt (Faraday + Lenz)
motional: EMF = B L v
transformer: Vₛ/Vₚ = Nₛ/Nₚ
generator: EMF = N B A ω sin ωt