Faraday's Law of Induction — How Changing Magnetism Makes Electricity

The discovery that powers the world

In 1831, Michael Faraday — a bookbinder's apprentice with almost no mathematics — discovered the principle behind the modern world: changing magnetism creates electricity.

Every power station, from hydroelectric dams to wind turbines, is Faraday's discovery at industrial scale: mechanical energy spins coils in magnetic fields, and changing flux becomes the voltage in your wall socket.

Magnetic flux: Φ = BAcosθ

Magnetic flux ($\Phi$) measures how much magnetic field threads through an area:

$$\Phi = BA\cos\theta$$

• $B$: field strength
• $A$: area of the loop
• $\theta$: angle between the field and the area's normal

Flux changes if any of the three change: stronger/weaker field, bigger/smaller area, or rotation. Faraday's law cares only about the change — never the static value.

Faraday's law

Faraday's law of induction:

$$\mathcal{E} = -N\frac{\Delta\Phi}{\Delta t}$$

• $\mathcal{E}$: induced EMF (voltage)
• $N$: number of coil turns — each turn contributes
• $\Delta\Phi/\Delta t$: how fast the flux changes

Fast changes and many turns mean big voltage. This is why generators spin coils rapidly inside strong fields with thousands of turns.

Lenz's law: the meaning of the minus sign

The minus sign is Lenz's law: the induced current flows in the direction that opposes the change producing it.

Push a north pole toward a coil → the coil becomes a north pole facing you, repelling the magnet. You're doing work against that repulsion — and that work becomes the electrical energy. Pull the magnet away → the coil flips polarity to attract it, again resisting your motion.

Without Lenz's opposition, pushing a magnet would generate current for free — a perpetual motion machine. The minus sign is energy conservation, written in calculus.

Generators: the law in action

A generator is Faraday's law rotating: spin a coil at angular speed $\omega$ in a uniform field $B$. The flux $\Phi = BA\cos(\omega t)$ oscillates, and its time derivative gives:

$$\mathcal{E} = NBA\omega\sin(\omega t)$$

Sinusoidal EMF — alternating current, straight from the mathematics. Grid frequency (50/60 Hz) is just the spin rate; transformers then step this AC up for efficient long-distance transmission (high V, low I, minimal $I^2R$ loss).

Worked mini-example

A 200-turn coil of area $0.05\ \mathrm{m^2}$ sits perpendicular to a field that rises from $0.1\ \mathrm{T}$ to $0.5\ \mathrm{T}$ in $0.2\ \mathrm{s}$. Find the induced EMF.

Flux change: $\Delta\Phi = A\Delta B = 0.05 \times 0.4 = 0.02\ \mathrm{Wb}$.

$$|\mathcal{E}| = N\frac{\Delta\Phi}{\Delta t} = 200 \times \frac{0.02}{0.2} = 20\ \mathrm{V}$$

The induced EMF is 20 V (direction given by Lenz's law: opposing the field increase). Note the method: flux change first, then divide by time, then multiply by turns.

Variables and units

Common mistakes

Real-world applications

• Every power station on Earth — coal, gas, nuclear, hydro, wind — ends with Faraday's law: spinning coils in magnetic fields generating the grid's electricity.

• Transformers step voltage up for transmission and down for homes using changing flux in shared iron cores.

• Induction cooktops drive alternating current through a coil; the changing field induces eddy currents directly in the pan — the cooktop itself stays cool.