Almost every watt of electricity in your home was born the same way: a coil of wire and a magnet moved past each other. The turbine in a power station — driven by steam, falling water or wind — spins magnets past coils, and out comes the current in your walls. The same physics runs in reverse and sideways all around you: the transformer on the pole outside steps the grid's voltage up and down; the induction cooktop heats a pan without ever touching it hot; the pad that wirelessly charges your phone pushes energy across a few millimetres of air; and the swipe-card reader, the electric guitar pickup, and the regenerative brake on an electric car all live off one single law.

That law is Faraday's law of induction, and stated fully it is astonishingly compact: a changing magnetic flux through a loop drives a voltage — an electromotive force (EMF) — around that loop. This page is the quantitative, first-year-undergraduate treatment. The school version tells you "a changing field makes a current"; here we pin down exactly how much, starting from the flux integral, deriving motional EMF two independent ways, and finishing with a coil's own reluctance to change — its inductance, and the energy stored in a magnetic field. One idea, followed all the way down.

Magnetic flux: how much field threads the loop

Before we can talk about a changing field we need a single number for "how much field is passing through this loop right now." That number is the magnetic flux \Phi_B. Picture the magnetic field \mathbf{B} as a bundle of field lines; the flux counts how many of those lines pierce the surface bounded by the loop. Formally it is a surface integral of the field's component perpendicular to the surface:

Here d\mathbf{A} is a tiny patch of the surface, pointing along its normal, and the dot product keeps only the part of \mathbf{B} that actually goes through the patch — a field skimming parallel to the surface threads nothing. When the field is uniform over a flat loop of area A, the integral collapses to a single product:

That \cos\theta is easy to forget and it is the key to the whole generator, so hold onto it: tilting a loop in a fixed field changes the flux through it even though neither the field nor the loop's area has moved.

Faraday's law: the rate of change of flux is an EMF

Faraday's experimental discovery, made precise, is that the EMF induced in a loop equals the rate at which the flux through it changes:

If instead of a single loop you wind N turns of wire, each turn sees the same flux and their EMFs add in series, so the whole coil delivers

Read that carefully. It is not the flux that drives the EMF — it is the rate of change of flux. A colossal but steady magnetic field pushed through a coil produces no EMF at all; a feeble field that is changing quickly can produce a large one. Since \Phi_B = BA\cos\theta, the product rule hands us three independent ways to make the flux change, and each is a real machine:

• Change the field B — switch a nearby current on or off, or move a magnet closer. This is the transformer, the induction cooktop and the wireless charger.
• Change the area A — slide a movable wire so the loop grows or shrinks. This is motional EMF, which we derive below.
• Change the angle \theta — spin the loop in the field. If it turns at a steady angular speed \omega so that \theta = \omega t, then \Phi_B = BA\cos(\omega t) and

That is the AC generator. A loop cranked round at constant speed puts out a sinusoidal voltage with peak value \mathcal{E}_0 = N B A\,\omega — the very shape of the mains supply. Notice the EMF is a sine while the flux is a cosine: the voltage peaks exactly when the flux is passing through zero and changing fastest, and it vanishes when the flux is momentarily at its maximum and standing still. The graph below lets you watch that quarter-cycle offset.

Lenz's law: what the minus sign is for

We have been carrying a minus sign in \mathcal{E} = -N\,d\Phi_B/dt without cashing it in. That sign is Lenz's law, and it is a statement of energy conservation dressed as a direction rule:

Work an example. Push the north pole of a bar magnet down towards a horizontal loop. The downward flux through the loop grows, so by Lenz the induced current flows to create an upward field inside the loop — which (right-hand rule) means the current runs anticlockwise seen from above. The loop's top face becomes a north pole and repels the incoming magnet. Pull the magnet back out and everything reverses: the current now flows the other way and attracts the magnet, trying to hold onto the flux it is losing.

Why must it be this way? Suppose the induced current instead helped the change — reinforced the growing flux. That stronger flux would drive a still larger current, which would strengthen the flux further, and you would have built a machine that makes electrical energy from nothing. Lenz's law is the universe refusing that free lunch: the induced current always fights the change, so you must do work against that opposition to keep the flux changing, and it is that work that becomes electrical energy. The minus sign is conservation of energy written in the language of circuits.

Motional EMF: the sliding rod, derived two ways

The cleanest case of a changing area is a conducting rod of length L sliding along two parallel rails at speed v, with a uniform field B pointing straight through the circuit (into the page below). Reveal the figure to see the rod sweep out fresh area as it moves.

Derivation 1 — the flux sweep. The circuit is the loop bounded by the two rails, the closed left end, and the moving rod. Its area grows as the rod advances. In a time dt the rod moves a distance v\,dt, sweeping out a thin strip of new area dA = L\,(v\,dt). With the field perpendicular to the loop (\theta = 0), the flux gained is d\Phi_B = B\,dA = B L v\,dt. Faraday's law then gives the magnitude of the EMF directly:

Derivation 2 — the force on the carriers. We can get the same answer without ever mentioning flux, using the magnetic force on a moving charge. Every free electron in the rod is being carried sideways at speed v through the field, so it feels a magnetic force of magnitude F = qvB directed along the rod. That force acts like a tiny battery, pushing charge to one end until the pile-up creates an electric field strong enough to balance it. The force per unit charge is vB, and the EMF is the work done per unit charge in carrying a carrier the full length L of the rod:

Two completely different pictures — geometry sweeping out area, and the Lorentz force on individual electrons — give the identical result \mathcal{E} = BLv. That agreement is no accident; it is one of the deep unifications that eventually pushed Einstein towards relativity.

If the rails are joined by a resistance R, this EMF drives a current I = \mathcal{E}/R = BLv/R. That current, flowing through the rod in the field, itself feels a force F = BIL = B^2L^2v/R — and by Lenz's law it points backwards, opposing the motion. To keep the rod moving at constant speed you must push against this drag, and the mechanical power you supply, P = Fv = B^2L^2v^2/R, equals exactly the electrical power \mathcal{E}^2/R dissipated in the resistor. Energy in, energy out — Lenz again.

Inductance: a coil's reluctance to change its own current

So far the changing flux has come from outside. But a current in a coil makes its own flux, and if that current changes, the coil induces an EMF in itself. The proportionality between a coil's current and the total flux linkage it produces is its self-inductance L:

Inductance is measured in henries, 1\ \text{H} = 1\ \text{Wb/A} = 1\ \text{V}\cdot\text{s/A}. The defining relation \mathcal{E} = -L\,dI/dt says an inductor resists changes in its current: try to ramp the current up quickly and it fights back with an opposing voltage. It is the electrical analogue of inertia. A large henry value means a coil that is stubborn about letting its current change.

For a long solenoid of N turns, length \ell and cross-section A — with n = N/\ell turns per metre — the interior field is B = \mu_0 n I, so the flux linkage N\Phi = N(\mu_0 n I)A gives

where V = A\ell is the volume enclosed. Inductance is pure geometry (and the material inside): it grows with the square of the turns, so doubling the winding quadruples the henries.

Finally, energy. Because the inductor opposes the rising current, you must do work to establish it, and that work is stored in the magnetic field. Integrating the power P = \mathcal{E}I = LI\,dI/dt as the current climbs from 0 to I gives

The energy is not "in the coil" so much as in the field — the same view that lets a magnetic field carry energy across empty space as light.

Worked examples

Example 1 — a changing field through a coil. A flat coil of N = 200 turns and area A = 4.0\times 10^{-3}\ \text{m}^2 sits square-on in a field that rises steadily from 0.10\ \text{T} to 0.50\ \text{T} in 0.20\ \text{s}. The angle is fixed at \theta = 0, so only B changes:

Example 2 — the sliding rod with a resistor. A rod of length L = 0.50\ \text{m} slides at v = 3.0\ \text{m/s} through a field B = 0.40\ \text{T}, with the rails closed by R = 2.0\ \Omega. The motional EMF, current, and retarding force are

Example 3 — peak voltage of an AC generator. A generator has N = 100 turns of area A = 0.020\ \text{m}^2 spinning at 50\ \text{Hz} (so \omega = 2\pi f \approx 314\ \text{rad/s}) in a field B = 0.25\ \text{T}. The peak EMF is

Example 4 — an inductor's back-EMF and stored energy. The current through a L = 0.30\ \text{H} inductor is ramped at dI/dt = 4.0\ \text{A/s}, reaching I = 2.0\ \text{A}. The self-induced EMF and the energy then stored are