11.5 Faraday’s law and electromagnetic induction

So far electric and magnetic fields have been separate: charges make $\mathbf{E}$ , currents make $\mathbf{B}$ . Faraday’s discovery is that the two are coupled the moment anything changes in time — a changing magnetic field creates an electric field. This is the third Maxwell equation, and the principle behind every generator, transformer, and inductive sensor.

The law of induction

A changing magnetic flux through a loop drives an electromotive force — a voltage — around it:

\varepsilon \;=\; -\frac{d\Phi_B}{dt}, \qquad \Phi_B = \int \mathbf{B}\cdot d\mathbf{A}.

where

$\varepsilon$: induced electromotive force V
$\Phi_B$: magnetic flux through the loop Wb
$\mathbf{B}$: magnetic field T

The flux can change three ways — a changing field strength, a changing loop area, or a changing orientation — and all three induce a voltage. In differential form the law becomes the third Maxwell equation,

\nabla\times\mathbf{E} \;=\; -\frac{\partial\mathbf{B}}{\partial t},

which says a time-varying magnetic field produces a circulating electric field — one that is not the gradient of any potential, unlike the electrostatic field of the earlier lessons.

Mean B = 1.00 T Loop area A = 0.50 m² Tilt θ = 0°

Faraday's law: ε = −dΦ_B/dt. A changing magnetic flux through a loop induces an EMF equal to the time-derivative of the flux, with a minus sign (Lenz's law: the induced current opposes the change). This is the operative principle of electric generators, transformers, and induction stoves. In integral form, ∮ E·dℓ = −∂Φ_B/∂t over any closed loop — the third of Maxwell's equations.

Lenz’s law and energy conservation

The minus sign is Lenz’s law: the induced current flows in the direction whose own magnetic field opposes the change that produced it. Push a magnet toward a loop and the induced current repels it; pull it away and the current attracts it back. Either way the induced effect resists the motion, so an external agent must do work to keep the flux changing — and that work is exactly the electrical energy delivered to the circuit. The minus sign is energy conservation written into the field equation; without it, an induced current would reinforce its own cause and generate energy from nothing.

This single principle runs the electrical world. A generator turns a loop in a magnetic field, converting mechanical work to electrical energy through the changing orientation. A transformer couples two coils so that a changing current in one induces a voltage in the other, trading voltage for current. Induction heats metals, brakes trains, and charges devices across an air gap. With Faraday’s law the fields are coupled in one direction — changing $\mathbf{B}$ makes $\mathbf{E}$ — and the final lesson supplies the missing reciprocal coupling that closes the equations and sets the fields free as waves.