Faraday's Law and Lenz's Law

Faraday's law simply says that whenever the amount of magnetic field passing through a loop of wire changes, a voltage appears around the loop and pushes current through it. It's named after Michael Faraday, who discovered the effect in 1831, and Lenz's law after Heinrich Lenz, who worked out which way the current goes. It's behind nearly all the electricity you use: every power-station generator is a coil turning in a magnetic field. The formula $\mathcal{E} = -N\,d\Phi_B/dt$ looks intimidating, but it's just $\mathcal{E}$ (the voltage that appears), $N$ (how many turns the coil has), $d\Phi_B/dt$ (how fast the magnetic field through the loop is changing) and the minus sign (the current pushes back against that change). It's one short sentence written in symbols; bear with it.

The experiment. Wire a coil to a sensitive meter and push a bar magnet into it. The needle kicks while the magnet moves, sits at zero while the magnet rests inside and kicks the other way as you pull it out. Move faster and the kick is bigger. Hold the magnet still and move the coil instead: the same kicks. A strong magnet sitting still does nothing at all. What matters is not the field but how fast the field threading the coil changes.

Magnetic flux. Measure "field threading the loop" exactly as electric flux measures field through a surface: $\Phi_B = \iint_S \mathbf{B}\cdot d\mathbf{A}$, a flux integral over any surface whose edge is the loop. For a uniform field through a flat loop, $\Phi_B = BA\cos\theta$, with $\theta$ the angle between $\mathbf{B}$ and the loop's normal. So there are exactly three ways to change it: change $B$, change $A$ or turn the loop.

Faraday's law. The EMF around the loop is the rate at which the flux falls:

A coil of $N$ turns is $N$ loops in series, each linking the same flux, so $\mathcal{E} = -N\,d\Phi_B/dt$. One volt is one weber per second.

Lenz's law: why the minus sign. The induced current makes its own magnetic field, and the minus sign says that field opposes the change in flux. Push a north pole into a coil and the coil's current makes a north pole facing it, repelling the magnet; pull it out and the coil makes a south pole that pulls back. It has to be this way. If the induced current helped the change, it would add flux, which would induce more current, which would add more flux: a runaway making energy from nothing. Instead you feel a resisting force, and the work you do pushing against it is exactly the electrical energy the coil delivers.

To find a direction: (1) which way does $\mathbf{B}$ thread the loop, and is that flux growing or shrinking? (2) The induced field points against a growth, or along with a shrinkage. (3) Point your right thumb along the induced field: your curled fingers give the current.

Motional EMF, two ways. A rod of length $L$ slides at speed $v$ along two rails in a field $B$ perpendicular to the rails' plane, closing a circuit. In time $dt$ the loop gains area $Lv\,dt$, so $d\Phi_B/dt = BLv$ and $\mathcal{E} = BLv$. Now look from inside the rod: each charge moves with the rod, so it feels the magnetic force $q\mathbf{v}\times\mathbf{B}$, of size $qvB$, pointing along the rod. The work per coulomb along the rod's length is $vBL$. Same answer: motional EMF is the magnetic force on the rod's charges.

With resistance $R$ in the circuit, a current $I = BLv/R$ flows, and the field pushes on that current with $F = ILB = B^2L^2v/R$ (forces on currents), against the motion, as Lenz demands. Keeping the rod moving takes power $Fv = B^2L^2v^2/R = I^2R$: every joule you put in comes out as heat in the resistor. Let go and the rod coasts to a stop. That's magnetic braking, and there's an animation further down that shows a rod doing exactly this.

Generators. Spin a flat coil of $N$ turns and area $A$ at angular speed $\omega$ in a uniform field. Its normal turns, $\theta = \omega t$, so

The EMF peaks at $NBA\omega$ exactly when the flux passes through zero: the coil is edge-on to the field and the flux is changing fastest. This sine is why the power grid runs on alternating current, and the AC circuits unit picks it up from here.

Faraday's law in field form. The EMF around a loop is the circulation of an electric field, $\mathcal{E} = \oint_C \mathbf{E}\cdot d\mathbf{l}$. For a loop that stays put,

Stokes' theorem turns the left side into $\iint_S(\nabla\times\mathbf{E})\cdot d\mathbf{A}$. The two surface integrals agree for every surface, so the integrands agree: $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$, one of Maxwell's equations. This is a real break from electrostatics. A static field has zero curl, zero circulation and a potential; an induced field circulates, so "the voltage between two points" now depends on the path taken, and no potential function exists.

The field around a changing solenoid. Inside a long solenoid of radius $a$ whose uniform field $B(t)$ is changing, symmetry says the induced $\mathbf{E}$ runs in circles around the axis, the same size all the way around. Apply Faraday's law to a circle of radius $r < a$: $E\cdot2\pi r = \pi r^2\frac{dB}{dt}$, so $E = \frac{r}{2}\frac{dB}{dt}$. Outside, the enclosed flux stops growing with $r$: $E\cdot2\pi r = \pi a^2\frac{dB}{dt}$, so $E = \frac{a^2}{2r}\frac{dB}{dt}$. There's an electric field out where there's no magnetic field at all. It answers to the flux the loop encloses, not to the field at the wire.

A common mistake. It's tempting to think a strong magnetic field through a loop must induce a big EMF, since the field is what does the inducing. It doesn't: a huge steady field induces nothing, because only the change in flux counts. The other usual slips are using the angle to the plane instead of to the normal, forgetting $N$, and getting Lenz backwards when the flux is shrinking. Then the induced field points the same way as $\mathbf{B}$, propping the flux up.

Every power station turns Faraday's law into electricity: steam, falling water or wind spins coils through magnetic fields, and $NBA\omega\sin\omega t$ comes out. An induction cooktop drives a rapidly changing field into the pan, and the induced eddy currents heat the metal directly. Wireless chargers and RFID tags pick up power from a neighboring coil's changing flux, and an electric guitar's pickup senses the vibrating steel string changing the flux through its coil. Regenerative braking runs an electric car's motor as a generator, turning the car's motion back into charge in the battery, with the Lenz force doing the braking.