FARADAY'S LAW OF INDUCTION

Every equation in this branch so far has been a photograph. Gauss and Ampère say what a static charge or steady current does right now; time never appears. In 1831, in the basement of the Royal Institution, Michael Faraday ran a cheap experiment that broke the stillness.

He wound two coils on opposite sides of an iron ring. The first, the primary, could be switched to a battery. The second, the secondary, ran to a galvanometer. Close the switch: the needle snapped over, then drifted back to zero. Open it: the needle snapped the other way. Leave the switch closed with steady current flowing: nothing.

That was the discovery. Not the current — the change in it. Push a bar magnet into a coil: needle twitches. Hold it still: nothing. Rotate a loop in a static field: needle. One rule governs all of it. The American Joseph Henry had found the same phenomenon the year before but published late; the law took Faraday's name and Henry got the SI unit.

To write the rule down we need one ingredient: the magnetic flux through a loop.

Pick a closed wire. Imagine a soap film filling it in — any surface whose boundary is the wire. Walk a little area element dA across the film, with a vector pointing perpendicular to it. The flux contribution from that bit is the dot product of the local magnetic field with dA: B · dA. Add up every contribution over the film; that total is the flux.

The symbol for "add over a surface" is the double integral ∬. Read ∬ B · dA as "the surface integral of B over the film" — total field threading through. The film's shape doesn't matter, only its rim: a flat disk and a bulging hemisphere sharing the same boundary give the same answer.

Units are teslas times square metres — collectively called webers (Wb). For a uniform B through a flat loop of area A whose normal makes angle θ with B, the integral collapses to Φ = B·A·cos θ. If the loop has N turns wound so each sees the same flux, the total flux linkage is N·Φ — and every turn contributes independently to the EMF.

Here is Faraday's law in plain words. The electromotive force around a closed loop — the voltage a test charge would gain going once around — equals minus the rate at which the flux through the loop is changing.

Two pieces need unpacking. "d/dt" is calculus shorthand for "how much is this quantity changing per second right now" — zero if Φ is constant, big if it whips around, negative if it is falling. EMF — electromotive force — is the line integral of E around the loop: the work E does per unit charge on a charge shuttled once around.

The minus sign is not a bookkeeping choice. It is Heinrich Lenz's 1834 observation: the induced EMF drives a current whose magnetic field opposes the flux change that caused it. The next topic, Lenz's law, unpacks this fully.

The cleanest thing about EQ.02 is what it doesn't care about — why Φ is changing. Move the magnet. Move the loop. Change a neighbouring current. Deform the loop. Tilt it. All of these are one equation.

Faraday's list reduces to three moves: change the field, change the area, change the angle.

Push the bar magnet toward the coil and the flux through each turn rises. dΦ/dt is positive; EMF is negative; the galvanometer deflects one way. Pull it out and the needle flips. Hold it still — zero. The faster you move, the bigger the deflection, exactly because EMF scales with how fast Φ is changing, not how large it is.

Here the field is uniform and constant — a grid of into-the-page crosses. The loop is a rectangle with a sliding bar on rails. Slide the bar right: area grows, Φ grows, dΦ/dt is positive, EMF is negative. Lenz says the induced current must oppose the increase, so it circulates counter-clockwise (viewed from the reader) — making a little out-of-page field inside the loop that cancels part of the growing into-page flux. Reverse v and everything flips.

The third move — changing the angle — is the heart of every AC generator: a loop rotated at angular frequency ω in a uniform field B gives Φ(t) = B·A·cos(ωt), and EQ.02 gives EMF(t) = B·A·ω·sin(ωt). That sinusoid is what comes out of every wall socket on Earth.

Faraday built the first continuous DC generator in 1831 from a copper disk, a hand crank, and a permanent magnet.

The disk sits between the poles of a magnet so B is perpendicular to the disk and uniform across it. Spin it at angular speed ω. A charge in the disk at radius r moves at v = ω·r; the Lorentz force qv × B on that charge is radial — pushing positive charge outward or inward depending on signs. Integrate the motional EMF from axle to rim and:

Unlike a rotating-loop generator, this output is steady DC — the geometry has rotational symmetry, so the axle-to-rim voltage does not oscillate. Plug in B = 1 T, ω = 100 rad/s, R = 0.1 m: 0.5 V. Small voltage, but a homopolar generator delivers colossal currents, which is why modern variants power rail guns and ship drives.

EQ.02 is a statement about a loop. Like Ampère's and Gauss's laws, Faraday's law has a local version — an equation that holds at every point, with no loops or surfaces in sight.

Ask at a point: how much does E circulate around the point itself? Imagine a tiny paddlewheel with an electric-field rotor; the curl of E, written ∇×E, tells you how fast it spins and about which axis. Straight-line flow has zero curl; a field that winds around a point has non-zero curl.

Faraday's law in differential form says the curl of E equals minus the time rate of change of B at that point:

In plain words: wherever the magnetic field is changing in time, an electric field curls around that place. Where B is steady, ∇×E is zero and E looks like the old gradient-of-a-potential field from §02.

Pair this with the local Ampère law ∇×B = μ₀ J from §03 and you have two of the four Maxwell equations in their dynamic form. The two curl equations are what turn electromagnetism into a theory with time in it.

In §07 we'll find that Ampère's law is almost right — missing a term proportional to ∂E/∂t that Maxwell will add. Together with EQ.04 that correction makes self-propagating waves of E and B. But that is the next module's door; we keep it closed for now.

Every generator on Earth runs on EQ.02. Coal, gas, hydro, nuclear, wind — the energy source differs, but the end of every chain is the same: spin a coil in a field, extract −dΦ/dt as AC EMF. The whole electrical grid is Faraday's 1831 experiment, scaled up.

Transformers chain two Faraday rules back-to-back: AC in the primary makes time-varying flux in an iron core, which threads a secondary whose EMF is N₂/N₁ times the primary's.

Induction cooktops run 20–60 kHz AC through a coil under a ceramic plate. A ferromagnetic pan on top sees time-varying flux; EQ.02 induces eddy currents in its base, which heat the pan directly — the cooktop itself stays cool.

Contactless chargers (Qi pads) are transformers with an air gap. Magnetic-stripe readers and RFID tags decode the microvolt EMF induced in a pickup coil as a stripe or chip moves past. Anti-theft tags flood the doorway with time-varying B; a tagged item's resonant loop re-radiates and triggers the alarm. All Faraday.

In FIG.22 we stare at the minus sign in EQ.02 — Lenz's contribution — and see how "oppose the change" gives us magnetic braking, the jumping ring, and every regenerative-brake system on a moving vehicle.