Magnetism and induction

A motor spins your fan. A pad charges your phone through a gap of air. An induction cooktop boils water in a pan while the glass beside it stays cool. All three run on a single link between electricity and magnetism - and the link has one strict rule that decides when it works at all.

You already know magnets: two bar magnets snap together one way and shove apart the other, just like charges, but with poles instead of plus and minus. For centuries magnetism and electricity looked like separate subjects. They are not. The hinge fact, discovered by experiment, is this: a moving charge makes a magnetic field, and a changing magnetic field pushes charges. Electricity in motion is magnetism.

The answer surprises almost everyone: none. A strong steady field sitting in the coil induces nothing. What induces a current is the field changing. Shove the magnet in and the meter kicks one way; pull it out and it kicks the other; park it anywhere and hold still, and the meter falls to zero no matter how close the magnet is. Only motion, only change, counts.

Play with it: only the change counts

Drag the magnet past the coil below, or press auto-sweep. Watch the meter needle: it swings only while the magnet is moving, and it swings the opposite way going in versus coming out. Then switch to the Motor tab and run a current the other direction to make the loop spin - the same link, run in reverse.

Flux, and Faraday's law

To make "the field through the coil" precise we need one idea: magnetic flux, written $\Phi$. Think of it as how many field lines thread through the loop. A strong magnet near a big coil means a lot of flux; far away, little. The experimental law tying flux to voltage is Faraday's law (a stated result^[1]):

Read it slowly. The induced voltage $\mathcal{E}$ is not proportional to the flux; it is proportional to how fast the flux is changing. That is the rate-of-change, the derivative again. A constant flux, however large, has zero rate of change, so it induces nothing - exactly what the widget shows when you hold the magnet still. The minus sign encodes Lenz's law: the induced current fights the change that made it, which is why the needle flips direction between pushing in and pulling out.

def induced_emf(flux_samples, dt):
    # flux_samples: the flux through the coil at each time step
    emf = []
    for i in range(1, len(flux_samples)):
        d_phi = flux_samples[i] - flux_samples[i - 1]
        emf.append(-d_phi / dt)     # Faraday: EMF = -dPhi/dt
    return emf

# magnet held still -> flux constant -> every difference is 0 -> no current
print(induced_emf([0.9, 0.9, 0.9, 0.9], 0.01))  # [0.0, 0.0, 0.0]
# magnet moving in -> flux rising -> nonzero, one sign the whole way
print(induced_emf([0.2, 0.5, 0.8, 0.95], 0.01)) # negative values (Lenz)

The same link, six devices

Every device in the anchor is one law used two ways. A motor runs it forward: send a current through a coil in a magnet's field and each wire feels a sideways force, so the coil spins. A generator runs it backward: spin a coil in a field and the changing flux induces a current - a motor and a generator are the same machine. A transformer and a wireless charger put two coils close: an alternating current in one makes a changing field that induces a voltage in the other, ferrying energy across a gap with no wire. An induction cooktop induces swirling currents directly in the metal pan, and those currents heat it by the same $I^{2}R$ you met in circuits. A hard drive's read head is a tiny coil: as magnetized patches on the spinning platter rush past, their changing field induces the blips it reads as bits^[2].

Check yourself