Eddy currents. When a magnet moves near a metal sheet, the magnetic flux through each part of the sheet changes. By Faraday’s law this induces an EMF, and because the sheet is a conductor, currents flow in closed loops in it: eddy currents. The EMF is proportional to the speed, ε = Bwv, and so is the current, I = ε/R.

Lenz’s law. The induced current always flows so that its magnetic field opposes the change that produced it. In front of the moving magnet the flux is increasing, so the loop there makes a field that pushes the magnet back: it faces the magnet with the same pole (a north pole facing the magnet’s north pole). Behind the magnet the flux is decreasing, so the loop there makes a field that pulls the magnet back: it faces the magnet with the opposite pole. Both loops pull the magnet backwards, so the force is always opposite to the velocity. The top view shows the direction of the currents: opposite senses in front and behind.

Turning the magnet over reverses the direction of every eddy current (and swaps the induced poles), but the force on the magnet is still opposite to its velocity and has exactly the same size: the braking does not depend on which pole is up.

ε = B w v, I = ε/R, F = B I w = (Bw)² v / R = b v, P = I²R = F v

Energy. The magnet slows down, and the kinetic energy it loses is turned into heat by the eddy currents in the resistance of the sheet: the heat is ∫Fv dt = ½m(v₀² − v²). The magnet is not stopped as if by friction: the force is proportional to the speed, so the speed falls off exponentially, v = v₀ e^−bt/m, and the braking becomes weaker as it slows. A good conductor (copper, aluminium) and a thick sheet give a small resistance, large currents and strong braking. Plastic does not conduct, so there are no eddy currents and the magnet is not slowed down at all. A weaker field (a bigger gap) gives much weaker braking, because the force depends on B².

In use: the brakes of some trains and roller coasters, which have no parts that wear out, and the induction cooker, where the same eddy currents heat a pan.

The magnet is drawn much larger than it is (a 4 cm square magnet is assumed), and the eddy current is a simple estimate for a thin sheet. The magnet stops at the ends of the track.