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Lenz’s law, eddy currents and magnetic braking
Lenz's law and how to use it to find the direction of an induced current, why it is a consequence of conservation of energy, eddy currents in metal plates and pipes, and their uses in magnetic braking, induction cooking and metal detectors.
Drop a strong magnet down a vertical copper pipe and it takes several seconds to come out the bottom, drifting down as if it were falling through honey. Copper isn't magnetic, and the magnet never touches the walls. Drop a steel bolt down the same pipe and it falls straight through. The magnet is being held up by currents it creates in the copper as it falls.
Lenz's law
Faraday's law tells you how big an induced emf is. Lenz's law, worked out by Heinrich Lenz in 1834, tells you which way it acts:
An induced emf always gives rise to a current whose magnetic field opposes the change in magnetic flux that produced it.
It is the minus sign in $\varepsilon = -N\Delta\Phi/\Delta t$. Notice exactly what is opposed: not the field, and not the flux, but the change in flux.
- If the flux through a coil is increasing, the induced current makes a field in the opposite direction, trying to keep the flux down.
- If the flux is decreasing, the induced current makes a field in the same direction, trying to keep it up.
Finding the current direction, step by step
- Which way does the external field point through the coil, and is the flux increasing or decreasing?
- Decide which way the induced field must point to oppose the change.
- Use the right-hand grip rule for a solenoid (fingers curl with the current, thumb points to the north end) to find the current that makes that field.
A handy shortcut: looking at the end of a coil, an anticlockwise current makes that end a north pole, and a clockwise current makes it a south pole.
The north pole of a bar magnet is pushed towards the left-hand end of a coil. Which way does the induced current flow, viewed from the magnet?
The magnet's field points out of its north pole, into the coil, from left to right. As it approaches, this flux is increasing.
The induced field must point from right to left inside the coil, so the left-hand end of the coil must be a north pole, facing the magnet's north pole.
A north pole facing you means the current is anticlockwise viewed from the magnet. The two north poles repel: the coil pushes back on the magnet.
Why it must be so: conservation of energy
Imagine Lenz's law were the other way round. Push a north pole towards a coil, and the coil makes a south pole that attracts the magnet. The magnet speeds up, so the flux changes faster, the current gets bigger, the attraction gets stronger, and so on. You would get ever-increasing kinetic energy and electrical energy from one small push. That would create energy from nothing.
Lenz's law is how conservation of energy shows up in electromagnetism. Because the induced current always opposes the change, you have to do work to change the flux, by pushing the magnet in or turning the generator. That work is what becomes the electrical energy in the circuit, and eventually heat. In the sliding rod example in 6E, the mechanical power you put in (Fv) exactly equalled the electrical power out (εI).
Eddy currents
In a coil, the induced current has a wire to follow. In a solid piece of metal, the induced currents swirl around in closed loops inside the metal itself. These are eddy currents. Like all induced currents they obey Lenz's law, so they always oppose the relative motion between the metal and the field.
Their energy ends up as heat in the metal (the current flows through the metal's resistance). That is useful when you want to brake something or heat something, and a waste when you don't.
- A solid aluminium plate swinging between magnet poles stops within a swing or two. Cut slots in it and it swings almost freely, because the slots break up the large current loops.
- A magnet falling down a copper pipe induces eddy currents in the rings of pipe below it (flux increasing, so they repel it) and above it (flux decreasing, so they attract it). The faster it falls, the bigger the currents and the braking force, until the braking force equals the weight and the magnet falls at a constant terminal velocity.
- Drop the strong magnet down the copper pipe. Read the terminal speed and the time. Describe the shape of the speed–time graph and explain it with forces.
- Try PVC, then aluminium. Why does PVC give free fall? Why does aluminium give a higher terminal speed than copper? (Aluminium's resistivity is about 1.6 times copper's.)
- Try the slotted copper pipe and explain the difference.
- Switch to the weak magnet. The braking force depends on B²: explain why, using Faraday's law for the size of the current and the motor effect for the force on it.
- For copper, compare the gravitational potential energy lost (shown as total) with the energy that ends up as heat. Where did the rest go?
Magnetic braking and other uses
A magnetic brake uses eddy currents to slow a moving conductor without touching it, converting kinetic energy into heat in the conductor.
- Rides and trains: many roller coasters and drop towers have copper or aluminium fins on the cars that pass between rows of strong permanent magnets near the end of the ride. Some high-speed trains use eddy-current rail brakes. These brakes have no parts to wear and need no power, so they work even in a blackout.
- Self-limiting: the braking force is proportional to speed, so it is strongest when it is most needed and fades to zero as the object stops. That makes magnetic brakes very smooth, but it also means they can't hold something stationary; a friction brake is still needed to stop completely.
- Exercise bikes and dynamometers: a metal flywheel spins near an electromagnet; turning up the current increases the resistance.
- Induction cooktops: a coil under the glass carries AC at 20–100 kHz. The rapidly changing flux induces large eddy currents in the base of a steel pan, heating it directly. The glass, which is an insulator, stays relatively cool.
- Metal detectors and coin sorters: eddy currents induced in a metal object create a field that the detector senses; different metals respond differently.
A 20 g magnet falls 1.0 m through a copper pipe and leaves the bottom at its terminal speed of 0.12 m s−1. How much energy was transformed to heat in the pipe?
Gravitational potential energy lost: $mgh = 0.020 \times 9.8 \times 1.0 = 0.196$ J.
$K = \tfrac{1}{2}mv^2 = \tfrac{1}{2}(0.020)(0.12)^2 = 1.4 \times 10^{-4}$ J.
Heat = 0.196 − 0.000 14 ≈ 0.20 J. Over 99.9% of the energy became heat via the eddy currents. (Air resistance is negligible at this speed.)
Test yourself
The south pole of a magnet is pulled away from the end of a coil. Viewed from the magnet, the induced current in the coil is
To oppose the leaving south pole, the coil attracts it with a north pole: anticlockwise when viewed from that end.
Lenz's law is a consequence of the law of conservation of
If the induced current aided the change, energy would be created from nothing.
A magnet is dropped through a long vertical copper pipe. Once it reaches terminal velocity,
Net force zero. Eddy currents still flow; the energy turned into heat comes from gravitational potential energy.
Slots are cut in an aluminium plate that swings between magnet poles. Compared with a solid plate, the slotted plate
The slots break up the large current loops, so less current flows and the braking force is smaller.
Why can't an eddy-current brake hold a roller coaster car stationary at the bottom of a slope?
No relative motion means no changing flux, so no eddy currents and no force.
Label the Lenz's law diagram.
Put the explanation of the magnet in the copper pipe in order.
Lenz's law states that the induced current opposes the in flux that caused it. When the north pole of a magnet approaches a coil, the near end of the coil becomes a pole. Lenz's law follows from the conservation of . Swirling induced currents in solid metal are called currents, and their energy is turned into . An induction cooktop heats a pan using a rapidly magnetic field.
Explain how magnetic braking works on a roller coaster car fitted with a copper fin that passes between rows of permanent magnets. In your answer, refer to Faraday's law, Lenz's law and the law of conservation of energy.
As the copper fin moves into the gap between the magnets, the magnetic flux through the part of the fin entering the field increases, and through the part leaving the field it decreases. By Faraday's law (ε = −NΔΦ/Δt), these changes in flux induce emfs in the fin, and because copper is a good conductor, large eddy currents circulate within it. By Lenz's law, the eddy currents flow in directions that produce magnetic fields opposing the change in flux; equivalently, the currents in the magnetic field experience motor-effect forces that oppose the fin's motion relative to the magnets. This decelerates the car without any contact. The faster the car moves, the faster the flux changes, so the larger the currents and the braking force: braking is strong at high speed and fades smoothly as the car slows, which is why a friction brake is still needed to hold it stationary. Lenz's law is required by conservation of energy: because the force opposes the motion, the car does work against it, and its kinetic energy is transformed into electrical energy of the eddy currents and then into heat in the fin, which is dissipated to the surroundings.