Drop a small, strong magnet down a vertical copper tube about a metre long, a little wider than the magnet. A steel ball of the same size falls through in under half a second. The magnet takes several seconds, drifting down as if the tube were full of honey. Copper is not magnetic, and the magnet touches nothing. Yet something holds it back.
The direction of the induced current
Faraday's law gives the size of the induced emf. Its minus sign carries the direction. In 1834 Heinrich Lenz put that direction into words. The induced current flows in the sense that makes its own magnetic field oppose the change of flux that caused it. If the flux through a loop is growing, the induced field points against it. If the flux is shrinking, the induced field points along it, trying to keep it.
The direction of an induced current is such that its own magnetic field opposes the change of flux that produces it. This is the meaning of the minus sign in \(\varepsilon = -N\,d\Phi/dt\).
Notice the word "change". The induced current does not oppose the flux itself. A loop sitting in a huge steady flux has no current at all. Only when the flux starts to change does a current appear, and then it fights that change. A current loop is a small magnet, as the chapter Magnetic Force on Currents and the Magnetic Dipole showed. So the induced current turns the loop into a magnet that resists the change.
A bar magnet, north pole first, is pushed towards a coil. What does the face of the coil nearest the magnet become?
In the figure above the magnet's field at the coil points along the axis, towards the coil. As the magnet comes closer, that flux grows. So the coil's own field points back at the magnet, and its near face becomes a north pole. Two north poles face each other, and the coil repels the magnet. Seen from the magnet, the induced current runs anticlockwise, as round any north face.
Now choose "pull out". The flux is falling, so the coil's field points along the magnet's field, trying to keep it. The near face becomes a south pole, the current seen from the magnet is clockwise, and the coil attracts the retreating magnet. Either way the coil pushes against the motion. You have to push the magnet in, and you have to pull it out.
Why it must be so
Suppose the rule were the other way round. A magnet pushed towards a coil would make its near face a south pole. The coil would pull the magnet in faster. The flux would then grow faster, the current would grow larger, and the pull would grow stronger. A tiny push would set off a runaway, with the magnet speeding up and the current growing, all without any work being done. That would create energy from nothing, which never happens.
So the induced current must oppose the change. Then work must be done to push the magnet in against the repulsion. That work is exactly the electrical energy that appears in the circuit, heating its wires. Lenz's law is not a separate law of nature. It is the conservation of energy applied to induction.
Now the copper tube is explained. Picture the tube as a stack of copper rings. As the magnet falls, the flux through the rings below it grows and the flux through the rings above it shrinks. Currents are induced in the rings. By Lenz's law the rings ahead repel the magnet and the rings behind attract it, so both slow it. Its weight is soon balanced, and it drifts down at a small, steady speed. Its lost energy becomes heat in the copper.
Currents induced like this in a solid block of metal, rather than in a wire, are called eddy currents. The next chapter, Eddy Currents, Generators and Inductance, studies them and their uses, from brakes to furnaces.
Why must an induced current oppose the change of flux that produces it?
A Delhi Metro train slowing into a station does not only rub brake pads against its wheels. Its traction motors are switched to work as generators, turned by the moving wheels. The induced currents are fed back to the overhead line for other trains to use. By Lenz's law those currents oppose the turning of the motors, so they slow the train. This regenerative braking returns a large share of the train's kinetic energy that pads would waste as heat.
A recipe for the direction
Every induced current can be found in two steps. First, find the flux through the loop from the outside source, its direction and whether it is growing or shrinking. The induced field must point against a growing flux and along a shrinking one. Second, curl the fingers of the right hand so that the thumb points along the induced field inside the loop. The fingers give the sense of the current.
(a) A circular loop lies in the page in a field into the page that grows as \(B = 2t\). Find the sense of the induced current. (b) A rectangular loop lies to the left of a long straight wire. The wire's current flows down the page and decays as \(I = 2/(1 + t)\). Find the sense of the induced current. (c) Two coils share an axis. The current in the first is switched on. How does the induced current in the second compare with the first? (d) What happens when the current in the first coil is switched off?
(a) The flux into the page grows, so the induced field points out of the page. A current makes a field out of the page inside it when it runs anticlockwise.
(b) A current flowing down the page makes a field into the page on its left. So the flux through the loop is into the page. It is shrinking. The induced field must point into the page to keep it, so the current is clockwise.
(c) Switching on makes the flux through the second coil grow, so its current runs opposite to the first. Opposite currents repel, so the two coils push apart.
(d) Switching off makes that flux shrink, so the second coil's current runs the same way as the first, trying to keep it. Like currents attract, so the coils pull together for a moment.
Values used: none; only directions are asked. The later numbers use \(B = 0.5\,\text{T}\) and a loop of \(0.5\,\Omega\).
Part (b) is worth a second look, because it is easy to get wrong. A decaying flux does not give the "opposite" current. It gives a current that keeps the flux, so the induced field points the same way as the fading field. Always ask whether the flux is growing or shrinking before choosing.
In Fig 3.1 current flows in coil 2 only while the current in coil 1 is changing. As the current in coil 1 grows, the current in coil 2 runs the other way round. While the current is steady, coil 2 carries nothing. As the current in coil 1 dies, the current in coil 2 runs the same way round. The needle kicks one way at switch-on and the other way at switch-off, just as Faraday saw in 1831.
A loop lies in the page. A field into the page through it is increasing. What is the sense of the induced current, seen from the front?
The north pole of a bar magnet is pulled away from a coil. What does the face of the coil nearest the magnet become, and what force does the coil exert on the magnet?
A loop leaving a field
Pull a square loop steadily to the right, out of a region of field into the page. While the loop is wholly inside, its flux is constant, so nothing is induced. While it straddles the edge, the area inside the field shrinks, and the flux into the page falls. The induced current must keep that flux, so it runs clockwise. Once the loop is wholly outside, the flux is zero and constant, and the current stops.
In the figure above, the clockwise current runs up the left side, the side still inside the field. The force on that side is \(I\vec L\times\vec B\), and up the page crossed into the page points to the left. So the field pulls the loop back, against the pull. The top and bottom sides feel equal and opposite forces, which cancel.
The numbers are small and neat. Only the left side cuts across the field, so the emf is \(0.5 \times 0.2 \times 1 = 0.1\,\text{V}\), and the current is \(0.1/0.5 = 0.2\,\text{A}\). The drag on the left side is \(BIl = 0.5 \times 0.2 \times 0.2 = 0.02\,\text{N}\). The pull does work at \(0.02 \times 1 = 0.02\,\text{W}\), exactly the heating \(I^2R = 0.04 \times 0.5 = 0.02\,\text{W}\). Reverse the field, or push the loop in, and the current reverses; the force still points against the motion.
A square loop of side \(10\,\text{cm}\) and resistance \(0.2\,\Omega\) is pulled out of a field of \(0.4\,\text{T}\). It moves at a steady \(2\,\text{m/s}\), its sides parallel to the edge. While it straddles the edge, what force is needed to keep it moving, in newtons?
A loop is pushed to the left into a region of field out of the page. While it straddles the edge, which way does the magnetic force on the loop point?
A loop of zero resistance takes Lenz's law to its limit. Any tiny emf would drive an enormous current, whose field cancels the change almost at once. So the flux through such a loop cannot change at all. Superconducting rings behave like this: a current set up in one keeps flowing for years, holding the flux fixed. The larger the resistance, the weaker the opposing current, and the more easily the flux changes.
Heinrich Lenz was born in Dorpat, now Tartu in Estonia. As a young physicist he sailed round the world on a Russian ship, measuring the sea. At the St Petersburg Academy he took up Faraday's discovery. In 1834 he stated the rule for the direction of induced currents that bears his name. He tied it to the forces between currents, which always resist the motion that makes them.
- Making the induced field always opposite to the outside field.It opposes the change. For a shrinking flux the induced field points along the outside field.
- Finding the sense of current from the wrong side.Say where you look from. A current anticlockwise from the front is clockwise from the back.
- Expecting a current while a loop is wholly inside a uniform field.No change of flux, no current. A loop leaving a field has current only while it straddles the edge.
- Letting the magnetic force help the motion.The force on an induced current always opposes the motion that causes it; otherwise energy would come from nothing.