Lenz's law

Pushing against the magnet

Push the north pole of a bar magnet towards a closed coil. You feel a small push back. Pull the magnet away, and you feel a small pull on it. This force tells us the direction of the induced current. In 1834 Heinrich Lenz stated the rule.

Lenz's law. The induced current flows in the direction in which its own magnetic field opposes the change of flux that produces it.

Read the words with care. The induced field opposes the change of flux. It does not oppose the flux itself.

• If the flux through the loop is growing, the induced field points against the original field. It tries to cancel the growth.
• If the flux through the loop is falling, the induced field points along the original field. It tries to keep the flux from falling.

In the top picture the north pole comes in. The flux through the coil points to the right, away from the north pole, and it grows. So the induced field must point to the left. The near face of the coil then acts as a north pole, and it pushes the magnet back.

In the bottom picture the magnet goes away and the flux falls. The induced field must point to the right, along the old flux, to keep it alive. The near face of the coil is now a south pole, and it pulls the magnet back.

Why must it be so? The energy argument

Suppose that the opposite were true. Let the north pole move towards the coil, and let the induced current make the near face a south pole. Then the coil would pull the magnet in. The magnet would speed up, the flux would change faster, and the current would grow. This would go on without limit, with no one doing any work. We would get energy from nothing. This never happens.

With Lenz's law the account is honest. You push the magnet against the repulsion, so you do work. That work becomes electrical energy in the coil. The current heats the wire, and the energy leaves as heat. So Lenz's law is the law of conservation of energy for induction.

One more point. Suppose that the coil is not closed. An emf is still induced, but no current flows. With no current there is no induced field and no force on the magnet. You move the magnet with no extra effort, and no energy is used. You pay for the energy only when a current flows.

Finding the direction in four steps

1. Original field. Find the direction of the original field through the loop.
2. Change of flux. Decide whether the flux through the loop is growing or falling.
3. Induced field. If the flux grows, the induced field points opposite to the original field. If the flux falls, it points the same way.
4. Current. Curl the fingers of your right hand so that the thumb points along the induced field. The fingers show the direction of the induced current.

For a loop drawn on a page, an induced field out of the page needs an anticlockwise current. An induced field into the page needs a clockwise current.

Three examples

Example 1. A loop lies in the page. A field points into the page and grows stronger. Which way does the current turn?

1. The original field points into the page.
2. The flux into the page grows.
3. The induced field must point out of the page.
4. An induced field out of the page needs an anticlockwise current.

If the field falls instead, the flux into the page falls. The induced field points into the page, and the current is clockwise.

Example 2. The same loop is in a steady field into the page, and a student stretches it so that its area grows. The flux grows, so the current is anticlockwise again. The loop resists being stretched.

Example 3. Two coils are wound on the same iron rod. Coil 1 is joined to a battery through a switch. Coil 2 is joined to a meter. When you close the switch, the flux through coil 2 grows, and its induced current makes a field in the opposite direction. When the current in coil 1 is steady, the meter reads zero. When you open the switch, the flux falls, and the induced current turns the other way round.

Set a positive rate. The field grows, and the current turns anticlockwise. Set a negative rate. The current turns clockwise. At zero there is no current. Now make the area larger. The arrows stay the same, but the emf in the readout grows.

The minus sign

Faraday's law and Lenz's law together make one equation:

The minus sign is Lenz's law. When a flux that points one way grows, $\Delta\Phi$ is positive, and the emf is negative. A negative emf drives a current whose field points the other way. So we find the size of the emf from Faraday's law, and the direction from Lenz's law.

A magnet that falls through a coil

Let a magnet pass through a coil that is joined to a meter. While the magnet comes in, the flux grows, and the emf has one sign. While it goes out, the flux falls, and the emf has the other sign. So the meter shows a pulse one way, then a pulse the other way.

Go faster. Each pulse gets taller, because the flux changes more quickly. Each pulse also gets narrower, because the magnet passes sooner. The flux linkage that passes is the same, so the area under each pulse does not change. This is $Q = N\Delta\Phi/R$ again.

Practise with new numbers

Try it in Python

The program lets a magnet fall from rest through a coil. It finds the emf from the change of the flux linkage, as Faraday's law says, and adds up the area under each pulse.

Common mistakes

• Thinking that the induced field always opposes the original field. It opposes the original field only when the flux grows. When the flux falls, it points the same way.
• Thinking that the induced field opposes the flux. It opposes the change of the flux.
• Forgetting the open circuit. A coil that is not closed has an emf but no current, and so no force on the magnet.

Remember these

• Lenz's law: the induced current has the direction whose field opposes the change of flux.
• Flux growing: the induced field is against the original field. Flux falling: it is along the original field.
• Lenz's law is conservation of energy. The work that you do against the force becomes electrical energy, and then heat.
• An open coil has an emf but no current and no opposing force.
• A magnet through a coil gives a pulse one way and then a pulse the other way. The areas under the pulses are equal.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A light aluminium ring hangs from a thread, closed all round. You push the north pole of a magnet towards it from the left. Does the ring move towards the magnet or away? Why?

2. A loop in the page is in a field that points out of the page. The field falls to zero. Which way does the induced current turn?

3. A loop lies in a steady field that points into the page. The loop is squeezed so that its area becomes smaller. Which way does the induced current turn?

4. A closed coil and an open coil are exactly alike. A magnet is pushed into each at the same speed. Compare the emf, the current and the force on the magnet in the two cases.

5. A magnet is pushed into a closed coil. The induced current is $0.30$ A for $0.40$ s, in a coil of total resistance $5.0\ \Omega$. How much heat is produced? What is the least work that the hand did?