Lenz's law — direction of induced current — Physics

Before this

Faraday's law — changing flux induces e.m.f.
Conservation of energy — the big idea
Electromagnets — current creates a magnetic field · coming soon

This leads to

A magnet that falls in slow motion

Hold a strong magnet at the top of a plastic pipe about a metre long, and let go. It drops through in less than half a second, as anything would. Now do the same with a copper pipe of the same size. The magnet takes several seconds to come out of the bottom. It drifts down as if the pipe were full of honey.

This is strange. Copper is not magnetic: a fridge magnet will not stick to it. The magnet never touches the walls. Yet something pushes up on it all the way down.

The last lecture gives half of the answer. As the magnet falls, the flux through each ring of copper below it grows, and the flux through each ring above it shrinks. By Faraday's law, an e.m.f. is induced in every ring, and because copper conducts well, currents flow around the pipe. But Faraday's law gave only the size of the e.m.f. It promised that the next lecture would explain the direction, and the minus sign that many books put in front. Here it is. The direction turns out to be the reason the magnet slows down.

Only one direction is possible

An induced current can flow one way round a coil or the other. Let us test both, using an idea you already trust completely: energy cannot be created.

Suppose you push the north pole of a magnet towards a coil. A current is induced, and a current in a coil makes the coil a magnet. So the face of the coil nearest your magnet becomes either a north pole or a south pole.

Possibility 1: the near face becomes a south pole. A south pole attracts your approaching north pole. The magnet would be pulled in, so it would speed up. A faster magnet changes the flux faster, which makes a bigger current, which pulls harder. Give the magnet one small push, and it would rush in faster and faster on its own, while the coil also heats up. Both the motion and the heat would come from nothing.
Possibility 2: the near face becomes a north pole. It repels the approaching north pole. To keep the magnet moving, you must push against this force, so you do work. That work becomes the electrical energy in the coil, and then heat in the wire.

Possibility 1 creates energy from nothing, so it never happens. Nature always chooses possibility 2. The induced current pushes back against whatever is causing it.

That is it. That is the whole idea. The direction of an induced current is not a new fact to learn. It is conservation of energy, applied to induction.

Lenz's law

The induced current flows in the direction that makes its magnetic field oppose the change in flux linkage that produces it.

This is the meaning of the minus sign in Faraday's law:

ε = - \frac{Δ ( N Φ )}{Δ t}

The minus sign says that the induced e.m.f. acts against the change. If the flux linkage is rising, the e.m.f. drives a current whose field tries to bring it down. If the flux linkage is falling, the current's field tries to hold it up. The law is named after Heinrich Lenz, a physicist working in St Petersburg, who stated it in 1834. (Source: Encyclopaedia Britannica, "Lenz's law".)

A tool for direction: the right-hand grip rule

To use Lenz's law, you need to link the direction of a current in a coil to the direction of the field it makes. Curl the fingers of your right hand round the coil in the direction of the current. Your thumb then points along the field inside the coil. This is the right-hand grip rule.

The end of the coil that the thumb points out of is its north pole, because field lines leave a north pole. A quick check follows from this. Look at one end of a coil. If the current flows anticlockwise as you look, your thumb points towards you, so that end is a north pole. If the current flows clockwise, that end is a south pole.

Finding the direction, one step at a time

Every Lenz's law question can be answered with the same four steps.

Step 1. Which way does the original field point through the coil?
Step 2. Is the flux through the coil increasing or decreasing?
Step 3. If it is increasing, the induced field points against the original field. If it is decreasing, the induced field points the same way as the original field, to try to keep it.
Step 4. Use the right-hand grip rule to turn the induced field into a current direction. Say which side you are looking from.

Then check with forces. An approaching magnet must be repelled, and a leaving magnet must be attracted. Either way, the coil resists the motion.

Worked Example 1 — A north pole approaches

The north pole of a bar magnet is pushed towards a coil, as in the figure. Find the direction of the induced current, seen from the magnet's side.

Top: as the north pole approaches, the magnet's field through the coil (to the right) grows, so the induced field points to the left. Bottom: seen from the magnet's side, the current is anticlockwise, and the near face is a north pole.

Step 1. Field lines leave a north pole, so the magnet's field passes through the coil from left to right.

Step 2. The magnet is getting closer, so this flux is increasing.

Step 3. The induced field must oppose the increase, so it points to the left, back towards the magnet.

Step 4. Point your right thumb to the left, towards the magnet. Seen from the magnet's side, your fingers curl anticlockwise. So the current is anticlockwise.

Check. The induced field leaves the coil on the left, so the near face is a north pole. It repels the approaching north pole, and you must push. ✓

Worked Example 2 — A south pole leaves

Now the magnet is turned round. Its south pole faces the coil, and the magnet is pulled away to the left. Find the direction of the induced current, seen from the magnet's side.

Step 1. Field lines go into a south pole. So near the coil, the magnet's field points to the left, towards the magnet.

Step 2. The magnet is moving away, so this flux is decreasing.

Step 3. The induced field must oppose the decrease. It tries to keep the flux, so it points the same way as the magnet's field: to the left.

Step 4. Thumb to the left, towards the magnet. Seen from the magnet's side, the current is anticlockwise.

Check. The induced field leaves the coil on the left, so the near face is a north pole. A north pole attracts the departing south pole, so you must pull to take the magnet away. ✓ Notice that the near face is a north pole in both examples, but for opposite reasons. The first repels, and the second attracts.

Stop and think

In Worked Example 2, the induced field points the same way as the magnet's field. Does that break Lenz's law, which says the induced current opposes?

Show the answer

No. Lenz's law opposes the change in flux, not the flux itself. The flux was falling. A field in the same direction as the magnet's field adds to the flux, so it slows the fall. That is opposing the change. The most common mistake with Lenz's law is to make the induced field point against the magnet's field every time. That is right only when the flux is increasing.

Practise the rule on eight cases. For each one, decide what the face of the coil nearest the magnet becomes.

Sort — what does the coil's near face become?

Each card describes a magnet and a coil. Put each card in the box that says what the face of the coil nearest the magnet becomes.

This box sorts eight cases by what the face of the coil nearest the magnet becomes.

Near face becomes a north pole: a north pole moved towards the coil (repels it); a south pole moved away (attracts it); the coil moved towards a north pole that is held still (repels it).
Near face becomes a south pole: a south pole moved towards the coil (repels it); a north pole moved away (attracts it); the coil moved away from a north pole that is held still (attracts it).
No induced current: a magnet held still inside the coil (no change); a north pole moved towards a coil whose circuit is broken (an e.m.f., but no complete circuit, so no current).

The rule: the near face always acts to resist the relative motion. It repels an approaching pole and attracts a leaving one. No change in flux, or no complete circuit, means no current.

You should have seen that the answer never depends on the magnet alone. It depends on what is changing. And when no current can flow, there is nothing to push back.

A loop falling into a field

Worked Example 3 — A loop falls into a field

A rectangular loop of wire falls into a region where a uniform field points into the page. Find the direction of the induced current as the loop enters, while it is completely inside, and as it leaves.

As the loop enters the field, the flux into the page grows, so the induced current flows anticlockwise and makes a field out of the page.

Entering. More of the loop is inside the field each moment, so the flux into the page increases. The induced field must point out of the page. With your thumb pointing out of the page towards you, your fingers curl anticlockwise. The current is anticlockwise.

Completely inside. The flux is large but steady. No change, so no current.

Leaving. The flux into the page decreases. The induced field points into the page, to keep it. The current is clockwise.

Check. The current flows only while the loop enters or leaves. Both times, the loop's energy is turned into heat in the wire, so the loop falls more slowly than it would without the field. ✓

Where the energy goes: a number to check

Lenz's law says that the person pulling the coil must supply the electrical energy. Let us check this with numbers from the last lecture.

Worked Example 4 — Pulling a coil out of a field

In the last lecture, a square coil of 20 turns and side 0.10 m was pulled at a steady 2.0 m/s out of a field of 0.50 T. The induced e.m.f. was 2.0 V. Now suppose the coil has a resistance of 4.0 Ω. Find the current, the power turned into heat, and the force needed to pull the coil at a steady speed.

Think first. The power the puller supplies must equal the power turned into heat. Power is also force × speed.

The current is $I = ε / R = 2.0/4.0 = 0.50$ A. The heating power is $P = I^{2} R = 0.5 0^{2} \times 4.0 = 1.0$ W.

The puller must supply 1.0 W at 2.0 m/s, so the force is $F = P / v = 1.0/2.0 = 0.50$ N.

Check. The lecture on the motor effect gives the force on a current in a field as $F = B I L$ for each wire in the field. Here only the trailing side of the coil, with its 20 wires, is still in the field. So $F = 20 \times 0.50 \times 0.50 \times 0.10 = 0.50$ N. ✓ Two completely different routes give the same force. That is energy conservation, seen in numbers.

Notice one more thing. Double the speed, and the e.m.f., the current and the force all double. The faster you pull, the harder the coil pulls back. This is why the magnet in the copper pipe does not keep speeding up.

Back to the copper pipe: magnetic braking

Now the slow magnet makes sense. Every ring of the pipe below the magnet has growing flux, so its current repels the magnet from below. Every ring above has shrinking flux, so its current attracts the magnet from above. Both push up. These swirling currents in a solid piece of metal are called eddy currents.

The braking force grows with the speed. So the magnet speeds up only until the upward force equals its weight. After that, it falls at a steady speed, and all the gravitational potential energy it loses becomes heat in the pipe. Run the model below to see the energy account.

Simulation — a magnet falling through a pipe

Choose a pipe, then press Play. This is a simple model: the braking force is proportional to the speed. The magnet's mass is 20 g and the pipe is 1.0 m long. The braking strengths are invented, but typical of a strong small magnet in a thick pipe.

This box shows a simple model of a 20 g magnet falling through a 1.0 m pipe, with a braking force proportional to its speed. The braking strengths are invented, but typical.

Pipe	Steady speed (m/s)	Time to fall 1.0 m (s)	Heat in the pipe (mJ)	Kinetic energy at the bottom (mJ)
Plastic	none	≈ 0.45	0	196
Aluminium	≈ 0.39	≈ 2.6	≈ 194.4	≈ 1.6
Copper	0.25	≈ 4.0	≈ 195.4	≈ 0.6

In every pipe, the magnet loses $m g h = 0.020 \times 9.8 \times 1.0 = 0.196$ J of gravitational potential energy. In the plastic pipe, all of it becomes kinetic energy. In the metal pipes, almost all of it becomes heat. Copper conducts better than aluminium, so its eddy currents are larger and the steady speed is lower.

You should have seen the energy account balance at every moment. In the plastic pipe, the lost potential energy all became kinetic energy. In the metal pipes, the magnet reached its steady speed almost at once, and from then on every joule of lost potential energy became heat in the pipe. Copper conducts electricity better than aluminium, so its eddy currents are larger, and it brakes harder.

Engineers use exactly this. Germany's ICE 3 high-speed trains, which entered service in 2000, were the first trains in regular service to brake with a linear eddy current brake: electromagnets on the train induce eddy currents in the steel rail, and nothing touches. Many roller coasters and drop-tower rides slow their cars with magnets passing close to metal fins. (Sources: Wikipedia, "Eddy current brake"; RSSB research summary on the ICE 3 brake.) But the braking force is proportional to the speed, so it fades to zero as the vehicle slows. An eddy current brake cannot hold anything still. Trains and rides always finish with a friction brake or a mechanical stop.

Three traps

Trap 1 — Opposing the field instead of the change

The induced field points against the original field only when the flux is increasing. When the flux is decreasing, it points the same way, to keep the flux up. Always ask Step 2 first: is the flux going up or down?

Trap 2 — "Clockwise" without saying from where

A current that is clockwise seen from one end of a coil is anticlockwise seen from the other end. An answer of "clockwise" alone can be marked wrong even when you understood everything. Always write where you are looking from: "anticlockwise, seen from the magnet's side".

Trap 3 — Expecting a push with no circuit

A changing flux always induces an e.m.f. But a current, and so a magnetic push back, needs a complete circuit. A coil with a broken circuit has an e.m.f. between its ends but no current, so a magnet passes it with no extra effort. A copper pipe with a long cut down its side brakes much less, because the currents can no longer flow all the way round.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

The south pole of a bar magnet is pushed towards a coil. (a) What does the near face of the coil become? (b) Seen from the magnet's side, which way does the current flow?

Hint 1
The coil must resist the approach.

Hint 2
A south pole faces you when the current, as you look, is clockwise.

Full answer
(a) A south pole, so that it repels the approaching south pole. (b) Check with the four steps. Field lines go into a south pole, so the magnet's field through the coil points towards the magnet. The flux is increasing, so the induced field points away from the magnet. Seen from the magnet's side, the thumb points away from you, so the current is clockwise. Check: clockwise as you look means a south pole faces you, which matches (a). ✓
A light aluminium ring sits loosely over the top of a vertical coil wound on an iron core. When the current in the coil is switched on, the ring jumps up into the air. (a) Explain why, using Lenz's law. (b) Compare the direction of the current in the ring with the direction of the current in the coil. (c) What happens if the ring has a narrow cut in it?

Hint 1
When the coil is switched on, the flux through the ring grows from zero.

Hint 2
For the ring's field to oppose the coil's growing field, which way must the ring's current flow?

Full answer
(a) Switching on makes the flux through the ring grow quickly. The current induced in the ring makes a field that opposes this growth, so the ring's field points against the coil's field. Two magnets facing each other with opposite fields repel, so the ring is pushed up. (b) The ring's current flows round in the opposite direction to the coil's current. (c) The cut breaks the circuit. An e.m.f. is still induced, but no current flows round the ring, so there is no force and the ring stays where it is. Check: the energy for the jump comes from the electrical supply to the coil, which must supply extra energy while the ring is being pushed away. ✓
A square coil of 50 turns and side 0.080 m, with a resistance of 2.0 Ω, is pulled at a steady 0.50 m/s out of a field of 0.25 T. (a) Find the induced e.m.f. and current. (b) Find the heating power and the force needed to pull the coil. (c) The field points into the page. Which way does the current flow, seen from the front?

Hint 1
Use $ε = NB Lv$ from the Faraday's law lecture, then $I = ε / R$ .

Hint 2
Power = $I^{2} R$ , and force = power ÷ speed. For (c), the flux into the page is falling.

Full answer
(a) $ε = 50 \times 0.25 \times 0.080 \times 0.50 = 0.50$ V, and $I = 0.50/2.0 = 0.25$ A. (b) $P = 0.2 5^{2} \times 2.0 = 0.125$ W, and $F = 0.125/0.50 = 0.25$ N. (c) The flux into the page is decreasing, so the induced field points into the page to keep it. Thumb into the page, away from you: the current is clockwise, seen from the front. Check: $F = NB I L = 50 \times 0.25 \times 0.25 \times 0.080 = 0.25$ N, the same as from the power. ✓
A stretch. In a copper pipe, a 20 g magnet falls at a steady 0.25 m/s. (a) What is the braking force on the magnet? (b) At what rate is heat produced in the pipe? (c) How much heat is produced while the magnet falls the full 1.0 m? (d) Explain why a stronger braking pipe still cannot hold the magnet still at one height.

Hint 1
At a steady speed, the forces on the magnet balance.

Hint 2
Heating power = braking force × speed. For (d), what is the braking force when the speed is zero?

Full answer
(a) At a steady speed, the braking force equals the weight: $0.020 \times 9.8 = 0.196$ N. (b) The heating power is $F v = 0.196 \times 0.25 = 0.049$ W, about 49 mW. (c) The magnet loses $m g h = 0.020 \times 9.8 \times 1.0 = 0.196$ J of potential energy. Its kinetic energy hardly changes, so nearly all 0.196 J becomes heat. Check: at 0.25 m/s the fall takes about 4.0 s, and $0.049 \times 4.0 \approx 0.196$ J. ✓ (d) The braking force exists only because the flux in the pipe is changing, and that needs the magnet to move. At zero speed there is no change, no current and no force. So the magnet always keeps falling, however slowly.

The bigger idea

Lenz's law is not really a separate law. It is conservation of energy, written for electromagnetism. Every induced current takes its energy from whatever causes the change, and the push back you feel is the price of that energy. The same idea runs through the rest of this topic. A generator is hard to turn when it supplies a large current, because the induced current opposes the turning. That is why a power station must burn more fuel when more lamps are switched on across a city.

There is a pattern here that appears all over physics. A system that pushes back against change is a stable system. Lenz's law gives circuits a kind of inertia: they resist sudden changes in flux, just as a mass resists sudden changes in velocity. Pre-University physics builds on this idea when a coil resists a change in its own current.

Key concepts in this lesson

Lenz's law states that the induced current flows in the direction that makes its magnetic field oppose the change in flux linkage that produces it.
Lenz's law follows from conservation of energy, because a current that helped the change would create energy from nothing.
The minus sign in Faraday's law, ε = −Δ(NΦ)/Δt, expresses Lenz's law.
When the flux increases, the induced field points against the original field; when the flux decreases, it points the same way.
The right-hand grip rule links a coil's current to its field: anticlockwise as you look means a north pole faces you.
Eddy currents in a conductor oppose its motion through a field, giving a braking force that grows with speed and turns kinetic or potential energy into heat.