Lenz’s law and energy
Which way does the current flow?
Faraday’s law tells us how big the induced emf is. In 1834, the German physicist Heinrich Lenz gave a rule for its direction.
Lenz’s law: the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.
Let us apply it. The north pole of a bar magnet moves towards a closed coil. The magnet’s field lines point into the coil, and their number through the coil is increasing.
To oppose this increase, the induced current must make its own field pointing back towards the magnet. Seen from the magnet’s side, the current flows anticlockwise.
A coil with an anticlockwise current, seen from one side, behaves like a magnet with its north pole on that side. So the near face of the coil becomes a north pole, and it repels the approaching north pole.
Now pull the magnet away. The flux through the coil decreases. The induced current now tries to keep it up: it flows clockwise, seen from the magnet, and the near face becomes a south pole. It attracts the magnet, opposing its withdrawal.
Notice: the induced current does not oppose the flux. It opposes the change in flux. Approach: it repels. Withdrawal: it attracts. Either way, it resists the motion.
The same rule works for a loop moving into a region of field pointing into the page. The flux into the page increases, so the induced current flows anticlockwise, making a field out of the page inside the loop.
When the whole loop is inside the uniform field and moving, its flux does not change, and there is no current. As it leaves, the flux decreases, and the current is clockwise.
If the coil is not closed, no current flows, but an emf is still induced. Its direction is the one that the current would have.
Lenz’s law and conservation of energy
Why must the induced current oppose the change? Imagine for a moment that it helped instead.
Push the north pole towards the coil. Suppose the near face became a south pole. It would attract the magnet and pull it in faster.
A faster magnet means a faster change of flux, a bigger current, and a stronger pull. The magnet would speed up by itself, and the current would grow without limit.
We would get ever more kinetic energy and electrical energy from one small push. That is a perpetual-motion machine, and it would violate the law of conservation of energy.
So nature takes the other choice. The near face becomes a north pole and pushes back. To keep the magnet moving, you must push against this force. You do work.
That mechanical work is not lost. It becomes electrical energy in the coil, and the current then heats the wire. That heat is Joule heating.
So Lenz’s law is not a separate law of nature. It is conservation of energy, applied to electromagnetic induction.
This is also the meaning of the minus sign in Faraday’s law, ε equals minus N dΦ_B by dt. The emf always acts against the change of flux.
Power stations rely on this. When homes switch on more fans and lights, more current flows in the generator’s coils, the opposing force grows, and the turbine must be driven harder, burning more coal or using more falling water.
Try it: Lenz’s law
- Push the north pole in. The near face of the coil becomes N (red) and the green arrow shows a force pushing the magnet back.
- Pull it out. The near face becomes S and the force now pulls the magnet back in. Either way the force opposes the motion.
- The orange arrow is the induced field inside the coil. It points against the magnet’s field while the flux grows, and along it while the flux falls.
- Pass the magnet right through the coil. Watch the emf reverse as the magnet passes the middle.
Pulling the magnet away
The north pole of a bar magnet faces a closed coil and is pulled away from it. Seen from the magnet’s side, what is the induced current in the coil?
- Anticlockwise, so the near face is a north pole that repels the magnet
- Clockwise, so the near face is a south pole that attracts the magnet
- Zero, since the magnet is moving away from the coil
- Clockwise, so the near face is a south pole that pushes the magnet away
Answer
B. The flux through the coil (towards it, from the magnet) is decreasing. The induced current tries to keep it up, so it makes a field in the same direction: the near face becomes a south pole, which means a clockwise current seen from that side. The south pole attracts the north pole, opposing its withdrawal.
Where does the energy come from?
A magnet is pushed into a coil connected to a resistor, and the resistor warms up. Where does this heat energy come from?
- From the magnetic energy of the magnet, which gets weaker each time
- From nowhere: induction creates energy
- From the work done by the person pushing the magnet against the opposing force
- From the Earth’s magnetic field
Answer
C. By Lenz’s law, the induced current opposes the motion, so the person must push against a force and does work. This mechanical work becomes electrical energy and then Joule heat in the resistor. The magnet is not used up. If the current helped the motion instead, energy would come from nothing, which would break conservation of energy.
Summary
- Faraday’s law with Lenz’s sign:
ε = −N dΦ_B/dt
- Lenz’s law: the induced emf drives a current that opposes the change in magnetic flux that produced it.
- North pole approaching → near face becomes N (anticlockwise seen from the magnet) and repels. North pole receding → near face becomes S (clockwise) and attracts.
- It opposes the change of flux, not the flux. A loop moving inside a uniform field has no change of flux and no current.
- Lenz’s law follows from conservation of energy: the work done against the opposing force becomes electrical energy and then Joule heat.