# Explaining Lenz's Law without conservation of energy

I was often told by my professor, using the following example, to demonstrate the relationship of conservation of energy and Lenz's Law.

If you push a conductor into a constant magnetic field. By Lenz's Law, voltage will be induced to oppose the "cause" which then resists the conductor from moving quicker which would be a violation of conservation of energy. In other words, if you push the rod very slightly, and the induced force turns out to push it even further, you generate infinite energy which then contradicts the conservation of energy law.

However, I wasn't happy enough by this answer, even though the fact is that you won't get infinite energy. I was thinking, whether it is something to do with the things happening on the atomic scale. Considering a more trivial example, a magnet moving towards a current loop,

so why would it produce a opposite pole to resist the change?

Is it something to do with the cutting flux of the magenetic field?

Why must a field be created to cancel the effect?

Why can't nothing happen?

Or is it just a matter of fact in nature?

I am finding this hard to understand how this really works. Is it a virtual particle?

Please explain the above phenomena without relating it to energy conservation law.

• In the image posted by you (motional EMF), you can consider the free electrons in the rod moving in the direction of $v$ and therefore the magnetic force acting on them due to their motion is sideways in a direction that would produce a current in the loop as if it tries to oppose the increase in the field within the loop. Work it out with Fleming's left hand rule to get the direction of force and verify. – Satwik Pasani Sep 19 '13 at 13:18

According to Lenz's law, polarity of induced emf will be such that it produces the current which opposes the change in magnetic flux which produced it. It directly follows from the law of conservation of energy.

Let's say you have a circular loop of conductor, if you bring the north pole of the magnet towards the loop, current induced in the loop will be anti-clockwise when viewed from the side of the magnet. Remember if current is in the anti-clockwise direction it acts same as the north pole. So, north pole-north pole repel each other. Thus, induced emf is opposing the change in magnetic flux. Therefore, work has to be done in order change the magnetic flux linked with the coil. This work done will be converted into electrical energy.

When you take the magnet away, when viewed from the side of magnet, current in the loop will be clockwise (acts as south pole) whereas magnet side near the loop will be south pole. So, even here south pole-south pole repel each other. Thus, induced emf is opposing the change in magnetic flux. Therefore, work has to be done even here in order to change the magnetic flux linked with the coil. As said above, this work done will be converted into electrical energy.

Thus, in the above two cases energy is conserved.

Let's say that you bring the north pole of the magnet towards the same circular loop, now let current induced be clock-wise (acts as south pole) when viewed from the magnet side. North pole- South pole attract each other, so no work is needed to change the flux linked with the coil. But when flux linked with the coil changes, there is emf induced. Here, electrical energy is produced without any work being done. It violates the law of conservation of energy. Similarly you can consider the other case.

Therefore, from Lenz's law it follows that electrical energy is produced by the expense of mechanical energy. For this to happen, induced emf should oppose change in magnetic flux.

You asked to explain with out using law of conservation of energy, but I felt the above explanation would explain your questions better.

Without directly referring to conservation of energy, we can say that a force that causes a change in a situation by moving something or by changing forms of energy invariably has an opposition to its action. If not, the force would be unnecessary and a trivial phenomenon. Indeed the perceived status quo itself would be an illusion.

So, in a mechanical system whereby for example a force pushes (slides) a mass along a surface, the mass being pushed exerts an equal and opposite force on the force pushing it (Newton's Third law of motion). This opposing force serves to preserve the status quo.

Inducing a current in a conductor via magnetic flux is an attempt to change the status quo. Lenz's law, while seemingly deliberately 'active' against the magnetic flux, is no less a passive reaction to the attempts to change a status quo.

The electrical phenomena described by Lenz's Law can readily be explained by noting that the radial symmetry of the electric field of a point charge changes to that of an prolate spheroid if it is moving at a speed $v$ since its electric field is shortened in the positive and negative directions of motion by the factor $1-v^2/c^2$ relative to the Coulomb equation which describes the field of the charge at rest. In the transverse direction of motion the electric field is increased by dividing by the square root of that factor. The transverse field also increases because of the increased line density due to the decrease of the electric field in the direction of motion.

These relations are derived from the Lorentz Transformation which specifies the electric field of a moving charge at any position $(x,y,z)$.

Thus when a wire conducting a current to the right is pushed toward a neutral wire the conduction electrons would follow a diagonal path up and to the right. Since these electrons "flatten" in the direction of motion and the ends of the ovoid increase their field intensity to the left, the mobile electrons in the neutral wire must also be driven to the left, in agreement with observation.

When the conducting wire is pulled away the tilt of its mobile electrons is reversed and their path is now down and to the right which forces the mobile electrons in the neutral wire also to the right, again in agreement with observation. According to Ampere's Force Law parallel currents will oppose the motion while the anti parallel currents produced when the conducting wire is pushed toward the neutral wire likewise oppose the motion.

It is not necessary to designate the Conservation of Energy Law as the cause of the behavior of Lenz's Law.

Special relativity also explains the behavior of Ampere's Force Law where like currents attract while unlike currents repel, in contrast to the Coulomb Law where like charges repel while unlike charges attract.

It is not necessary to designate the Lorentz Force Law ($v\times B$) as the cause of the behavior of Ampere's Law.

## protected by AccidentalFourierTransformJul 22 '18 at 14:20

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