Lenz' law and magnetic moment

Question

Lenz Law says that the induced current will circulate in a direction such that it produces a magnetic field that is opposite to the change of applied magnetic field. For example, if the loop is a circle on the screen, and the applied magnetic field is increasing and pointing out of the screen, then the induced current will be clockwise. So that the field produced by the induced current is pointing into the screen.

We also know that the magnetic moment due to the current loop follows the right-hand rule. So I think the magnetic moment created by the induced current is point into the screen.

But energy-wise, the interaction energy between the magnetic moment will be

$$U=-\vec\mu \cdot\vec B$$

It means the energy is positive between the moment by induced current and the applied field. So does this mean Lenz Law favors higher energy? Does this make sense? I must have missed some critical parts or misunderstood some critical concepts. Can you help?

Answer 1

You can think of this loop of wire and the passing increasing magnetic field through it as a spring. To compress this spring (or extend it) one needs to perform work on the system. That is, if one tries to take the spring from the state of equilibrium to the state of increased potential energy, he needs to do work. This work cannot be done instantaneously. To be more explicit, one cannot change the extension of a spring infinitely quickly. What is more, the energy acquired by the spring is equal (assuming no heat losses) to the work performed on it.

As to the system in question, the Lenz's law simply ensures that one performs work on the system when he tries to change it. That is, it also makes sure that the system's energy cannot be changed instantaneously. Not only that, because it creates an opposing force, it provides the agent with an opportunity to do the work. Otherwise, there would be nothing to "press" against!

When one increases the field through the loop, it sets up a magnetic moment which interacts with the increasing field in a way that opposes the increase of the magnetic field. Hence, to keep increasing this field (keep compressing this spring), one needs to perform enough work to overcome the strength of this interaction (potential energy). One cannot give this loop of wire energy without doing some kind of work. There is no free lunch in the universe. This is, of course, better known as conservation of energy.