A question about magnets, energy, and forces

Question

Following my first lecture on magnetism today, I'm left with just one question my lecturer couldn't answer.

Imagine a 1 kg magnet supporting its own weight stuck to a vertical surface, resisting almost 10 newtons of gravitational force pushing it down, by exerting a force of attraction strong enough to crate enough friction to prevent the gravitational force from accelerating it.

I've done a bit of web searching, and sort of understand or accept that although it's exerting this force it isn't moving in any direction, so no work is being done/no energy is needed.

So how does a magnet use this force of attraction to move metal objects towards it, or push other magnets away from it. Here items are moving, work is being done, where does the energy come from, and does this energy run out?

Answer 1

The short answer (meaning I'm not going to work it out) is that the energy of the static field changes: this is the only possible answer.

Imagine a steel ball bearing on a horizontal surface, where you suddenly place a magnet near by. The ball accelerates and sticks to the magnet. You must pull on the magnet with force $\vec F$, but it is through a distance $\vec x = 0$ so that the work is indeed zero--yet the ball accelerated, so something was doing work. Moreover, to return to the initial conditions, you need to do work: you must pull the ball away from the magnet with a force applied over a distance.

What happens is the energy density of the magnetic field is less when it's inside the ball, so it wants to put as much of it as it can inside the ball, so it pulls the ball in closer to do that.

Since you've only had one lecture, you probably haven't covered the magnetic field $\vec H$ (as opposed to $\vec B$), which is used when the permeability $\mu$ differs from that of free space, $\mu_0$.

Answer 2

Here items are moving, work is being done

Conservative forces are only conservative if you consider the entire path when the object returns to its original location. You are considering only half the path. That will invariably lead to confusion.

For instance, apples fall off trees, and definitely pick up kinetic energy in the process. Yet gravity is still conservative, which is immediately obvious when you expend energy to pick up the apple off the ground.

Likewise, a magnet will cause motion to start and, in your example stop when it hits the magnet. But that's not the entire path - if you remove the metal object from the magnet and return it to its original location, that obviously takes effort.

So to avoid such confusion, we define some sort of "virtual energy" term, for instance, "potential energy". We define that term such that it is equal and opposite to the kinetic energy you gain moving through that same path. The total of potential and kinetic energy is unchanging at all times, in both the gravity and magnetic cases.