Can Magnets do Work?

In our high school physics, we learn that magnetic force does not do work. But why do bar magnet stick together. Our hands feel force and they really move in that direction. It seems that some sort of work is done.
- PChan (age 22)
HK

I love this question, partly because a few years ago my distinguished friend and colleague Sid Nagel and I spent a few hours discussing it, with no thought that there would ever be a chance to more or less publish the discussion.

(And then some years later Tom Lemberger from Ohio State found a basic error- so this is now fixed up.)

Let's break up the answer into two parts: the (easy) quantum case and the (subtle) classical case.

First, particles such as electrons, protons, and even neutrons are themselves tiny magnets, each with a fixed magnitude of magnetic moment. There's a term in the energy proportional to μ_^.B, the dot product of the magnetic moment and the field. That term depends on position, since B depends on position. So it acts just like any other potential energy term, and is responsible for work. In other words, if you drop an intrinsically magnetic particle into a field, the field definitely does work on it. Since a lot of the magnetism in ordinary permanent magnets comes from this intrinsic spin magnetism of the electrons, there's a lot of plain ordinary work done by magnetism as two magnets pull or push on each other. To that extent, your school taught you wrong.

Second, though, they did have a reason to say what they said.  If you look at a classical charged particle (no intrinsic magnetism) moving in a magnetic field, the magnetic field does no work on it directly. You know that because the force is proportional to vXB, where v is the particle velocity. That vector cross product is always at right angles to v, so F_^.v=0, i.e. no work is done on the particle.

OK, so here's where it gets interesting. We know that you can have a magnetic moment from an ordinary current going around a loop, and it can get pulled into a magnetic field just the way some permanent magnet would. Work gets done on it. Isn't it done by the magnetic field? And didn't we just show that couldn't happen?

I should put some drawings in here,  but meanwhile here's words. Say that the magnetic field (from whatever source) is pointing mostly in the z direction, but getting weaker with increasing z, i.e. spreading out radially in the xy plane. This is just the standard picture of the field from a solenoid or cylindrical bar magnet aligned with the z axis. You've got a ring of conductor symmetrically arranged round the z axis with electronic current running around the loop. Let's say that it's a very good conductor, so the current isn't just running down over the time we're interested in, but not a superconductor so we can temporarily not worry about quantum effects.

Let's say that the direction of the current is such that the loop is pulled into the stronger part of the field. The reason that the field along z can get stronger near the source is precisely that the field is spreading out in the xy plane. So there's a little radial field. Take the cross product with the tangential electron velocity and you get a force in the negative z direction on all the electron current. That's at right angles to the current, so there's still no work done. But the electrons can't leave the wire. They bounce off the bottom (low-z) side, imparting momentum to the wire, i.e. exerting force on the wire. As soon as the wire starts to move, that force (in the -z direction) is along the motion of the wire, so it's doing work. The electrons are doing work on the wire, by whatever (non-magnetic) force causes them to bounce off the surface of the wire and stay inside.
 
What happens to the electrons' energy? They are now all moving, on average, in the -z direction, with the wire. That drives a magnetic force on them (again from the radial part of B) that slows down the tangential current. Energy is flowing from the moving electrons into the overall motion of the wire. The magnetic field causes that without actually doing any work directly on the electrons.

Yet somehow, energy flows from the magnetic field into the motion of that loop, because the total magnetic field energy is still μ_^.B and that decreases as the loop is pulled into the solenoid. Here's the key. As the loop starts to move the magnetic field that it makes moves too, i.e. it changes. But a changing magnetic field is always accompanied by an electric field, an "EMF". That EMF does do work on electrons. It extends out to the solenoid and drives a changing current in the solenoid, changing the solenoid magnetic field. That makes another EMF that can do work on the electrons in the loop. The net flow of energy goes like this:

magnetic field--> electric field--> charge carriers (electrons) --> mechanical motions.

Mike W.
 

(published on 05/18/2011)

Can magnetic field/force do work on a current carrying loop? maybe it could possibly do work but INDIRECTLY? Or the main cause of work is the magnetic force/field on a loop because it deflects the charge? It makes no sense to me how magnetic fields/force do no work on a loop of wire carrying-current that resemble's a motor but more simplified... I know as a fact magnetic fields/forces do NO WORK on a charged particle... But a particle in motion through a conductor creates a electric dipole, thus the B field could possibly do work? Thanks!
- Miyze (age 20)
Colorado

As it happens, we've answered just this question before, but it may have been hard to find. So I've marked it as a follow-up.

Mike W.

(published on 08/06/2012)

I'm sorry but I didn't understand your answer. could you explain it more perspicuous. thanks and something else: when a magnetic crane lifts the carcass of a junked car,what is doing work?
- ali (age 20)
Iran

Here's a shorter summary of the argument.

1. For most magnetic materials, including iron, the magnetism largely comes from electron spins which are intrinsically magnetic – it's not from electric currents. The usual saying about magnetic fields not doing work is  false in this case.

2. When all the magnetism comes from classical currents, the magnetic field does no work directly on the currents. However, by steering the electrons in new directions it can cause them to bounce off things and do work. Once the current carrying loops start moving, they create electric fields that do work on the currents. The energy flows from the magnetic field to the electric field to the currents and the mechanical motions.

With regard to the crane, work is done at many points where mechanical energy is transmitted from one part of the machine to another. Probably you're asking about the actual magnet and the car. The car is made of steel, with magnetism coming from aligned electron spins. The magnet simply does work on it, as in our point (1).

Mike W.

(published on 12/24/2012)

So am I correctly concluding from your explanation that in the case of the current loop in the magnetic field, the energy going into the movement of the conductor is originally coming from whatever force is driving the current?
- Daniel (age 18)
Belgium

That's what I wrote in an incorrect original version. That can be partly true depending on what maintains the currents and how they change as the loops move, but the big term is just that the net magnetic field energy changes as the magnets move.

Mike W.

(published on 10/20/2015)

I find your explanations here very helpful; these are questions I've long pondered. However, I suspect your argument on electrons being "intrinsically magnetic", and thus subject to work directly from magnetic fields, to be flawed. I quote a footnote from the classic textbook on electromagnetism by David Griffiths (the curious reader will find it in chapter 6.1.3):[in reference to the application of an external magnetic field increasing the orbital momentum of an electron]"I said earlier that magnetic fields do no work, and are incapable of speeding a particle up. I stand by that. However, a changing magnetic field induces an electric field, and it is the latter that accelerates the electrons in this instance."With this in mind, the consideration of the magnet lifting the car will have to be revised, no? Is it maybe the act of moving the magnet close to the car that ends up lifting the car, as it induces an electric field capable of doing work?
- Hans (age 24)
Bergen

Griffiths is referring to the orbital contribution to the angular momentum and magnetic moment. That's different from the intrinsic spin contribution to which we were referring. My son taught out of Griffiths last semester and found that passage somewhat irritating because it ignored spin. That's a bit odd since Griffiths also has a nice quantum text, but we all have our quirks.

Mike W.

(published on 01/21/2016)

One doubt still lingers. Suppose we have two situations. In the first, there is an electromagnet pulling a current carrying loop and in the second a permanent magnet is pulling a current carrying loop. In the first case, as you explained the magnetic field merely acts as a mediator. The real energy(work) is provided by the battery. The battery gets a little discharged as the loop is pulled. But who takes this place of the battery in case of permanent magnet? Who is the real source of energy?
- Riddhi (age 24)
Newark, Delaware, U.S.A

What I wrote originally wasn't right. There's a term in their energy that looks like the dot product of the magnetic moment and the magnetic field for either the intrinsic spin moment or the moment form circulating currents. As the loop gets pulled toward the magnet, that field potential energy is reduced. So either type of magnet can supply the energy to make some mechanical motions. For purely classical currents, the path of that field energy to the motion goes through an electric field, since magnetic fields do no work on classical moving charges.

The big practical difference between the permanent magnet and the electromagnet is that the electromagnet runs down as the energy goes to just heating up the wires in which the current flows. So Its field energy has to keep being renewed from some source of electrical power, e..g. a big battery.

Mike W.

(published on 10/24/2016)