Can Magnets do Work?
Most recent answer: 05/18/2011
- PChan (age 22)
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.
(published on 05/18/2011)
Follow-Up #1: Can magnetic fields do work?
- Miyze (age 20)
(published on 08/06/2012)
Follow-Up #2: more magnetic work
- ali (age 20)
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).
(published on 12/24/2012)
Follow-Up #3: source of energy for magnets
- Daniel (age 18)
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.
(published on 10/20/2015)
Follow-Up #4: magnetism and work
- Hans (age 24)
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.
(published on 01/21/2016)
Follow-Up #5: work and permanent magnets
- 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.
(published on 10/24/2016)
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