What energy is used by a magnet to attract another magnet?

People sometimes ask, “What is the source of the energy expended by a magnet to attract another magnet, giving it kinetic energy? After all, the magnet doesn’t get colder.” The energy aspects of a magnet accelerating another magnet can indeed seem mysterious.

Here is a brief summary of an argument presented in the mechanics section of our “Matter & Interactions” textbook that shows that it must be energy in the magnetic or electric or gravitational field that is the source of the increased kinetic energy. Consider two similar stars A and B far from each other, initially at rest. They accelerate each other, moving toward each other faster and faster. It is customary (and gives the correct results) to take the two stars as the system of choice and quantify the energy of the system (neglecting the constant rest masses) as kinetic energy plus gravitational potential energy of the two stars. The change in the energy of the system plus the change in the energy of the surroundings is zero (energy conservation). Since there is nothing in the surroundings, the change of kinetic energy plus change of potential energy is zero, from which we can calculate the final kinetic energy of the two stars.

If we choose star A as the system, the surroundings consist of star B. The kinetic energy of our chosen system, star A, increases, so for energy to be conserved the energy of the surroundings must decrease. However, the kinetic energy star B increases, not decreases. How can one deal with this paradox?

We state that there is energy in the “field,” field being a concept to be introduced in the second-semester course, and we’re even able to calculate the amount by which the field energy changes despite not even knowing what it is. Keep in mind that single objects don’t have potential energy, only pairs of interacting particles in a system have potential energy. Therefore the change K in the kinetic energy of the system (star A) plus the change K in the kinetic energy of star B (part of the surroundings) plus the change in the field energy (also part of the surroundings) must be equal to zero, so the change in the field energy is -2K. We then comment that calculations based on the concept of potential energy give the correct result, but that in general there’s an issue. The case of an electron and a positron attracting each other is entirely similar to the two-star case, but the field energy is associated with electric and magnetic fields.

The article on the net work done by magnetic forces shows that forces exerted by a magnetic dipole can do positive work on another magnetic dipole as long as these forces also do the same amount of negative work. In the case illustrated with a VPython program in that article, the rotational kinetic energy of a magnetic dipole increases with decreasing rotational kinetic energy, and vice versa. A question was raised about how we might understand the energetics of two magnets attracting each other. In response to this and others’ questions about this, Ruth Chabay and I had a conversation that resulted in some understanding, which I want to share.

The mystery of where does the energy come from to accelerate a magnet is no more (and no less) mysterious than the mystery of where does the energy come from for an electron to accelerate a positron. We don’t typically think about where the energy comes from when electric charges attract each other, but for some reason we do puzzle over the attraction of two magnets. In both cases the answer is field energy.

A VPython program models an electron and positron starting from rest and attracting each other. In a large cylindrical region of space, as the particles approach each other, the field energy is summed repeatedly over a volume much larger than the particle separation. As a computational convenience, the change in the total kinetic energy is repeatedly calculated from the change in electric potential energy, but it could also have been done in terms of iterative use of force and momentum, without using the concept of potential energy. The change in the kinetic energy plus the change in the field energy is seen to be zero as expected. The program also treats the case of two electric dipoles attracting each other, and for calculational purposes one can think of this as modeling two magnetic dipoles, each modeled as two magnetic monopoles. (In the case of the two moving charges, their magnetic field is ignored, and in the case of the two moving magnetic dipoles thought of as consisting of magnetic monopoles, their electric field is ignored, because the speeds are small.)

A tricky part of the calculation is to avoid the singularity of the infinite field near a point charge. Steve Spicklemire suggested a simple scheme, which is to model the point charge as a thin hollow uniformly charged sphere, so that this charge’s own contributions inside that sphere are zero. The space is divided into a large number of volume elements in the form of cyindrical shells, each of length and wall thickness equal to the diameter of the sphere. Since the sphere nearly fills its volume element, it’s a good approximation to say that the field contributed by this charge is zero throughout the volume element, and only the other charges contribute field in this volume element.

I exploit the axial and near-mirror symmetry of both situations, the two-charge case and the two-dipole case. I sum E2 only in the right half plane, because the net field in a volume element points in opposite directions in two “mirror” containers while the magnitude is the same. So I get the sum of E2 for the entire volume simply by multiplying by two the sum for the right half plane.

I sum E2 throughout a large cylinder which is subdivided into cylindrical shells of length d and thickness d. Looking end-on at a cylinder with inner diameter r and outer diameter r+d (and depth d), I compute E at location r+d/2 (and at the longitudinal center of the shell), take that as a good average to the field everywhere in the shell, and multiply E2 by the volume of the shell, dV = \pi ((r+d)^2 - r^2)d. The field energy content of this volume element is 0.5\epsilon_0 E^2 dV.

At first I couldn’t get the two-dipole sum to stay zero. It turned out that my choice of a volume that was supposed to represent all space was too small. When I made the total volume significantly bigger, the change in particle energy plus the change in field energy stayed (nearly) zero.

There’s another aspect to this computation worth pointing out. In M&I, as in other introductory treatments of electric field, we choose an object with charge q, calculate the field E contributed by all the other charges in the universe, and calculate the force F on the chosen object as F = qE. There is another way of dealing with electric field, of particular importance in plasma physics, which is to calculate the field due to all charges in the universe, and then draw conclusions from the pattern of this field. I calculate the field everywhere, made by all of the charges, and from this field I calculate the total field energy. This is a different way of thinking about electric field.

Jarrett Lancaster tells me that it is possible to calculate analytically the change in field energy for two charges approaching each other. I asked how the singularities are handled, and he explained that when you set up the integrals it is possible to identify a component that though infinite doesn’t change. He also said that it is useful to recognize that E^2 = (\vec{E}_1 + \vec{E}_2) \centerdot (\vec{E}_1 + \vec{E}_2) = E_1^2 + 2\vec{E}_1 \centerdot \vec{E}_2 + E_2^2, since the integrals of E_1^2 and E_2^2 do not change. He wasn’t sure whether the two-dipole case has an analytical solution, but it seems likely. Nevertheless, the numerical approach described here has the advantage that it can work for more complex situations than the ones described, though of course some of the symmetry properties exploited here might not be valid.

Bruce Sherwood

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