# Electric potential energy between charge and moving magnet

The setup for one magnet

Introduce a magnet:

Inertial Frame 1 (or lab frame)

A frame where a magnet is seen to move with uniform velocity v and carries a uniform polarization P while a point charge Q is seen to be stationary at time t=0.

Inertial Frame 2 (or material frame)
A frame where a magnet is observed to have a uniform magnetization M', while possessing no polarization (P'=0), and is seen to be stationary at time t'=0.

The polarization of the Magnet in Inertial Frame 1
This polarization manifests as an apparent charge separation.

The polarization of the Magnet in Inertial Frame 2
This polarization does not exist so no apparent charge separation manifests.

The electric field of the Magnet in Inertial Frame 1
This electric field E exists. It is the sum of both a solenoidal field and an irrotational field. The polarization P is responsible for the irrotational field only.

The electric field of the Magnet in Inertial Frame 2
This electric field E' does not exist (E'=0).

Introduce a point charge:

The point charge Q in Inertial Frame 1
This charge Q is not a point-like current element (yet was and will be).

The point charge Q in Inertial Frame 2
This charge Q is a point-like current element.

The setup for a system of magnets

Suppose that instead of one magnet we had a system of magnets, analyzed in Inertial Frame 1. Suppose that in this system of magnets, all the solenoidal fields contributing to the magnets' electric field E were canceled. So we have a constant magnetic field B due to this ensemble of magnets. What remains are only the irrotational fields contributing to the electric field E. So we have polarizations Pi due to magnets with velocities vi and "rest" magnetizations M'i.

Let's consider a special case of the above: a perfectly-symmetric cylinder magnet of finite length lined up with the z-axis and spinning only on the z-axis at constant angular velocity. The polarizations Pi observed in the non-rotating frame are cylindrically-symmetric in their distribution and are perpendicular to the axis of rotation at all points within the cylinder magnet. The result is an effective density of charge on the surface of the cylindrical magnet, with an equal and opposite charge within the interior of the magnet.

Let's consider a special case of the above: the cylinder magnet is now also centered on the origin of a cylindrical coordinate system, where the geometric center of the cylinder is located at point (0,0,0), while the charge Q is located at any point outside the cylinder where z=0. In the non-rotating frame, the charge Q is subject to the electric field E which lacks any component in z.

Let's consider a special case of the above: let's have the motion of the charge Q start off with an arbitrarily small velocity with no z component. The magnetic field of the cylinder magnet causes the charge Q to gyrate in the plane z=0. The magnetic field performs no work on the charge. The cylinder magnet is rotating, so it does produce an electric field E. The charge Q experiences an additional force, radial to the z-axis.

Considering different rates of rotation

Let's consider a set of special cases of the above: each member of the set of special cases differs by the rotational frequency of the cylinder magnet. The electric field of the cylinder magnet depends on terms on the order of O(v/c), but the magnetic field of the cylinder does not depend on terms on the order of O(v/c). The Barnett effect which is of order O(v/c) is excluded by the additional requirement that the magnetic susceptibility is zero within all points of the cylinder magnet.

Let's now consider a case which transitions between members of this set of special cases: the cylinder magnet is rotated around the z-axis at a steadily-increasing rotational frequency. The result is that the magnitudes of polarizations Pi increase with the rotational frequency and the charge Q is subjected to a stronger and stronger electric field E.

Let's consider a special case of the above: the electric field E at plane z=0 is directed so as to cause the charge Q to move away from the axis, towards infinity. Thus, the amount of work per unit distance traveled that the electric field E performs on the charge Q can in principle be integrated along the path undertaken by that charge. The work done is a function of the rotational velocity of the cylinder magnet. Therefore, the electric potential energy of the charge Q with respect to the cylinder magnet is a direct function of the cylinder magnet's polarizations Pi.

It would stand to reason that in order to increase the electric potential energy of Q with respect to the cylinder magnet, some additional sort of mechanical resistance to rotational frequency changes of the magnet should exist which must somehow depend on the location and magnitude of charge Q. However, in the literature, there is the concept of hidden momentum. A corollary of hidden momentum is hidden forces, or simply the time-derivative of the hidden momentum.

Some relevant articles:
The bottom line: It is unclear to me that as to whether the magnet would experience an additional sort of mechanical resistance to rotational frequency changes because of the position and magnitude of charge Q, requiring an external force to counteract, rather than some internal "hidden force".

Sincerely,

Kevin M.

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Dale
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2021 Award
I read through 3 or 4 scenarios before I got tired of reading, without ever seeing a point or a question. Why don’t you choose one scenario which most simply demonstrates the question or point and get to the point quickly?

I read through 3 or 4 scenarios before I got tired of reading, without ever seeing a point or a question. Why don’t you choose one scenario which most simply demonstrates the question or point and get to the point quickly?

Ok. So last night I came up with idea of describing my scenario in terms of other scenarios, like some sort of special case of a special case.

So for the vast majority of the scenarios, what I am essentially trying to do is picture a magnet and a charge, with the magnet being a relatively simple cylindrical magnet, having it spin on its axis of symmetry, and a having a charge being placed in a part of the electric field that is easier to think about (i.e. the plane where the electric field lines do not curve, the plane which is equidistant from both ends of the magnet).

Then I notice that electric potential energy of the charge seems dependent on rotational speed of the magnet, and I wonder about where the energy for this potential energy is supposed to come from. Does it comes from the kinetic energy of rotation of the magnet as a whole, or from some energy associated with charges inside the magnet, possibly hidden?

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Dale
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2021 Award
So of the vast array of scenarios, which single scenario best exemplifies that?

So of the vast array of scenarios, which single scenario best exemplifies that?

The one I described in my reply, implicitly.

Dale
Mentor
2021 Award
Ok, this is too annoying. I am done. Best of luck to you

mfb
Mentor
In general electric potential energy is not a good quantity in dynamic electromagnetic systems.

In general electric potential energy is not a good quantity in dynamic electromagnetic systems.

In that case, perhaps one can consider the electric field observed in a particular frame of interest, and be concerned not with the exact quantity of electric work that is done, but rather be concerned with whether that electric work is simply positive or negative, larger or smaller, and how that might depend on the electric field strength, which in turn depends on the electric polarization of the rotating magnet, as per the problem which I presented. That this electric work done on a bare electric charge Q even depends on the rotation of the magnet, begs the question as to what energy source expended when that work is done. Does it take energy of rotation of the magnet, or does it get it from the magnet's own electromagnetic field energy?

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