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After some persistent trouble, I finally succeeded in registering and logging in. Hurrah.

I cannot often use internet, but I will try to return on this forum at least once a week, ususlly in nor near the weekend.

I have a question about electromagnetism and I searched first using the words "moment of inertia of the electric field". The search returned nothing, so here is my conundrum. (It is not my homework assignment, or anything like that, as I am 53 years old; it is something I recenty thought of myself.)

Consider a spherical conductor of radius R, carrying an electric charge Q. There will be an electric field around it, and this field will contain some energy. Therefore, by Eintein's familiar formula it will contain some mass. This mass will increase the work needed to change the velocity of the sphere. (mvv/2) It will also, by its moment of inertia, increase the work needed to change the rotation of the sphere. (Iww/2)

The size of these phenomena should be calculable from Q, R and some constants of nature. There will be no electric field inside the conductor, and the field outside can be subdivided into an infinite number of equally thin spherical shells concentric with the sphere. By Coulomb's Law, the field strength in every shell will be proportional to the charge and inversely proportional to the square of the shell's radius r. Proportional, that is, to Q/rr. The energy density will be proportional to the square of the field strength, hence to QQ/rrrr. But the volume of the shell will be proportional to rr, so the energy contained in it will be proportional to to QQ/rr.

Adding up the energies contained in all the shells yields a total energy which is proportional to QQ/R. This is, of course, as it should be. It is the energy one had to expend in gathering together the charge from infinity onto the surface of the sphere. This energy had to go somewhere; it has gone into the field surrounding it.

But this energy has mass, and therefore adds to the moment of inertia relative to any axis passing through the center of the sphere. Here, too, every shell has its own contribution to make. Its moment of inertia is proportional to its mass multiplied by rr. Its mass, however, is proportional to QQ/rr. Therefore, its moment of inertia is proportional to QQ and independent of r. Every one of the infinite number of shells contributes the same amount. Adding up these amounts yields an infinite moment of intertia.

How can I escape from this crazy result? I can think of only three counterarguments.

1. The situation has spherical symmetry; therefore rotating the sphere does not change the field strength anywhere, which makes it meaningless to say that the field is rotating with the sphere.

But any scratch or bump on the sphere's surface would create an inhomogeneity in the field, which would rotate with the sphere. Moreover, rotating the sphere might set up a circular current, causing a magnetic field to form, and changing the situation "outside" from what it would be for a nonrotating sphere. (Such a current would certainly exist if the charge had been sitting on the surface of an insulating sphere.)

2. The field does not propagate at infinite speed; therefore charging the sphere cannot set up a field extending to infinity in finite time.

But this suggests that the moment of inertia would steadily, and linearly, increase without bounds. One would be able to ascertain how long ago the sphere was charged by measuring the torque needed to set it spinning.

3. It is impossible to gather a charge from infinity; therefore the charge on the sphere came from somewhere else, where an equal but opposite charge must now exist. The fields from the two charges will cancel at large distances from both, and anything contained in these fields will therefore have a finite value.

But this would still allow one to create a device carrying a huge moment of inertia relative to the axis separating the two charges while expending just a tiny amount of energy. If one considers two charges Q and -Q, both sitting on a sphere of radius R with the midpoints of these spheres separated by a distance D, the energy would for large values of D/R approach a value proportional to QQ/R, but the moment of inertia would be roughly proportional to QQD.

I wonder whether there is another counterargument.

(I also wonder whether, if I log out, I will again be able to log in; so if you never hear from me again, I probably wasn't able to. But I'll try anyway tomorrow.)

Almanzo

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# Finally logged in; moment of inertia of the electric field.

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