From another Physicsforums thread I found this outstanding webpage:
http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html" [Broken]

which contains this startling statement:
In electromagnetism, conservation of charge forbids the existence of an oscillating electric monopole.

Suppose you place a highly charged ball in space and oscillate it with an insulating piston connected to a simple mechanical engine which is in turn connected to a non conducting inertial mass.

Do you not have an oscillating electric monopole?

Or, consider a very long straight wire connected at both ends to small spheres. A current is generated in the wire by a "pioint sized" generator. Charges are built up on the two spheres, but they are so far apart you can easily detect the field from the nearest sphere independently of the other.

Effectively, measurably, an electric monopole?

Would the waves from these travel at c?

Is there a solution to Maxwell's equations for these?

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Do you not have an oscillating electric monopole?

Nope. Calculate the monopole moment before and after the expansion. Same number, right?

What expansion? We assume the highly charged conductive sphere is attached by a non-conductive connecting rod to the mechanical engine. The center of the conductive sphere, and the center of all other masses, all lie along the same line. The physical motion of the sphere is longitudinal, along this "mass center line". The sphere does not significantly change shape. What expansion?

Thought of a better example. Fewer moving parts. Two black holes orbit one another. Doesn't matter if they are equal mass or not. One has electric charge, the other does not. Electric monopole radiation? Yes, it would be very low frequency. Yes, their orbits decay due to gravitational waves. But the orbit could last many cycles. Black holes are permitted to carry charge. Mass, charge, linear and angular momentum and nothing else.

Staff Emeritus
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No, that's dipole radiation. Again, calculate the monopole moment at any two places in the orbit. Same number, right?

I fail to comprehend your response at all.

"Moment" in physics generally simply means means "force", albeit usually a vector rather than a layman's scalar. Googling "monopole moment" verifies this. The "monopole moment" for an electric field reduces to Coulombs law for a simple sphere of charge, measured or "sensed" at some distance, in a frame at rest with respect to the sphere.

But we are not at rest with respect to our sphere. We are at some distance, well OUTSIDE any orbit for the black hole example, at a SINGLE observation station, in an inertial frame at rest with respect to the Galactic background, and our sphere is moving! So we measure all the effects of a moving electric charge - Lorentz contraction of the charge distribution, some amount of magnetic field, et cetera. And if we CONTINUE to sense the fields at this SAME point WELL OUTSIDE the orbiting black holes or the mechanically driven sphere, we measure TIME-VARYING electromagnetic effects.

This is not, by definition, radiation? Maybe weak and low frequency, but not zero!

Staff Emeritus
2021 Award
"Moment" in physics generally simply means means "force", albeit usually a vector rather than a layman's scalar.

I'm sorry. I thought you understood multipole expansion of radiation. See http://en.wikipedia.org/wiki/Multipole_moment The way this works is that a radiation multipole (dipole, quadrupole, etc.) arises from a changing multipole moment - for example, a changing magnetic dipole will produce magnetic dipole radiation.

This is not, by definition, radiation?

No, it's not. Radiation has a specific definition - a changing field because something whooshes by is not "radiation by definition".

I am pleased that you finally understand my question. My math is very insufficient to understand multipole radiation. I was only able to understand the parts of articles on electric moments which explained that the zero-order term in a mathematical expansion for exterior electric moments is the monopole moment and is effectively Coulomb's law. My poor math is the reason I precluded all discussion of multipoles with words.

So to use your words, what does one call the changing fields caused by a monopole whooshing by?

Do these changing fields propagate at velocity c?

Is there a solution of Maxwell's partial differential equations for these changing fields?

And a belated thank you for being the only responder to this bizarre question.