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I've been reading up on electric charges that oscillate in space, creating radio waves and the like, and it's got me wondering.

If an otherwise stationary spherical electric charge oscillated in time, transitioning back and forth between $q = 0$ and $q = -x$ (where $x$ is some arbitrary value), what would be the shape of the resulting magnetic field?

Mauricio
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WaveInPlace
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  • Monopoles do not radiate: https://physics.stackexchange.com/questions/139819/why-does-a-monopole-not-radiate-energy-in-electodynamics – Mauricio Mar 02 '23 at 14:09

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Maxwell's equation require charge conservation if they are to be internally consistent. Your charge is not conserved, so there is no possible solution to your problem.

mike stone
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  • Could that could be solved by having a second, counter oscillating charge, such that the total charge in the system is always 0? – WaveInPlace Mar 02 '23 at 14:13
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    Only if they are connected by a wire that transfers the charge. Then you will need to include the magnetic field of the wire. Thi is hoiw a dipole antenna works. – mike stone Mar 02 '23 at 14:36
  • Why can we not have an oscillating electro-chemical reaction of a spherical shape so that the ions and electrons move radially back and forth, something in analogy with the Belousov-Zhabotinskiy reaction, and thereby have an electro-chemical (very low frequency) antenna. This way charge is conserved both locally and globally. – hyportnex Mar 02 '23 at 15:02
  • Hyportnex, that would still involve movement in space, and like connecting two charges with a wire changes the question. I'm not sure oscillation solely in time has a well-characterized physical parallel.

    As a thought experiment though, I'm not sure why the two charges would have to be physically connected to obey Maxwell's laws, rather than that just limiting the minimum size of the system to a volume that contains both charges. The classical view of light, for example, doesn't appear (to my uneducated eye) to follow Maxwell's laws for any system less than a full wavelength in size.

     – WaveInPlace Mar 02 '23 at 16:19
  • You need to satisfy $\dot \rho+ \nabla\cdot {\bf j}=0$ at all points in space and time in order for for Maxwell to have solutions. You cannot have a $\dot \rho$ without a current ${\bf j}$ therefore. – mike stone Mar 04 '23 at 16:06