Ring of Charges

A single charged particle moving in uniform circular motion undergoes centripetal acceleration and radiates light. The phenomenon is well-understood. A charged particle spit out in a jet by a supermassive black hole glows with this light. The Advanced Light Source at UC Berkeley exploits this light. And this light was missing from the atom, invalidating the Bohr theory of electron orbits.

But it’s also well-known that a conducting ring sustaining a constant current, such as a superconducting coil below the transition temperature, does not radiate, though it’s just a superposition of many of these single charges, arranged in a symmetric way. This latter case can be understood as a statics problem. If the ring is modeled as a continuous charge density, its configuration at any point in time cannot be distinguished from any other point in time, therefore, the fields also cannot change. Static fields do not radiate.

I wanted to understand how the transition happens, how, as we add more particles, the light turns off.

jackson

Using the equation for the electric field derived from the Lienard-Wiechert potential for a radiating point charge in the non-relativistic limit (Jackson E&M), I plotted the field at a considerable distance from the sources (x,y)=(5,10). As I was only interested in the dynamic component of the field, the plotted values have their means subtracted.

Here’s the field for one charge:


Two charges, evenly spaced:


Three charges:


Four charges:


Five charges (here we’re starting to run into MATLAB’s floating point number precision limit):


The single charge case can be understood as a rotating dipole, confirmed by the radiation pattern. Add a second charge and we’ve canceled the net dipole moment, only a quadrupole remains. The magnitude of the field is reduced by some factor involving v/c. The next order in the multipole expansion is an octupole, with the magnitude of the field further reduced by the same factor. We see this geometric progression as we add more charges:

plot1

Another thing to notice is that a ring of N charges has N-fold rotational symmetry and the period of the emitted radiation is reduced by a factor of N in each case. As N approaches infinity, the ring approaches a continuous charge density and both the amplitude and period of the emitted radiation go to zero.

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