Why does a charge moving in a circular coil not lose energy?

Question

We know that an accelerating charge radiates energy. So, a charge moving in a circular coil should also moving out energy as its velocity, being a vector, changes constantly. So, it should also radiate energy and thus its kinetic energy should also decrease. Why doesn't it happen? And how is it different from Rutherford's Model (Is it because of the presence of a battery in the circular coil?)

Answer 1

In principal, they do radiate. It is just that the power loss is negligible. Consider a worked example using the Larmor formula, where we have a copper wire of radius $1~\mathrm{mm}$ coiled with a radius of curvature of $1~\mathrm{cm}$. Through this coil is flowing a current of $1\,\mathrm{A}$. The free charge carrier density of copper is about $8.5\times10^{22}\,\mathrm{cm}^{-3}$. From the cross section of our wire, we can find the total linear free charge carrier density to be about $2.7\times10^{21}~e\,\mathrm{m}^{-1}$. Coupled with our current, this corresponds to a drift velocity of about $2.3~\mathrm{mm}\,\mathrm{s}^{-1}$. Such a velocity in circular motion with the above radius corresponds to an acceleration of about $0.53~\mathrm{mm}\,\mathrm{s}^{-2}$. Using the Larmor formula with these quantities, we can estimate the total radiated power loss to be on the order of $10^{-39}~\mathrm{W}\,\mathrm{m}^{-1}$, which is completely inconsequential.

The ultimate answer as to why these charges do not radiate consequentially is that the accelerations they experience are trivial. The drift velocities encountered in typical situations are much too low, and the curvatures much too mild. Even at an acceleration of 1 billion $g$'s, we would still only expect a radiated power of something like a picowatt per meter of our above wire. This is an absolutely enormous acceleration to achieve mechanically (through the geometry of the wire alone), but it is trivial in circumstances where we typically think of charges radiating (calculate for yourself the typical acceleration of a charge in a dipole antenna or a cyclotron).