Why Doesn’t a Steady Current Loop Radiate?

The answer must be that the radiation is canceled by destructive interference between the radiation fields of the large number N of electrons that make up the steady current. A single electron in uniform circular motion emits electric dipole radiation, whose power is proportional to the square of the acceleration a = v/r, and hence to (v/c). But, the electric dipole moment vanishes for two electrons in uniform circular motion at opposite ends of a common diameter; quadrupole radiation is the highest multipole in this case, with power proportional to (v/c). It is suggestive that in case of 3 electrons 120◦ apart in uniform circular motion the (time-dependent) quadrupole moment vanishes, and the highest multipole radiation is octupole. For N electrons evenly spaced around a ring, the highest multipole that radiates in the Nth, and the power of this radiation is proportional to (v/c). Then, for steady motion with v/c 1, the radiated power of a ring of N electrons is very small. Verify this argument with a detailed calculation. Recall the basic expression for the vector potential of the radiation fields,