The notion that an accelerating charged particle emits electromagnetic radiation started with an 1895 article by Joseph Larmor based upon the derivation by Hendrik Lorentz of the force experienced by a particle due its charge field being dragged through the ether. It was not appreciated at the time but Larmor's analysis presumed that the velocity of the charge was small relative to the speed of light in a vacuum. Larmor's analysis was followed by a more comprehensive analysis by Alfred-Marie Liénard in 1898 and, independently, by Emil Wiechert in 1900. Their analyses were compatible with Einstein's Theory of Special Relativity published in 1905. Liénard and Wiechert based their analyses on the vector and scalar potentials of the electric field of a particle.

But all of these analyses were based upon a charged particle being a point particle, a singularity. A point particle is a dangerous concept because a charged point particle would have infinite energy. It would take infinite energy to just produce one such particle. The energy of all the stars in the Universe would not be be sufficient to produce one. The danger of the concept of a point charged particle is that it would lead to producing nonsensical theoretical results.

The Theory

Larmor's analysis leads to the electromagnetic radiation being emitted by a particle of charge Q under going an acceleration of A being proportional to

Q²A²

For the radiation to be the same for deceleration of −A as for an acceleration of +A sounds suspiciously like nonsence. But the crucial matter is the dependence upon Q².

Consider a charge that is spread over some space. If the charge of Q is distributed over M pieces then each of the M points radiate an amount proportional to (Q/M)². The M pieces altogether have a total radiation of M(Q/M)²=Q²/M. If M goes to infinity, as it would for a spatially distributed charge no matter how small the region of distribution, then the radiation goes to zero.

There is and has been a good deal of skepticism among physicists such as Ricard Feynman and Sir James Jeans concerning radiation by accelerated charged particles.

This issue is always raised in connection with models of atoms involving charged pparticles traveling in circular orbits. The above shows this is a nonissue for these models unless the charged particles are assumed to be point particles.

Conclusion

Real particles have spatially distributed charges. Spatially distributed charges do not radiate. Period!