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You will see statements like...

1. “Maxwell's Equations say that accelerating an electric charge results in the emission of electromagnetic radiation” (they actually say no such thing).
2. “According to classical physical theory an electron moving in an orbit would gradually radiate its energy and fall into the nucleus”.

While there are no issues with electrons losing or gaining energy in interactions with the atom’s electric fields, the argument is suspect when applied to simple acceleration outside the presence of such fields. Although quantum physicists commonly believe that classical physics predicts the behaviour described in the first paragraph above, for classical physicists the issue remains unsolved, because all theories of this have so far failed since they violate the principle of Conservation of Energy. There are three hurdles...

1. The basic theory concerning the mechanism
2. The Principle of Conservation of Energy
3. The Conservation of Momentum
4. Relativity - there can be no “special viewpoints”

You will come across theories of this mechanism in some textbooks. The argument goes like this:-

1. An accelerating charge radiates electromagnetic energy.
2. It takes extra energy to accelerate the particle above that required to drive the kinetic energy because of the radiative energy loss.
3. This extra energy increases the perceived inertia of the charged particle, so more work has to be done in accelerating it.

However, there is a serious problem with this concept. Try this thought experiment. Take a charged particle at rest in the local reference frame. Accelerate it to a second rest frame that is moving with respect to the local one. Then decelerate it back into the local rest frame:-

During acceleration, according to the above argument, the total energy supplied must be the sum of the kinetic energy and the radiated energy... Ein = Ekin +Erad1   So far so good, just as claimed. Now for the deceleration. From the point of view of the charged particle, however, deceleration is simply acceleration in another direction. It’s all relativity, and there is no “special” frame of reference. Hence the particle radiates during the deceleration, and the inertia of the particle is again increased during deceleration by the radiated energy, adding to the kinetic energy returned. Eout = Ekin+Erad2 To make the result simpler, let the deceleration profile match the acceleration profile, so that Erad1 is numerically equal to Erad2... Eout = Ekin+Erad2 = Ekin+Erad1 = Ein Hence all the energy put in as Ein is returned as Eout . But we have also released two bursts of radiation. The total energy in the system starts out as zero, and ends up as Eout - Ein + Erad1 + Erad2 =Erad1 + Erad2 Hence there is an increase in net energy, and the concept that simple accelerating charges radiate violates the Conservation of Energy.

You can believe that accelerating charges radiate, or you can believe in the principle of Conservation of Energy, but not both.

Another way of looking at it is this. An electron cannot distinguish between acceleration and deceleration (which is simply acceleration in the opposite direction. So if an electron always radiates electromagnetic energy under acceleration, and increases its inertia, it must equally do so under deceleration, since there are no “special” viewpoints in a relativistic universe, and deceleration to the electron is simply acceleration in a different direction. Hence for the same acceleration/deceleration profile the work expended during acceleration must be returned in full during deceleration, and a radiation burst is thrown in for free!

 Bear in mind we are talking about simple acceleration of a “free” charged particle.

This appears to be a generic constraint for a relativistic universe where there are no “special viewpoints”, and energy is lost from the system in a non-recoverable fashion.

If a theory one day manages to pass the Conservation of Energy constraint, it must then deal with the conservation of momentum. The radiated energy must either be radiated in-line forwards and/or backwards, or else in all directions in the plane normal to the acceleration. Any other option violates conservation if the electron is not deflected sideways during acceleration. This being the case, there is no mechanism that will permit an electron to spiral into the nucleus since it would require momentum loss to be along the direction of motion relative to the nucleus, rather than in-line with the acceleration; since motion is relative, Relativity, in disallowing any special viewpoint, disallows this too.

And Relativity constraints?

If you take the example above, and view the first acceleration from the rest frame the particle is in after its acceleration rather than the original reference rest frame, you will see a kinetic energy E_kin which after deceleration (as it is from this viewpoint) into this new rest frame yields inertial energy E_kin + E_rad after releasing a further E_rad radiation energy during the deceleration. In other words, for the initial acceleration to obey the Conservation of Energy it must be viewed from a very specific rest frame. Relativity does not allow this.

It may originally have come from the phenomenon of Bremsstrahlung (German for “braking radiation”) which is the radiation that occurs when charged particles are fired at high energy at a metal target, and was at one time assumed to be a radiation caused as a direct result of heavy braking. Now that Bremsstrahlung is better understood it is considered to be the interaction of the charged particles with the atomic fields of the atoms in the target, or with magnetic fields in cyclotronic or synchrotronic radiation.

There are also claims in some text books that electric radio aerials radiate by the acceleration of electrons up and down the aerial. However, according to Maxwell’s equations, while the electrons are in motion from one end of the aerial to the other they induce a magnetic field; if the aerial is vertical this field is horizontal, ringed round the aerial. When the electrons have finished their drift motion a quarter of a cycle later, and the voltage across the dipole is at its peak, the aerial generates an electric field between the excess of electrons at one end of the aerial and the excess of protons at the other; for a vertical dipole the electric field so induced is vertical. Another quarter of a cycle later the electrons are drifting towards the opposite end of the dipole, generating a magnetic field in the opposite direction to before as they do so. The last quarter of the cycle involves the electrons now being gathered at the other end of the dipole, generating an electric field in the opposite direction to before. It is this alternating and crossed electric and magnetic fields that directly drive the Maxwellian radiation. In fact, aerial design uses this field generation mechanism as the design parameters, not the supposed radiation from acceleration of electrons.

The last scenario sometimes quoted is where a charged particle enters the poles of a magnet in a cloud chamber. It will spiral in the field in a decreasing loop, until it comes to rest. Again, this is an interaction with an external magnetic field, not simple braking; the disturbance in the magnetic field creates a spin wave (a sort of travelling precessional magnetic resonance) across the surface of the magnet’s poles, which in turn dissipates the energy via the Barkhausen effect (a sort of magnetic micro-hysteresis).

Free electrons do not radiate under simple acceleration. Bound electrons can of course radiate from interactions with external electric and magnetic fields.