# How does electromagnetic radiation affect the velocity of a charged particle?

I've heard that the acceleration of a charged particle releases electromagnetic waves. So let's say there is a charged electron moving forwards in a region with a downwards magnetic field. If the magnetic field is a certain strength, it should cause the particle to travel in a circular arc. This kind of motion is centripetal acceleration, and since the particle is accelerating and is charged, it should release radiation. Is this correct? And if so, how is the kinetic energy of the charged particle affected by this release of radiation?

• This is correct. The particle will radiate and has to lose kinetic energy. As a result the radius of its circular motion will decrease. This is one of the limiting factors for circular accelerators and storage rings. – CuriousOne Apr 27 '15 at 23:32
• @CuriousOne Ah I see, so circular accelerators must constantly apply an electric field to counteract this loss of kinetic energy? – Sully Chen Apr 27 '15 at 23:51
• Absolutely. Machines like the LHC are basically very large storage rings. Even after the particles have been accumulated and accelerated, one still needs to run the RF cavities, the active parts of the accelerator, to compensate for the beam losses due to these synchrotron radiation losses. The losses in the LHC are not that bad, on the order of 0.2W/m, I believe, but electron storage rings like the LHCs predecessor LEP can have losses in the kW/m range, which is a lot of x-ray radiation. Search for "lep synchrotron spectrum" and you can find lots of good stuff about it. – CuriousOne Apr 28 '15 at 1:07
• @CuriousOne Wow, fascinating, thanks for the info! – Sully Chen Apr 28 '15 at 2:20

since the particle is accelerating and is charged, it should release radiation. Is this correct?

This is the way experiments with radiation are often explained - radiation comes from places where acceleration of charge occurs.

Also, orbital motion of charged particles in a cyclotron was found to be always accompanied with EM radiation coming off apparently from the ring where the electrons are in circular motion.

Theoretically, this is based on a particular version of EM theory where the fields are purely retarded. This version is most common and intuitive and I suppose it is not a great mistake to say it is correct, but it does not follow that is what actually happens.

There are several possibly correct explanations - other views of the experiments, like those of Tetrode, Fokker, Frenkel, Feynman-Wheeler etc. where the radiation present/detected is not just the retarded (in particle originating) field of the charged particles. For example, the half-retarded, half-advanced field has two important parts - the retarded half compatible with the above view originates in the particle and spreads away to infinity, but the other one, advanced, comes from the infinity and collapses on the particle. This view is not usual, but it was considered seriously by the above people and it was never falsified.

And if so, how is the kinetic energy of the charged particle affected by this release of radiation?

That depends on whether the charge distribution is extended with finite density or point-like.

If the charge has finite spatial density everywhere, then the Poynting theorem can be derived. Further, if the fields are retarded, the Larmor formula can be derived which says the energy that comes off the particle per unit time is proportional to acceleration squared. If the particle has stable inner structure, per energy conservation this approximately equals to the loss of kinetic energy per unit time as well (there is always discrepancy due to EM energy in the vicinity of the particle) and so the particle loses kinetic energy until its motion changes into rectilinear motion or the particle stops moving.

On the other hand, if the particle is truly a point, the Poynting theorem cannot be derived and the Larmor formula cannot be derived. Furthermore, they cannot be consistently applied, even if we took them verbatim from the above case.

If the fields are retarded, such particle produces outgoing changes in the EM field, but these do not necessarily need to have any effect on the motion of the particle.

The cyclotron/synchrotron radiation definitely carries and is capable to release a lot of energy. This means that for these explanations to be applicable, either the particles themselves are extended charges or they are pointlike but then radiation comes off many of them synchronously, so they effectively form extended distribution in the physical sense.

Since the charges in cyclotrons are accelerated in so-called bunches (billions of electrons) and since no inner structure of electron was ever found, I think the second explanation is more likely - the beams of energy stream come off many particles close to each other rather than from one particle.

It will decelerate causing its speed to decrease, and because of $r=mv/qB$ the radius will decrease as well and you will get a spiral motion.

This deceleration due to radiation is known as the Abraham-Lorentz force of radiation backreation. Using these equations you can more precisely derive the spiral motion. This effect is also responsible for the failure of the classical picture of the atom, because of the collapse, which is saved by the uncertainty principle.