Why doesn't a charged particle moving at a constant speed radiate?

[alikim]

If I stand by a flying with constant speed charged particle, at my location the electric field will change as the particle get closer and further from me. So this change in electric field should create magnetic field and so on, producing an EM wave?

 

[Ibix]

The simplest argument is energy conservation. If the charge radiated, where would the energy carried by the radiation come from? If the charge is accelerating it comes from whatever is doing the acceleration, but if it is moving at a steady speed there is no energy source.

Looking at the fields, the electric field is radial from the charge and the magnetic field is axisymmetric about the direction of motion. If you compute the Poynting vector you will see that the energy flows along with the charge - unsurprisingly given that the field remains localised around the charge. So there's no flow of energy away from the charge. In short, an EM wave is a particular pattern of changing electric and magnetic fields, and although there are changing electric and magnetic fields here they are not the right pattern to be an EM wave.

It's worth noting that you can always use the Fourier transform to write a changing EM field as a sum of inifinitely many sine waves. So it is possible to write this field as a sum of waves, but they interfere destructively away from the source so no energy is radiated even if you view the EM field this way.

 

[A.T.]

If I stand by a flying with constant speed charged particle, at my location the electric field will change as the particle get closer and further from me. So this change in electric field should create magnetic field and so on, producing an EM wave?

Note that even if a particle undergoes constant proper acceleration, it doesn't radiate in its own rest frame.