Magnetic Field from a Single Electron

1. Feb 21, 2015

kq6up

I understand that an electric field from a string of electrons traveling in a wire gives a steady (magneto-static) field. Since this field is static, it will not cause a EM wave to propagate away from it. I also understand that individual charges that are not accelerating are not radiating since steady motion is frame dependent, and all interesting physics should be frame independent (Noether's Thm).

However, it seems if you were an observer measuring a magnetic field of an electron traveling past you. That magnetic field would be changing in the lab frame. Therefore, the dB/dt term would not be zero. This seems like it should cause a propagating EM wave by my thought experiment. Could someone help me with this apparent contradiction.

Thanks,
Chris

2. Feb 21, 2015

Einj

The magnetic and electric field are tightly related with each other. In particular, if you have an electron travelling with constant velocity and you see a magnetic field then if you go to the electron rest frame, the Lorentz transformation will mix the magnetic and electric field and the final result is that you'll just see an electric field.

3. Feb 21, 2015

kq6up

Indeed, in the rest frame of the electron there is only E field in a steady state. What explains the lack of radiation in the lab frame?

Chris

4. Feb 21, 2015

Einj

You only have radiation if the electron is accelerating.

5. Feb 21, 2015

Staff: Mentor

If you look at the Lienard Weichert potential you see that the radiation terms are proportional to acceleration, not velocity. So although there is a non-zero dB/dt, the energy and fields from that term do not radiate.

http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

6. Feb 21, 2015

kq6up

Yes, that is what I clarified in my first post. I know that it doesn't. I just need the mathematical explanation as to why not as I try to visualize what is going on with Maxwell's Equations in such a case. I will check out the Wiechert potentials.

Regards,
Chris

7. Feb 22, 2015

tech99

8. Feb 22, 2015

tech99

The magnetic field of the moving electron contains stored energy. It is therefore reactive and does not represent power radiated. It is equivalent to the magnetic reactive near field of an antenna. If an electric field is being used to accelerate the electron, then that has a reactive electric component corresponding to the electric reactive near field of an antenna. In a sinusoidal case, the accelerating electric field and the magnetic field of the electron are 45 degrees out of phase; half the energy is stored and half radiated.