Electromagnetic Four-Vector for Uniformly Moving Charge

[ApplePion]

The general formula for the electromagnetic four-vector produced by a moving charge is the Lienard Wiechert formula, which involves the retarded position of the charge. However, in the special case where the motion of the charge is a uniform velocity motion, the result becomes extremely simple, with the retarded position no longer appearing. For example, the electric potential for a uniform velocity charge located at the origin becomes simply Q/R where R is NOT the retarded position, but actually the current position.

I need to convince someone of this, and it is unlikely he can be convinced by me calculating it for him. Can anyone supply me with a specific reference from a "respected" source where the result is clear. The actual physics calculation need not be clear--I am not really trying to reason with the person as much as to show him that some respected source agrees with me.

Thanks.

 

[vanhees71]

Try Jackson, Classical Electrodynamics. Of course, the solution with the Lienard-Wiechert potentials (retarded propagator) is the same as Lorentz boosting the Coulomb field of a point charge at rest to the frame where it is moving with constant velocity.

 

[ApplePion]

Thanks Vanhees.

Your derivation is absolutely correct. There are actually two (or more) ways to derive it, with your way being the best because it is simple and intuitive. There is also a brute force way by just substituting into the Lienard Wiechert formula.

As far as I can tell, Jackson actually does not do it. Griffiths does it using the brute force method, and is not as clear as I would like it to have been.

So I am still looking for a "convincing" source.

 

[vanhees71]

Another very nice book is

F. Rohrlich, Classical Charged Particles, World Scientific

It gives a comprehensive overview of "microscopic" classical electrodynamics, including a convincing treatment of the self-consistency problem ("radiation reaction") for accelerated charges, interacting with their own electromagnetic field.

 

[Meir Achuz]

Franklin's 'Classical Electromagnetism' derives the E field for constant velocity in Sec. 15.3, and shows that this result is equivalent to the L-W field in Sec. 15.4.3.