(1 (+-) v/c) factor in the Lienard Wiechert potentials

[jason12345]

Does anyone know how either Lienard or Wiechert justified the (1-v/c) factor that appears in their potential formula for a moving charge?

 

[Meir Achuz]

The derivation is in many textbooks.

 

[jason12345]

The derivation is in many textbooks.

How do you know their interpretation is the same as that of Lienard and Wiechert?

In my copy of The Quantum Theory of Radiation by Heitler, for example, he uses the idea of a collapsing spherical wavefront with velocity c centred on the field point that samples the charge within a volume dv. He then says that within a time dt as the spherical wavefront moves a distance dr, charge is added or removed from the volume dv, compared to if it the charge inside was static and hence gives rise to the additional term rho v/c.

Yet this charge entering/leaving the volume element is compensated by charge leaving/entering another volume element so the net effect is that there is a change in the charge density which is compensated by the change in the volume occupied by the charge as the wavefront sweeps through it. This means the total charge sampled by the wavefront should be conserved, whereas the Lienard Wiechert expression shows that it isn't.

Hence, I'm interested in knowing what arguments Lienard and Wiechert originally used.

 

[Meir Achuz]

I think you are referring to a factor $$(1-{\hat r}\cdot{\vec v}$$ introduced into the expression for the power radiated into a solid angle, and not a factor in the L-W potentials. Heitler's description is a bit convoluted, but gets the right factor.
A simpler derivation (probably not L's or W's) is that that factor equals
$$\frac{dt}{dt_r}$$ where $$t_r$$ is the retarded time.