Why the Liénard-Wiechert Potential is the Result of a Flawed Derivation

The velocity dependence of the static electric potential of a point charge is attributed to A. Liénard and E. Wiechert who, independently of each other, derived.it in 1898 and 1909 respectively, although this was in fact already developed decades earlier by Gauss, Ampere und Weber. Educational texts nowadays usually derive the velocity dependence from the scaling of the spatial distribution function for the charge density introduced by the retardation associated with the finite propagation speed of the electric field/potential. Obviously, distribution of the charges over a different spatial scale must lead to a change of the charge density over this region, such as to preserve the total charge (i.e. the normalization of the distribution function in mathematical terms). However, this circumstance is completely neglected in all derivations of the potential of a moving charge, which is the sole reason why the velocity dependent term appears.
Rather than discussing this at the hand of the full general equation for the potential, we will use a one dimensional simplified equation here that demonstrates the independence of the potential of a point charge on its velocity

Proof

We consider the potential by two point charges with charges q/2 each situated at distance x' on the x-axis symmetrical to the origin.
The potential at point x (assumed >x') is then (assuming 'natural' i.e.. cgs units)

(1)   Φ(x) = q/2/(x-x') + q/2/(x+x')

Assuming a finite propagation speed c between points x' and x, the retardation condition is

(2)   x-x' = c.(t-t')

where (x',t') is the location and time of emission, and (x,t) the location and time of detection. For simplicity we take the convention

(3)   x=ct

that is, the zero point of the time variable is chosen such that a signal from the origin is submitted at t=0, arriving thus at location x at time t=x/c.
We than have obviously

(4)   x'=ct'

For the location of a uniformly moving charge, we have furthermore the constraint

(5)   x'=d+vt'

with d the distance from the origin at t'=0.
From (4) and (5) we have then

(6)   x'=d/(1-v/c)

and inserting this into (1)

(7)   Φ(x) = q/2/(x-d/(1-v/c)) + q/2/(x+d/(1-v/c))

which can be written as

(8)   Φ(x) = q.x/(x2-d2/(1-v/c)2)

If we let now d→0 (i.e. joining the two particles with charge q/2 into one particle with charge q at the origin), we obtain

(7)   Φ(x) = q/x

In other words, the potential of a point charge is depends only on the charge q and the retarded distance, but is independent of its velocity.

Conclusion

We have seen from a simple example that the potential of a point charge does not depend on the velocity as claimed erroneously in Classical Electrodynamics in terms of the Liénard-Wiechert potential. For a spatially extended charge distribution, the apparent relative positions of the charges, and thus the resulting potential, will obviously in general depend on the velocity because of the retardation (see Eq.(8)), but for sufficiently large distances, the potential will then still approach that of a point charge at rest at the retarded position.


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Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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