(1) ∫ d^{3}**r** ρ(**r**,t) = N

We re-arrange now the spatial positions of the particles to yield a different distribution function ρ'(

(2) ∫ d^{3}**r** ρ'(**r**,t) = N

(3) ρ'(**r**,t) = k(t)^{.}ρ(**r'**(**r**,t))

Inserting (3) into (2) yields then

(4) k(t)^{.}∫ d^{3}**r** ρ(**r'**(**r**,t)) = N

(5) k(t)/λ(t)^{.}∫ d^{3}**r'** ρ(**r'**,t) = N

(6) λ(t) = |det(∂**r'**/∂**r**)|

(7) (∂**r'**/∂**r**) = ∂**r' _{i}**/∂

From a comparison of (5) with (1) it follows thus

(8) k(t)=λ(t)

so
(9) ρ'(**r**,t) = λ(t)^{.}ρ(**r'**(**r**,t))

(10) x'(x,t) = λ^{.}x

(11) ρ'(x,t) = λ^{.}ρ(λ^{.}x,t))

In the next section we will apply this result to the case of the evaluation of the retarded potential of a system of moving charges.

(12) Φ(**R**) = q^{.}∫ d^{3}**r** ρ(**r**) / |**R**-**r**|

If instead we have a time dependent distribution due to particle motion and assume a finite propagation speed of the potential, the total number of particles (and thus the total charge here) is preserved, but the instead of ρ(

(13) Φ(**R**) = q^{.}λ(t)^{.}∫ d^{3}**r** ρ(**r'**(**r**,t)) / |**R**-**r'**(**r**,t)|

. Note that in (13) the density distribution is still formally written as a function of t (that is, unretarded in time), but it is a function relating to the retarded positions

Changing the integration variable to

(15) Φ(**R**) = q^{.}∫ d^{3}**r'** ρ(**r'**(**r**,t)) / |**R**-**r'**(**r**,t)|

(16) Φ(**R**,t_{0}) = q^{.} / |**R**|

In general, the potential depends obviously on the retarded positions

In order to illustrate the effect that the retardation has on the apparent charge density distribution and the resulting potential, we shall consider here a one dimensional scenario with a number of point charges at locations x within a finite length L and this whole configuration moving with speed v with regard to the observation point X (assumed >x) on the same line. The retardation condition in this case is

(17) X-x' = c^{.}(t-t')

(18) x'=ct'

For the location of a uniformly moving charge, we have furthermore the constraint(19) x'=x+vt'

where x is the position of the charge at t'=0, from which we get immediately(20) x'=x/(1-v/c)

In Fig.1 below, this result is shown graphically for a number of charges between x=-L/2 and x=+L/2 , at the top for the charges at rest, and at the bottom according to (20) for the whole configuration moving with speed v=0.5Fig.1: Retarded vs. unretarded density distribution

The expanded scale of the retarded distribution is due to the time layering of the picture caused by the different distances of the charges to the observation point X, which means that (assuming all charges are moving towards the observer) a signal from a distance closer than the origin has to be emitted at a later time t' in order to be observed at time t at point X, whereas a signal from a distance further than the origin has to be emitted at an earlier time. The retarded distribution is therefore spread over a corresponding range of retarded times. This of course is nothing new as such, and the derivation of the retarded potentials in the literature (following on from the work of Liénard [1] and Wiechert [2]) is indeed based on this circumstance, but it has been apparently overlooked that the expanded scale of the retarded distribution (by a factor 2 in this example) goes along with a corresponding decrease of its density, thus keeping the total charge constant, as already shown formally in Sect.II. Trying to keep the charge density in the bottom section in the diagram identical to that at the top whilst maintaining the expanded scale would be equivalent to doubling the number of charges for the retarded distribution (or alternatively doubling the value of the elementary unit charge). The Liénard-Wiechert potential, as derived in the literature throughout, is thus solely the result of a violation of charge conservation in this sense.

Of course, for a spatially expanded distribution as in this case, the apparent change of the spatial scale due to the retardation will result in a velocity dependent potential observed at a point X, but this velocity dependence disappears if the scale of the charge distribution becomes infinitesimally small. For instance, if we consider just the two outermost charges in Fig.1 i.e. if

(21) ρ(x) = δ(x-L/2) + δ(x+L/2)

and thus(22) ρ(x') = δ(x'-L/2/(1-v/c)) + δ(x'+L/2/(1-v/c))

so
(23) Φ(X) = q^{.}∫ d^{3}x' ρ(x') / |X-x'| = q/[X-L/2/(1-v/c)] +q/[X+L/2/(1-v/c)]

(24) Φ(X) = 2q/X

that is the classic Coulomb potential of a charge 2q located at the origin, regardless of the speed of that charge with regard to the observer at X.
(25) **E**(**R**,t) = -∇Φ(**R**,t) - 1/c^{.}∂**A**(**R**,t)/∂t

(26) **A**(**R**) = q/c^{.}∫ d^{3}**r'** **v**(**r'**(**r**,t))^{.}ρ(**r'**(**r**,t)) / |**R**-**r'**(**r**,t)|

If we want to calculate the field of an individual charge, we have to set the density distribution

(27) ρ(**r'**(**r**,t)) = δ^{3}(**r'**-**s**(t'(t)))

(28) Φ(**R**,t) = q/ |**d'**(**R**,t)|

(29) **A**(**R**,t) = **v**(**s**(t'(t)))/c^{.}q / |**d'**(**R**,t)|

(30) **d'**(**R**,t) = **R**-**s**(t'(t))

(31) ∇Φ(**R**,t) = -q/λ(**R**,t) ^{.} **d'**(**R**,t)) / |**d'**(**R**,t))|^{3}

(32) 1/c^{.}∂**A**(**R**,t)/∂t = q/c^{2} ^{.} [ **a'**/|**d'**(**R**,t)) + 1/λ(**R**,t) ^{.}**v'**v'^{.}cos(θ)/ |**d'**(**R**,t))|^{2} ]

(33) λ(**R**,t) = 1 - v'/c^{.}cos(θ)

Furthermore,

With (25), the electric field thus becomes

(34) **E**(**R**,t) = q/λ(**R**,t) ^{.} **d'**(**R**,t)) / |**d'**(**R**,t))|^{3} - q/c^{2}/λ(**R**,t) ^{.}**v'**v'^{.}cos(θ)/ |**d'**(**R**,t))|^{2} - q/c^{2}^{.}**a'**/|**d'**(**R**,t))

It is furthermore of crucial importance that the acceleration term does not contain the velocity dependent factor 1/λ(

One should also note that despite the factor 1/λ(

(35) E_{r} = **E**(**R**,t)^{.}**u'** = q/λ(**R**,t)/|**d'**(**R**,t))|^{2} - q^{.}v'^{2} cos^{2}(θ)/c^{2}/λ(**R**,t)/ |**d'**(**R**,t))|^{2} =

= q/λ(**R**,t)/|**d'**(**R**,t))|^{2} ^{.}(1-v'^{2} cos^{2}(θ)/c^{2}) =

= q/|**d'**(**R**,t))|^{2} ^{.}(1+v'^{.} cos(θ)/c)

(36) E_{t} = |**E**(**R**,t)×**u'**| = q/|**d'**(**R**,t))|^{2} ^{.}v'^{2}/c^{2}^{.}cos(θ)^{.}sin(θ)/(1 -v'/c^{.}cos(θ))

(A.1.1) ∇Φ(**R**,t) = ∇(q/ |**d'**(**R**,t)|)

(A.1.2) d' = |**d'**(**R**,t)|)

(A.1.3) ∇Φ(**R**,t) = ∂Φ/∂d' ^{.}∇_{R}d' = -q/d'^{2} ^{.}∇_{R}d'

(A.1.4) **R** = (X,Y,Z)

(A.1.5) x'=X-s'_{x} ; y'=Y-s'_{y} ; z'=Z-s'_{z}

(A.1.6) **d'**(**R**,t) = (x',y',z')

(A.1.7) ∇_{R} = (∂/∂X , ∂/∂Y , ∂/∂Z)

(A.1.8) d' =√(x'^{2}+y'^{2}+z'^{2})

(A.1.9) ∂d'/∂X = 1/2d' ^{.}[ 2x'^{.}(1-∂s'_{x}/∂X) - 2y'^{.}∂s'_{y}/∂X - 2z'^{.}∂s'_{z}/∂X ]

(A.1.10) d' = c^{.}(t-t')

(A.1.11) ∂s'_{x}/∂X = ∂s'_{x}/∂t' ^{.}∂t'/∂X = -v'_{x}/c ^{.}∂d'/∂X

(A.1.12) ∂s'_{y}/∂X = ∂s'_{y}/∂t' ^{.}∂t'/∂X = -v'_{y}/c ^{.}∂d'/∂X

(A.1.13) ∂s'_{z}/∂X = ∂s'_{z}/∂t' ^{.}∂t'/∂X = -v'_{z}/c ^{.}∂d'/∂X

After inserting (A.1.11)-(A.1.13) into (A.1.9) we obtain after a little algebra using the fact that (see (A.1.6))

(A.1.14) x'^{.}v'_{x} + y'^{.}v'_{y} + z'^{.}v'_{z} = **d'**^{.}**v'**

(A.1.15) ∂d'/∂X = x'/d' + **d'**^{.}**v'**/d'/c ^{.}∂d'/∂X

(A.1.16) ∂d'/∂X = x'/d'/(1-**u'**^{.}**v'**/c)

(A.1.17) **u'**=**d'**/d'

Analogously we get for the Y and Z components of the gradient of d'

(A.1.18) ∂d'/∂Y = y'/d'/(1-**u'**^{.}**v'**/c)

(A.1.19) ∂d'/∂Z = z'/d'/(1-**u'**^{.}**v'**/c)

(A.1.20) ∇_{R}d' = **u'**/(1-**u'**^{.}**v'**/c)

(A.1.21) ∇Φ(**R**,t) = -q/(1-**u'**^{.}**v'**/c) ^{.}**d'**/d'^{3}

(A.1.22) **A**(**R**,t) = **v'**/c ^{.}Φ(**R**,t)

(A.1.23) 1/c^{.}∂**A**(**R**,t)/∂t = 1/c^{2} ^{.}∂(**v'**^{.}Φ(**R**,t))/∂t =

= 1/c^{2} ^{.}[ Φ(**R**,t)^{.}∂**v'**/∂t + **v'**^{.}∂Φ(**R**,t))/∂t ] =

= 1/c^{2} ^{.}[ 1/d'^{.}∂**v'**/∂t + **v'**^{.}1/d'^{2}^{.}∂d'/∂t ]

In the way of a total time derivative, the first term in the bracket describes the change of the vector potential due a change of the current (i.e.

Therefore, the derivative ∂

(A.1.24) ∂**v'**/∂t = ∂**v'**/∂t'

(A.1.25) ∂d'/∂t = 1/2d' ^{.}[ -2x'^{.}∂s'_{x}/∂t) - 2y'^{.}∂s'_{y}/∂t - 2z'^{.}∂s'_{z}/∂t ] =

= 1/2d' ^{.}[ -2x'^{.}∂s'_{x}/∂t') - 2y'^{.}∂s'_{y}/∂t' - 2z'^{.}∂s'_{z}/∂t' ] ^{.}∂t'/∂t =

= -**u'**^{.}**v'**^{.}∂t'/∂t

(A.1.26) ∂t'/∂t = 1 -1/c^{.}∂d'/∂t

(A.1.27) ∂d'/∂t = -**u'**^{.}**v'**/(1 -**u'**^{.}**v'**/c)

(A.1.28) 1/c^{.}∂**A**(**R**,t)/∂t = q/c^{2} ^{.}[ **a'**/d' + **v'**^{.}(**u'**^{.}**v'**)/(1 -**u'**^{.}**v'**/c) ]

(A.2.1) 1/4π^{.}∫∫_{4π}dS^{.}**n**^{.}**E** = q

(A.2.2) dS = R^{2}^{.}dΩ

(A.2.3) 1/4π^{.}∫∫_{4π}dΩ^{.}R^{2}^{.}**n**^{.}**E** = q

(A.2.4) R^{2}/2^{.}_{0}∫^{π}dθ^{.}sin(θ)^{.}[F_{d'}(θ)+F_{v'}(θ)+F_{a'}(θ)] = q

(A.2.5) F_{d'}(θ) = q/R^{2}/(1-v'^{.}cos(θ)/c)

(A.2.6) F_{v'}(θ) = -q/R^{2} ^{.}v'^{2}^{.}cos^{2}(θ)/c^{2} /(1-v'^{.}cos(θ)/c)

(A.2.7) F_{a'}(θ) = -q/R ^{.}a'^{.}cos(θ)/c^{2}

(A.2.8) F_{d'}(θ)+F_{v'}(θ) = q/R^{2}^{.}(1-v'^{2}^{.}cos^{2}(θ)/c^{2}) /(1-v'^{.}cos(θ)/c) =

= q/R^{2}^{.}(1+v'^{.}cos(θ)/c)

(A.3.1) 1/λ(**R**,t) = 1/(1 -**u'**^{.}**v'**/c) = 1/(1-v'^{.}cos(θ)/c)

(A.3.2) Φ_{nn}(**R**,t) = q/λ(**R**,t) ^{.}1/|**d'**(**R**,t)|

(A.3.3) **A**_{nn}(**R**,t) = q/λ(**R**,t) ^{.}**v'**/c ^{.}1/|**d'**(**R**,t)|

We state here without going into the details of the calculation that in this case, following essentially the same procedure of calculating the spatial and time derivatives by applying the chain and product rules accordingly (only this time including the additional factor 1/λ(

(A.3.4) **E**_{nn}(**R**,t) = q/λ^{3}(**R**,t) ^{.}1/|**d'**(**R**,t)|^{2} ^{.}[ (**u'**-**v'**/c)^{.}(1-v'^{2}/c^{2}+**d'**^{.}**a'**/c^{2}) - d'^{.}**a'**/c^{2}^{.}(1-**u'**^{.}**v'**/c) ]

Even though (A.3.4) may be consistent with Gauss' law, this can at best be taken as an average statement regarding charge invariance, which however as such is only a necessary condition but by no means a sufficient one. And the erroneous factor 1/λ(

Apart from the incorrect additional factor 1/λ

In contrast to the electric field obtained from the properly normalized charge distributions (see (35), (36)), the electric field based on the usual Liénard-Wiechert potentials shows a diverging behaviour as

(A.3.5) E_{nn,r} = **E**_{nn}(**R**,t)^{.}**u'** = q/λ^{3}(**R**,t) ^{.}1/|**d'**(**R**,t)|^{2} ^{.}(1-v'^{.}cos(θ)/c) ^{.}(1-v'^{2}/c^{2}) =

= q/|**d'**(**R**,t)|^{2} ^{.}(1-v'^{2}/c^{2}) /(1-v'^{.}cos(θ)/c)^{2}

For the transverse electric field (A.3.4) yields instead (again neglecting the acceleration terms)

(A.3.6) E_{nn,t} = |**E**_{nn}(**R**,t)×**u'**| = q/|**d'**(**R**,t)|^{2} ^{.}|v'^{.}sin(θ)|/c ^{.}(1-v'^{2}/c^{2}) /(1-v'^{.}cos(θ)/c)^{3}

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

email:

email:

See also my sister site https://www.plasmaphysics.org.uk