Let us imagine an electron at x = 10 nano-light-seconds (nLS = 0.3 m)
of a coordinate system (y = 0, z = 0). The flux through the y-z-plane
(x = 0) divided by the permittivity of space is therefore equal to
half the elementary electron charge.
Now let us assume that the electron at rest is captured at the time
t_0 = 0 by a fast moving neutral particle and that afterwards the
electron moves at v = 0.9 c to the origin of the coordinate system.
A change in electric field of the plane can occur at the earliest at
t_1 = t_0 + x_0 / c = 10 ns at the orgin of the coordinate system.
At t_1 the electron has reduced its distance from the plane to
x_1 = 1 nLS. When an effect from x_1 propagating at c can reach
the plane, the electron's position is x_2 = 0.1 nLS. When an
effect from x_2 can reach reach the plane, the position is already
x_3 = 0.01 nLS, and so on.
Time in Position Duration Effect radius
nano-sec. in nLS until crossing in nLS
t_0 = 0 x_0 = 10 dT_0 = 11.11111 r_0 = 4.8432
t_1 = 10 x_1 = 1 dT_1 = 1.11111 r_1 = 0.48432
t_2 = 11 x_2 = 0.1 dT_2 = 0.11111 r_2 = 0.048432
t_3 = 11.1 x_3 = 0.01 dT_3 = 0.01111 r_3 = 0.0048432
t_4 = 11.11 x_4 = 0.001 dT_4 = 0.00111 r_4 = 0.00048432
The duration of the movement from x_n to the cross-point x = 0 is
dT_n = x_n / v. During this time dT_n the electric field and flux
on the plane can only change within a circle around the cross point
with the "effect radius" r_n = sqrt ((dT_n * c)^2 - x_n^2).
So whereas the electric field at the origin of our coordinate system
continuously increases to infinity until the electron crosses the
plane, the region where the flux no longer can change increases
continuously in the same way. At least superficial reasoning leads
to the conclusion that the flux through the whole plane increases
to infinity instead of remaining constant (and changing sign at
In the original theory of Coulomb, there is no paradox, because the
flux increase near the cross-point is instaneously compensated by a
flux decrease in more distant regions because of decreasing angles
There's no paradox either in standard EM. Charge is relativistic
invariant and Gauss' law works just as fine for a moving charge.
Consider the electron to be at rest and now move the plane
instead. The flux through the plane is that of half an electron
always. This is in the rest frame.
Now consider the viewpoint from the plane. The EM fields
transform according to Special Relativity:
E'x = Ex
E'y = gamma * (Ey - v/c Bz)
E'z = gamma * (Ez + v/c By)
Now only E'x goes through the plane, while E'y and E'z are
parallel to the plane. Thus: The flux is again that of half an
We can go a bit further and look at the plane through which
E'y goes. We forget Bz because it's zero in the electron's
rest-frame. The electric field E'y as seen by the moving plane
is now increased by a factor gamma. However, the plane sees
the world Lorentz contracted by a factor gamma as well.
So, again: The flux through the plane is that of half an electron.
If you think of the electric field in terms of the old fashioned
concept of lines of force, then the electron has radiating from it one
unit charge worth of these lines, uniformly distributed about 4pi
steradians. As the charged particle moves towards the plane these
lines of force will tilt since the changes in the fields propagate at
the speed of light. There will thus be a "kink" in the lines that
propagates outward from the time/place where the charge suddenly
Since the fields propagate at the speed of light, but the charge must
move at less than the speed of light, the lines of force will always
have the same direction of tilt towards or away from the plane, just
the magnitude of the slope changes. That is, a line of force that is
crossing the plane initially will always be tilted towards the plane.
Every line of force that initially passes through the plane will still
pass through the plane for any positon of the plane between the
original position and the particles location.
The only way you could get the "paradox" you are describing is if the
charge could move faster than the speed of light and thus get ahead of
the lines of force. Then the slope of some of the lines would change
sign (i.e. tilt the opposite direction) and then Gauss' Law would be
violated. This, of course, is never observed.
This argument is identical to Hans', but is a little more intuitive
In my first response I showed why Gauss's law also works for a
moving electron via special relativity.
I do see now why you get the wrong result. You let the electric
field E propagate away from the electron with c. To get the correct
answer one should let the electric potential V propagate away with
c and THEN differentiate to get the correct electric field E.
This was first recognized by Lienard and Wiechert around 1900.
With this method you can obtain the EM fields of an electron
making arbitrary motions and accelerations.
>> In the original theory of Coulomb, there is no paradox, because the
>> flux increase near the cross-point is instantaneously compensated by
>> a flux decrease in more distant regions because of decreasing angles
>> of incidence.
> There's no paradox either in standard EM. Charge is relativistic
> invariant and Gauss' law works just as fine for a moving charge.
> Consider the electron to be at rest and now move the plane
> instead. The flux through the plane is that of half an electron
> always. This is in the rest frame.
In this case, the flux can change at all points of the plane at
the same time. In my example however, an electron is at rest at
[x=10, y=0, z=0] and starts at t = 0 moving at v = 0.9c towards
[0, 0, 0]. Until the electron crosses the plane [0, y, z] at
t = 10/0.9, the flux through the plane can only change within the
circle y^2 + z^2 <= r^2 with r = sqrt((10/0.9)^2 - 10^2) = 4.84.
The assumption seems reasonable that from t = 10 on (when the
information that the electron is no longer at rest at [10,0,0] can
reach the plane), a flux change through the plane spreads from
[0,0,0], reaching the circumference y^2 + z^2 = r^2 of the circle
at t = 10/0.9.
Any flux increase near the center [0,0,0] of the y-z-plane however
must be compensated by a flux decrease at more distant regions,
because the flux integral through the circle cannot change before
the electron crosses the center. So any flux increase near the
center must wait (how long?), until a flux decrease in more
distant regions is possible.
As far as I've learned now, the logical conclusion of such an
"infinite flux paradox" is avoided by the assumption that the
velocity of a moving charge leads to extrapolated/anticipated flux
changes: The instantaneous change of the flux integral through a
plane from -flux to +flux at the moment a charge crosses the
plane, is caused primarily NOT by local effects of the crossing
charge BUT by extrapolated effects of former positions of the
But does this solution not entail another problem: The charge
can be stopped just before crossing the plane. Because this
information cannot reach the whole circle of the plane where the
flux is about to change, the flux integral through the circle
and therefore through the whole plane will nontheless change,
thus leading to a contradiction with Gauss' law for electricity?
Isn't it astonishing that in dynamic situations, Gauss' law
is as well valid under "retardation with c" (Maxwell) as under
"instantaneous interaction" (Coulomb), despite very different
mechanisms/maths (very simple/transparent in the case of Coulomb,
and rather complicated/opaque in the case of Maxwell)?
Has the old reproach that Maxwell starts with not-mediated
actions-at-a-distance and ends with the claim that the effects
are produced without such actions-at-a-distance actually been
cleared up? According to Heinrich Hertz 1890 this procedure
leads to the unsatisfying feeling that either Maxwell's result
or his reasoning leading to this result should be wrong.
"Maxwell geht aus von der Annahme unvermittelter Fernkräfte,
.. und er endet mit der Behauptung, dass diese Polarisationen
sich wirklich so verändern, ohne dass in Wahrheit Fernkräfte
die Ursachen derselben seien. Dieser Gang hinterlässt das
unbefriedigende Gefühl, als müsse entweder das schliessliche
Ergebnis, oder der Weg unrichtig sein, auf welchem es gewonnen
wurde. " (Heinrich Hertz, "Über die Grundgleichungen der
Elektrodynamik für ruhende Körper", Gesammelte Werke, 1894)