🚀 go-pugleaf

Maxwell's first equation can be written as well in differential as
in integral form. A special variant of the integral form is its
parallel-planes version:

  The difference between the integrals of the electric flux through
  two parallel (infinite) planes is equal to the net charge enclosed
  between the two planes.

Now let us assume one single electric charge and a series of parallel
planes p_1, p_2, p_3, ...

              :      :      :
              |      |      |
              |      |      |
     +        |      |      |      ...
              |      |      |
              |      |      |
              :      :      :
  charge     p_1    p_2    p_3     ...

We assume that there are no other net charges enclosed between the
planes. Therefore the flux is identical through all these planes.

The paradox arises if we move (accelerate) the electric charge in
direction to these planes during an arbitrary period of time t.

For every t there is an infinite number of planes p_i with i > N
through which the flux cannot have changed because of retardation
(propagation at c of changes in the electric field).

Because of the parallel-planes version of the 1. Maxwell equation
no change in flux through plane p_i ist possible without an equal
change through p_(i+1). By simple induction we conclude that no
change in flux is possible (not only through planes p_i with i > N
but also) through p_N, p_(N-1), p_(N-2), ..., p_3, p_2, p_1.

In other words: the first Maxwell equation leads to the conclusion
that electric fields cannot change when moving (accelerating) their
causing electric charges. Isn't this a wonderfully concise and
convincing paradox?