Several months ago, there was a week in which I spent all my time thinking about how to find the charge distribution of conducting disc(I end up with the distribution of cylindrical symmetric ellipsoid, that can’t be specialize into a disc, with sphere and needle as the two extreme case). Then I discuss with my friend hoping to find out another way to solve it, but we end up discussing about relativistic electrodynamics instead, which is triggered by the fact that if a sphere moving with relativistic velocity it will turns into an ellipsoid(without flight time effect) . During this discussion, somehow we found a simple and very interesting paradoxical case of relativistic electrodynamics.
As shown below there are two point particles with charge q each, the distance between them for this instance is L . Then a relativistic human, for some reason, using his hands, starts moving particle b away from a with constant relativistic velocity , while holding the particle a at rest .
At this instant, the force on b due to a is obviously
Now comes the problem, the force on a due to b is different
Which means that the relativisticman receives different force on his left and right hand. Which means…
The relativisticman can fly away like a superman just by moving his hands!!!
How can it be possible assuming that the relativisticman exists??
The explanation is as follows
The electric field due to a alone at a random point p is
The electric field due to b alone at a random point p is
b will also produce magnetic field
using geometry relation , and , we can eliminate the dependence of in
The total poynting vector of 2 charges is
the second term is constant over time, so we are not interested in it. By neglecting the this term, the momentum hidden in the electromagnetic field becomes
substituting ,, and we end up with
by symmetry we know that the momentum will be in the same direction with , so we can put , substituting this and the integral becomes
The integral of has the form of
Which can be solved without much effort by using wolfram alpha
After LOTS of mess, the integral becomes
We can see that the second integral is odd, then we get a surprisingly simple result
the rate of change of the momentum is
which is equal to . So it means that the force difference exerted by the Relativisticman is only used to change the momentum of electromagnetic field. Since the mechanical momentum is not changing at all, the Relativisticman must receives the same change in momentum but in the opposite direction. Thus the man can fly away like superman!