Lorentz transformation of Electron’s field
The only relativistic theory I know for the field of a moving point charge is by applying a Lorentz boost
to the Electromagnetic tensor:
We choose c=1.
φ is the ‘rapidity’.
The tensor in the boosted frame is F'μν =
|Aμα = ||
where C = cosh φ and S = sinh φ .|
For a motionless point change at the origin sufficient to produce a field of magnitude 1 at distance 1 the field is:
Consulting this result:
See this for calculation of F'01.
N.B. the coordinates x, y and z are still those of the point charge’s frame!
|F'μν = (x2 + y2 + z2)−3/2
t = Ct' − Sx', x = Cx' − St', y = y', z = z'.
and we have:
The magnetic field is everywhere perpendicular both to the velocity and the displacement vector from the observer to the charge.
The point charge is traveling in the x direction.
Note that the x component of the magnetic field (F32) is 0 everywhere.
At the event (t', x', y', z') = (0, 0, 0, 1) the electric field is
(0, 0, C) and the magnetic field is (0, S, 0).
At the event (0, 1, 0, 0) the electric field is (C−2, 0, 0) and the magnetic field is (0, 0, 0).
|F'μν = ((Cx' − St')2 + y'2 + z'2)−3/2
|0||−(Cx' − St') ||−Cy' ||−Cz'
|Cx' − St' ||0||−Sy'||−Sz'