Relativistic Electromagnetism - More Rigorous Analysis

More Rigorous Analysis

Consider the electric field of a single, infinite plate of positive charge, moving parallel to itself. The field must be uniform both above and below the plate, since it is uniform in its rest frame. We also assume that knowing the field in one frame is sufficient for calculating it in the other frame.

The plate however could have a non zero component of electric field in the direction of motion as in Fig 2a. Even in this case, the field of the infinite plane of negative charge must be equal and opposite to that of the positive plate (as in Fig 2b), since the combination of plates is neutral and cannot therefore produce any net fields. When the plates are separated, the horizontal components still cancel, and the resultant is a uniform vertical field as shown in Fig 1.

If Gauss's law is applied to pillbox as shown in Fig 1, it can be shown that the magnitude of the electric field between the plates is given by:

where the prime (') indicates the value measured in the frame in which the plates are moving. represents the surface charge density of the positive plate. Since the plates are contracted in length by the factor

then the surface charge density in the primed frame is related to the value in the rest frame of the plates by:

But the electric field in the rest frame has value σ / ε₀ and the field points in the same direction on both of the frames, so

The E field in the primed frame is therefore stronger than in the unprimed frame. If the direction of motion is perpendicular to the plates, length contraction of the plates does not occur, but the distance between them is reduced. This closer spacing however does not affect the strength of the electric field. So for motion parallel to the electric field E,

In the general case where motion is in a diagonal direction relative to the field the field is merely a superposition of the perpendicular and parallel fields., each generated by a set of plates at right angles to each other as shown in Fig 3. Since both sets of plates are length contracted, the two components of the E field are

and

where the y subscript denotes perpendicular, and the x subscript, parallel.

These transformation equations only apply if the source of the field is at rest in the unprimed frame.