hown above at left is a simple electrostatic field, created in this case by a pair of closely spaced charged plates. If we put the plates in motion to the right they get length-contracted, but the total charge on each is an invariant so the surface charge density increases by the Lorentz factor, and therefore so does the field strength. On the other hand, if the plates are moving vertically then the distance between them gets length-contracted, but for closely spaced plates the distance has no effect on the field strength so the field strength is the same as in the original reference frame. In summary, the components of E perpendicular to the boost are greater in the boosted frame, but the component parallel to the boost is the same in both frames.

Now consider the electrostatic field of a point charge at rest, shown at left below. At every point on the circle (actually a sphere), the field has the same strength and points directly away from the particle.

If we put this system in motion to the right (shown at right), two things happen. The first is that the sphere gets length-contracted, flattened in the direction of motion. The second is that the components of the field perpendicular to the motion get stretched by the very same Lorentz factor. Therefore the field still points directly away from the point charge, but it's not the same in all directions: it's weaker in front of and behind the particle, and stronger to the sides, as shown below.

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