Question

The electric field of a point charge at rest in an inertial frame is static in the frame. Any location outside the charge follows the Maxwell equations in the vacuum, and as a consequence the wave equation. It is easy to see that both the Laplacian of the field and its time second derivative are zero. The wave equation is fulfilled trivially.

If this charge is moving with constant velocity in a lab, the electric (and magnetic) field are calculated by a Lorenz boost. In the lab's frame, the E-field is not linear with time for a given point, due to the Coulomb law, so the second derivative with respect to time is not zero. It seems that the wave equation is fulfilled not trivially by a function f(r,t).

If it fulfills the wave equation it is a EM wave, or not? If not, what is the form of this function?

Answer 1

The electric field of a point charge at rest is static and does not fulfill the wave equation trivially. When the charge is moving with constant velocity, the electric field is not linear with time and may not fulfill the wave equation.

Step-by-step explanation:

The question is asking about the behavior of the electric field of a point charge at rest and when it is moving with constant velocity. When the charge is at rest, the electric field is static and does not change with time. In this case, the wave equation is fulfilled trivially and the field is not considered an electromagnetic wave.

However, when the charge is moving with constant velocity in a lab, the electric and magnetic fields are calculated using a Lorenz boost. In this case, the electric field is not linear with time for a given point, and the second derivative with respect to time is not zero. The wave equation is not fulfilled trivially by the field in this case.

If the field fulfills the wave equation, it is considered an electromagnetic wave. If not, it may have a different form depending on the specific conditions.