My question is on the nature of oscillating electric field. From Maxwells equation, we know that the solution of the electric and magnetic field for an empty space are always oscillating fields. I learned that oscillating fields are connected to oscillating charges. However, what about the case when the the charge is not oscillating but rather, lets say, just linearly accelerating over a given distance. The field gets disturbed once the particle accelerate suddenly but what I don't understand is why this would still cause an oscillating field rather than just a locally distribution of the field. Hope my question is clear.
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By linearity of the Maxwell's equations, you can always add an empty space oscillating wave solution to any solution you come up with in the presence of charge. Your field will be uniquely determined by the initial conditions and the source distribution (current and charge). Note that empty space EM waves do not typically oscillate, they are just propagating waves. It's just convenient to represent in Fourier space where they are decomposed in harmonic time dependence. In general, you would therefore expect that a generic initial condition would lead to propagating waves.
For your specific example, the fields can be calculated from the general Liénard Wiechert formula. Note that it does not require initial conditions, it gives you a default solution that is the "simplest" in some sense. Also, there is no sign of oscillation, but the retardation effect translates the finite time propagation.
Hope this helps.
