Determination of a unique electric and magnetic field produced by a point charge

Question

Let's suppose I have a point charge moving arbitrarily in space. As far I know, Maxwell's equations don't determine a unique electric and magnetic field, they only determine a family of electric and magnetic fields. What other information do I need in order to know uniquely the electric and magnetic field produced by a moving (the particle is moving arbitrarily) point charge?

I'm not asking for a method for solving it, I just want to know what other information (apart from Maxwell equations) do I need.

Answer 1

Maxwell's equations are linear, which means that if you have a (particular) solution, you can add any solution of the homogeneous Maxwell equations (i.e. electromagnetism in vacuum, without source, aka light) to it and get another solution. Furthermore, any solution is obtained that way. The way to pick one solution over another is by choice of initial conditions. This could be set at some finite time, or in the distant past or future, or even some combination of these.

A particularly natural choice here would be that there is no incoming radiation from the distant past. This could be implemented by solving by using Green's functions, which give solutions for a source localized at a point in spacetime, and choosing the retarded Green's function, which is zero at all times before the source.

Answer 2

If we know the position of the charged particle at any time, that is, if we know the value of the function $\mathbf r(t)$ for all $t$, one way to find the EM fields is based on knowledge of the fields at one time instant $t_0$. Let the field be such that

$$ \mathbf E(\mathbf x, t_0) = \mathbf C_E(\mathbf x),~~\mathbf B(\mathbf x, t_0)= \mathbf C_B(\mathbf x)~~\forall \mathbf x. $$

With knowledge of the functions $\mathbf C_E(\mathbf x), \mathbf C_B(\mathbf x)$, using the temporal Maxwell equations

$$ \nabla \times \mathbf E = -\frac{\partial \mathbf B}{\partial t} $$ $$ \nabla \times \mathbf B = \mu_0 \mathbf j + \mu_0 \epsilon_0 \frac{\partial \mathbf E}{\partial t} $$ and $$ \mathbf j(\mathbf x,t) = q\mathbf v(t)\delta(\mathbf x-\mathbf r(t)) $$ we can find $\mathbf E(\mathbf x, t), \mathbf B(\mathbf x, t)$ for any $t$.