How to choose the boundary condition for Maxwell's equations in the vacuum?

I need to solve the Maxwell's equations with sources in the vacuum numerically. The simplified problem is as following.

A charged particle moving along the $$z$$ direction with speed $$v_z$$. Then, it can create a magnetic field at the $$(x,y)$$ plane. From the Jackson's textbook, I have already known the analytic solution, which is

$$B_{\phi}=\frac{Q}{4\pi} \gamma v_z \frac{r}{(r^2+\gamma^2(z-v_z t)^2)^{3/2}},$$ where I have changed the coordinates $$(x,y)\rightarrow(r,\phi)$$ and $$r^2=x^2+y^2$$.

However, I do not know how to solve it numerically. The master equation is the wave equation for $$B_{\phi}$$, with the source $$\nabla\times J_{ext}$$ and $$J_{ext}= Q \delta(x)\delta(y)\delta(z-v_z t)$$.

For the initial condition, I guess that I can use the analytic form of $$B_\phi(t_0, r, \phi, z)$$.

Then, how to choose the boundary conditions?

• the solution you mention does not need to be solved numerically, just replace xyz and calculate B. – Wolphram jonny Mar 13 at 15:45
• @Wolphramjonny Thanks! I have already known the analytic solutions. But, as I said, this is the first step for my program. The whole project including several new terms in the Maxwell's equations, is too complicated to be solved analytically. That is why I need to test it by considering the simplest cases. – MPHY Mar 15 at 1:11