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?

  • $\begingroup$ the solution you mention does not need to be solved numerically, just replace xyz and calculate B. $\endgroup$ – Wolphram jonny Mar 13 at 15:45
  • $\begingroup$ @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. $\endgroup$ – MPHY Mar 15 at 1:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.