1
$\begingroup$

I'm working on solving Maxwell's equation numerically and have implemented Yee's algorithm in Matlab. In order to check if the algorithm is implemented succesfully, I need an analytical solution to the problmen. So given the Maxwell's equations on the form

$$\frac{\partial}{\partial t} \: E(x,y,z,t) = c \: \nabla \times B(x,y,z,t) \tag{1}$$ $$\frac{\partial}{\partial t} \: B(x,y,z,t) = - c \: \nabla \times E(x,y,z,t), \tag{2}$$ what is the analytical solution?

My implementation of Yee's algorithm lets me choose the boundary conditions and the initial conditions freely.

I've learned that there exists solutions for a homogenous and lossless medium, but I'm having trouble finding them.

That'll be great if anyone can provide any good litteratur/reference on the subject. Thanks!

Edit: I'm looking for a general solution to the problem.

$\endgroup$
  • 2
    $\begingroup$ Do you need a general solution, or if all you need is something to test your algorithm would some standard solution like a dipole radiator or a waveguide be enough $\endgroup$ – By Symmetry Mar 4 '15 at 12:02
  • $\begingroup$ Yes, I need a general solution. I'm keen on testing my implementation for different BC/IC, so a general soultion which can be adapted for each situation is my goal. $\endgroup$ – harisf Mar 4 '15 at 12:13
  • $\begingroup$ @harisf I don't exactly understand why are you having problems in finding a general solution. There is an article in Wikepedia that solves your two equations exactly. You take the derivative of eq. $(1)$ by time and substitute in it the eq. $(2)$. You obtain the wave equation, whose solutions are known, and any linear superposition of these solutions is also a solution. The boundary conditions will help you to fix the constants in the superposition. So, what's the problem? $\endgroup$ – Sofia Mar 4 '15 at 12:57
  • $\begingroup$ Oh, I didn't even think of that, @Sofia. However, I understand converting the equations into the wave equation, and finding a fitting solution. But, wouldn't I then just find a solution of the wave equation? I mean to ask, how do I convert back to solutions for the Maxwell's equation, if possible? Or will the solutions for the Maxwell's equation and the wave equation coincide in this case? If so, why? $\endgroup$ – harisf Mar 4 '15 at 13:16
  • $\begingroup$ @harisf simply: check if the solutions for E and B satisfy your eqs. (1) and (2) and you will convince yourself. Pay attention that your equations are homogeneous, i.e. no currents, no charges.The fact that the solutions of the wave-equation are the solutions if your (1) and (2) was proved again and again. $\endgroup$ – Sofia Mar 4 '15 at 13:35

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.