# Analytical solution to Maxwell's equations in 3D

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.

• 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 – By Symmetry Mar 4 '15 at 12:02
• 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. – harisf Mar 4 '15 at 12:13
• @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? – Sofia Mar 4 '15 at 12:57
• 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? – harisf Mar 4 '15 at 13:16
• @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. – Sofia Mar 4 '15 at 13:35