How Can You Derive the Wave Equation from Maxwell's Equations in a Vacuum?

[heyo12]

a really hard one here. would appreciate help on how to do this question:
a physical system is governed by the following:

curl E = -\frac{\partial B}{\partial t},
div B = 0,
curl B = J + \frac{\partial E}{\partial t},
div E = \rho
where t = time, and time derivatives commute with \nabla

.........
how could you show that \frac{\partial p}{\partial t} + div J = 0
.........
when \rho = 0 and J = 0 everywhere how can you show that:
\nabla^2E - \frac{\partial^2E}{\partial t^2} = 0
and
\nabla^2B - \frac{\partial^2B}{\partial t^2} = 0

 

[berkeman]

Welcome to the PF, heyo12. One of the rules we have here is that you must show us your work before we can offer tutorial advice. We do not do your work for you.

So can you show us how you would approach this type of problem?

 

[heyo12]

sure. i can totally understand what you just said. and i totally support those rules. however, i don't seem to have a clue how to start this question. the only guess i can make is that it is related possibly to divergence and stoke's theorem. but this is a guess

i would appreciate if you could give me a little clue, so that i can work from that. once i get a hint or 2 i'll try working out the rest and show you what I've done.

thank you very much :)

 

[George Jones]

The first of the equations that you are to prove involves J. Use one of the given equations to isolate J.

By the way, do you recognize the given equations?

 

[berkeman]

You've listed some basic E&M equations like Maxwell's equation and the continuity equation, and are asked to show something where the charge density and current density are zero. The hint is that those equations are a form of the wave equation in free space:

http://en.wikipedia.org/wiki/Wave_equation

One caveat -- wikipedia.org is not a bad resource for basic stuff, but as you get into more complex stuff like your question, wikipedia.org can have bugs or errors or other problems in its information. So although I'm pointing you to that page, please keep in mind that I'm not saying that it is 100% accurate. You can use a google search (or just look up the wave equation in your E&M textbook) for more/better information.