Simplified Maxwell's Equation Proof

[Von Neumann]

Problem:

I'm trying to crudely prove the following:

\frac{{\partial}B}{{\partial}x}=-\mu_{o}\epsilon_{o}\frac{{\partial}E}{{\partial}t}

Solution (so far):

I can get the derivation, but the minus sign eludes me somehow...

Integrating over a thing rectangular loop of length l and width dx, start with the following,

\oint{B{\cdot}dl}=\mu_{o}\epsilon_{o}\frac{\partial{\Phi_{E}}}{\partial{t}}

Then,

\oint{B{\cdot}dl}=(B+dE)l-Bl=dEl

Also,

\Phi_{E}=EA=E(dx)(l)

∴\frac{\partial{\Phi_{E}}}{\partial{t}}=\frac{\partial{E}}{\partial{t}}dxl

Equating the equations above,

dEl=\mu_{o}\epsilon_{o}\frac{\partial{E}}{\partial{t}}dxl

dE=\mu_{o}\epsilon_{o}\frac{\partial{E}}{\partial{t}}dx

\frac{\partial{E}}{\partial{x}}=\mu_{o}\epsilon_{o}\frac{\partial{E}}{\partial{t}}

Any advice is greatly appreciated.

 

[csopi]

Check this:

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