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