Derivation of electromagnetic waves

[FunkyNoodles]

I've seen derivations for c=E/B and c=1/√μ₀ε₀, but I don't seem to get the directions right. i.e. I end up with a negative sign in one of the equations. The derivations I've seen do not use vector calculus.
One derivation I've seen is in this video. But in this video I don't know how the direction of integration is determined, as that would solve my problem; it seems to contradict Lenz's law. Any help would be grateful!

 

[robphy]

Can you show some work... so we know which equation has the negative sign?
Do you know what the "curl" is?

 

[FunkyNoodles]

http://imgur.com/KzXQADm
Here's my poor limited understanding of em waves without using vector calculus (I've heard of "curl" or "divergence", but don't really know what they are). So my question is that two rectangles have different directions of integration. For example, for the top graph, magnetic field is increasing at the instance the rectangle is taken, so according to Faraday's law, shouldn't the direction of integration be reversed, as the net electric field needs to generate a magnetic field that opposes the change? Or maybe this is just an obvious mistake...

 

[robphy]

The integration loops are chosen so that
the normal to the upper loop (for $E_y$) points along $\hat x \times \hat y=\hat z$
and
the normal to the lower loop (for $B_z$) points along $\hat z \times \hat x=\hat y$

The circulation of $\vec E$ is positive... with your right-hand, the electric field curls* counter-clockwise (with its normal along $\hat z$), which follows the sense of the integration loop. By Faraday, with its minus-sign, that is $-\frac{\partial B_z}{\partial t}$.
*(With your right hand, have your right-hand fingers point along the longer Electric Field vector, then bending to curl around the loop to the shorter Electric Field vector.)

The circulation of $\vec B$ is negative... with your right-hand, the magnetic field curls clockwise (with its normal along $-\hat y$), which is opposite the sense of the integration loop. By Ampere, that is $\frac{\partial E_y}{\partial t}$.

So, both Faraday and Ampere say that the fields in that interval $dx$ must decrease in the next instant... which is consistent with the entire waveform advancing along the positive-x axis.

 

[FunkyNoodles]

Oh, I see, the curl is what I was missing. Thanks, that helped a lot.