Discussion between Semiclassical and crobar

9:24 PM

Q: How to get polarised electromagnetic TE wave differential equation from Maxwell's Equations?

I wish to understand how the following equation: $\frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} + n^2 k_0 E_x = \frac{\text{d} (\ln \mu)}{\text{d}z}\frac{\partial E_x}{\partial z} $ where $n^2 = \epsilon \mu$ and $k_0 = \frac{\omega}{c} = \frac{2 \pi}{\lambda_0}$...

calculus

Semiclassical

Which quantities can be assumed to be constant here?

crobar

$\omega$, $\mu$, $\epsilon$ and $c$ are constants

Semiclassical

It's simple, I think: differentiate (2b) and (2c) with respect to $z$ and $y$ respectively, then substitute the resulting partial derivatives into (1a).

crobar

If you flesh this out into an answer, I'd be happy to accept.

Semiclassical

Wait: Based on that first equation, the permeability $\mu$ is certainly not constant.

crobar

9:24 PM

Well, $\mu$ is a property of a material, and for some materials it is approximately constant, and others, a function of $H$. I'm happy just to consider the first case, constant $\mu$.

Semiclassical

My point is that that derivative $d \ln \mu / dz$ will vanish unless mu depends on z

And so you'll only get the RHS of the equation if you treat mu as constant

*LHS

crobar

Yes, and that is what I've been getting in my attempts, so it must be the problem.

Semiclassical

right. so you need to keep partial derivatives of mu

(well, not partial since it only depends on z)

crobar

ok, thanks, this is a good starting point for me to try and figure it out!

Semiclassical