TE mode in rectangular wave guide

[dingo_d]

Homework Statement

I'm following Jackson and solving the problem, but I came to a bump.

I've found $$B_z$$ component of the field (for the lowest mode $$\omega_{10}$$), and since it's TE mode I should use:

$$\vec{E}_t=-\frac{\omega}{ck}\vec{e}_3\times\vec{B}_t$$
$$\vec{B}_t=\frac{ik}{\gamma^2}\nabla_t B_z$$

So I get that $$B_y=0$$ because the $$B_z$$ doesn't depend on the y in the lowest mode.

But how do I get $$E_y$$ and not $$E_x$$ component (The Jackson says that $$E_y$$ is the non vanishing component)?

It should be $$\vec{E}_x=-\frac{\omega}{ck}\hat{z}\times\vec{B}_x$$ and $$\vec{E}_y=-\frac{\omega}{ck}\hat{z}\times\vec{B}_y$$ and since $$\vec{B}_y=0$$ then $$\vec{E}_y=0$$ should be zero, no?

It's probably due to this cross product but I cannot see why :( Help...EDIT:

Or is it because $$\hat{z}\times\hat{x}=\hat{y}$$ and that means I get the y component of the field? Because it's kinda silly way of marking this (the component itself without the unit vector is just a number and we mark it with the same letter - $$E_x$$ goes with $$B_x$$, but it's in fact the $$E_y$$ because of the $$\hat{y}$$ - it seems kinda messed up :\)...

 

[NateD]

Homework Equations\vec{E}_t=-\frac{\omega}{ck}\vec{e}_3\times\vec{B}_t \vec{B}_t=\frac{ik}{\gamma^2}\nabla_t B_z The Attempt at a SolutionThe cross product \hat{z}\times\hat{x}=\hat{y} so the E_y component is non-vanishing, while the E_x will be zero