- #1
dingo_d
- 211
- 0
Homework Statement
I'm following Jackson and solving the problem, but I came to a bump.
I've found [tex]B_z[/tex] component of the field (for the lowest mode [tex]\omega_{10}[/tex]), and since it's TE mode I should use:
[tex]\vec{E}_t=-\frac{\omega}{ck}\vec{e}_3\times\vec{B}_t [/tex]
[tex]\vec{B}_t=\frac{ik}{\gamma^2}\nabla_t B_z [/tex]
So I get that [tex]B_y=0[/tex] because the [tex]B_z[/tex] doesn't depend on the y in the lowest mode.
But how do I get [tex]E_y[/tex] and not [tex]E_x[/tex] component (The Jackson says that [tex]E_y[/tex] is the non vanishing component)?
It should be [tex]\vec{E}_x=-\frac{\omega}{ck}\hat{z}\times\vec{B}_x[/tex] and [tex]\vec{E}_y=-\frac{\omega}{ck}\hat{z}\times\vec{B}_y[/tex] and since [tex]\vec{B}_y=0[/tex] then [tex]\vec{E}_y=0[/tex] should be zero, no?
It's probably due to this cross product but I cannot see why :( Help...EDIT:
Or is it because [tex]\hat{z}\times\hat{x}=\hat{y}[/tex] 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 - [tex]E_x[/tex] goes with [tex]B_x[/tex], but it's in fact the [tex]E_y[/tex] because of the [tex]\hat{y}[/tex] - it seems kinda messed up :\)...
Last edited: