Jackson E&M: Power loss in waveguides for TE & TM modes

[IcedCoffee]

Hi. I'm having trouble with calculating

(8.59)

from (8.58)

($\vec H_{//}$ is simply $\vec H$ in Jackson, but that shouldn't matter.)

in Jackson. Specifically I think I'm not sure about parallel component of H field here.

For example, shouldn't I have two terms for the TE case in that

when $\hat n = \hat x$ (for simplicity), $\hat n \times \vec B = - B_z \hat y + B_y \hat z = B_z \hat y + \frac {ik} {\mu \epsilon \omega^2 - k^2} \frac { \partial B_z } {\partial y} \hat z$ (from 8.26b)

(which, by the way, does not include $\frac {\partial B_z} {\partial x}$ as in Jackson)

And similarly, shouldn't TM mode have only one term since $B_z$ is zero in TM mode?

 

[LightHero]

Thanks in advance.Yes, you are correct in your understanding of the equations. For the TE case, there should be two terms in the equation, one for the magnetic field component that is perpendicular to the direction of propagation, and one for the field component that is parallel to the direction of propagation. The TM case should have only one term since there is no magnetic field component that is perpendicular to the direction of propagation.