THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 401 Assignment # 8:
 
GUIDED WAVES
 
Wed. 1 Mar. 2006 - finish by Wed. 8 Mar.

1. (p. 405, Problem 9.25) - Group Velocity:¹Assuming negligible damping ( $\gamma_j \approx 0$), calculate the group velocity ( $v_g \equiv d\omega/dk$) of the waves described by Eqs. (9.166) and (9.169):
   
   $$\tVec{E}(z,t) = \tilde{E}_0 \, e^{-\kappa z} \, e^{i(kz-\om . . . . . . \over \omega_j^2 - \omega^2 - i \gamma_j \omega} \right] \; .$$
   
   
   Show that v_g < c, even when v > c.

2. (p. 411, Problem 9.27) - No TE₀₀: Show that the mode TE₀₀ cannot occur in a rectangular wave guide. [Hint: In this case $\omega/c=k$, so Eqs. (9.180) are indeterminate, and you must go back to Eqs. (9.179). Show that B_z is a constant, and hence - applying Faraday's law in integral form to a cross section - that B_z = 0, so this would be a TEM mode.]

3. (p. 411, Problem 9.28) - TE Modes:²Consider a rectangular wave guide with dimensions 2.28 cm x 1.01 cm. What TE modes will propagate in this waveguide, if the driving frequency is 1.70 x 10¹⁰ Hz? Suppose you wanted to excite only one TE mode; what range of frequencies could you use? What are the corresponding wavelengths (in open space)?

4. (p. 411, Problem 9.30) - TM Modes:³Work out the theory of TM modes for a rectangular wave guide. In particular, find the longitudinal electric field, the cutoff frequencies, and the wave and group velocities. Find the ratio of the lowest TM cutoff frequency to the lowest TE cutoff frequency, for a given wave guide. [Caution: What is the lowest TM mode?]