1. Consider a lossless rectangular waveguide with metallic walls, with dimensions a = 40mm, b = 30mm, filled with a dielectric having $\mu = \mu_0$, $\epsilon = \epsilon_0 \epsilon_R$, $\epsilon_R = 4$. The frequency of the electromagnetic field is 6.5 GHz.
1.1 Calculate all modes that can propagate.
1.2 Calculate the wavelength of TE$_{2,0}$ mode.
1.3 Calculate the group velocity of the slowest mode.
Consider the mode filters plotted in the following figure
(#1)
$a/2$
5a/6
(#2)
$a/2$
1.4 Discuss which modes that can propagate if it is realized the mode filter #1, that is a very thin slot along z axis on the upper wall in the position x = a/2
1.5 Discuss which modes that can propagate if it is realized the mode filter #2, that is a combination of an electric wall placed in xz plane in the position x = 5a/6, and a very thin slot along z axis on the upper wall in the position x = a/2.
