k = ω√((εμ)/(2))[√(1+((σ)/(εω))^(2))+1]^((1)/(2)), κ = ω√((εμ)/(2))[√(1+((σ)/(εω))^(2))-1]^((1)/(2)).
And we know that the skin depth, d = (1)/(κ).
a) For a good conductor σ ≫ ωε. Approximate the skin depth in this situation.
b) For a poor conductor σ ≪ ωε. Approximate the skin depth in this situation.
In class, we mentioned that for a TE wave we have the following differential eq:
(1)/(x)(d^(2)x)/(dx^(2)) = -k_(x)^(2)
a. Show that the following is a solution:
x(x) = Asin(k_(x)x) + Bcos(k_(x)x).
b. Due to the boundary conditions, we saw that k_(x) = (mπ)/(a) and k_(y) = (nπ)/(b) where a, b are the dimensions of the waveguide cavity and m, n are arbitrary numbers.
i. By plugging k_(x) and k_(y) into -k_(x)^(2) - k_(y)^(2) + ((ω)/(c))^(2) - k^(2) = 0 show that
k = √(((ω)/(c))^(2) - π^(2)[((m)/(a))^(2) + ((n)/(b))^(2)])
ii. Find the cutoff ω for TE_(23) where m = 2 and n = 3.
iii. Suppose a rectangular waveguide has dimensions a = 2.2 cm and b = 1.3 cm. If the driving frequency is 1.5 × 10^(10) Hz, determine the TE modes that would propagate.