3.  The cutoff frequency of an air-filed rectangular waveguide is 2.4 GHz for the TE₁₀ mode.  What would be the cutoff frequency if the same guide were filled with a lossless nonmagnetic material whose dielectric permittivity is six times that of air?
Answer: 980MHz

5. A rectangular waveguide with dimensions a=2b= 5cm operates at 11 GHz.  Assuming that the guide is filled with air, find the highest possible operating mode and the intrinsic impedance at that mode.
Answer: TE_31/ TM₃₁ = 2093ohms(Ω) / 67.91ohms Ω

7.  In an air-filled rectangular waveguide, the cutoff frequency of a TE₁₀ mode is 5GHz, whereas that of TE₀₁ mode is 12 GHz.  Calculate
(a) the dimensions of the guide
(b) the cutoff frequencies of the next three higher TE modes
Answer: (a) a = 3cm b = 1.25cm
(b) TE₂₀(10Ghz), TE₁₁(13GHz), TE₃₀(15GHz)

9. Calculate the dimensions of an air-filled rectangular waveguide for which the cutoff frequencies for TM₁₁ and TE₃₀ are both equal to 12 GHz.  At 8GHz, determine whether the dominant mode will propagate or evanesce in the waveguide.
Answer: b= 1.32cm, a=3.75cm, will propagate.

11.  A TM mode operating at 5GHz, is propagated in an air-filled rectangular waveguide.  If
                E_z=  sin (2π x/a) sin (π y/b) cos(ωt-10z) V/m  (* note that this is not in phasor form, but instant)
find: (a) the cutoff frequency
        (b) the field components Hz and Ex
Answer: (a) 4.977 GHz
(b) 0,     9.2 x 10^-4 (2π/a) cos(2πx/a) sin(π y/b) sin(ωt-10z) V/m

12.  In an air-filled square waveguide with a =1.2 cm,
                E_x=  -10 sin (2ω y/a) sin(ωt-150z) V/m
find: (a) the  mode of propagation
        (b) frequency of operation
        (c) the field components Hz and Ez
Answer: (a) since m=0 it has to be TE.   n=2  so it's TE₀₂.
    (b) Since a=b, f_c02= (c/2π) sqrt(k_x² +k_y² ) = 25GHz ,  where k_y=0.
            β=150 rad/m (is given from inside the sin(ωt-150z),
            k²= β²+ k_x² +k_y² = ω²με
              then f of operation is=26.00665 = 26 GHz .
     (c) E_y=0=E_z=H_x
            H_y = -7.286 sin(π y/a) sin(ωt-150z)   mA/m
            H_z= 25.43  cos(2π y/a) cos(ωt-150z)   mA/m
 

13.  In an air-filled 5 by 2 cm waveguide has,  at 15 GHz,
                E_z=  20 sin (50π y) sin (40πx) e^{-jβ z} V/m
find: (a) the  mode of propagation

Answer: (a)  TM₂₁