Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

7.8.18 Problems and solutions

The problems of this section use the results collected in the recap. Throughout, the cavities are filled with a lossless homogeneous medium ( $ε = ε_{r} ε_{0}$ , $μ = μ_{0}$ ), the phasors carry $e^{j ω t}$ , and

f_{mn p} W_{101} P_{L} f Q_{010} Q = \frac{v}{2} (\frac{m}{a})^{2} + (\frac{n}{b})^{2} + (\frac{p}{d})^{2}, = \frac{ε ∣ E _{0} ∣ ^{2} ab d}{8}, = \frac{R _{s} ∣ E _{0} ∣ ^{2}}{2 η ^{2}} [(\frac{λ}{2 d})^{2} ab + (\frac{λ}{2 a})^{2} b d + \frac{a d}{2}], R_{s} = \frac{π f μ _{0}}{σ _{c}}, = \frac{v}{2 π} (\frac{x}{a})^{2} + (\frac{p π}{d})^{2}, = \frac{η}{R _{s}} \frac{x _{01}}{2 ( 1 + a / d )}, = \frac{ω _{0} W}{P _{L}}, Δ f = \frac{f _{0}}{Q},

with $x_{01} = 2.405$ , $x_{11} = x_{01}^{'} = 3.832$ , $x_{11}^{'} = 1.841$ , $x_{21}^{'} = 3.054$ and $J_{1} (x_{01}) = 0.519$ . The constants are those of this course, $ε_{0} = 1 0^{- 9} / (36 π) = 8.842 \times 1 0^{- 12} F/m$ , $μ_{0} = 4 π \times 1 0^{- 7} H/m$ , $c = 3 \times 1 0^{8} m/s$ and $η_{0} = 120 π = 377 Ω$ , with copper $σ_{c} = 5.8 \times 1 0^{7} S/m$ . In air two shortcuts are useful: $c /2 = 15 GHz cm$ and $c /2 π = 4.775 GHz cm$ .

7.8.18 Problems and solutions

In this part

Problems · 5

Home exercises · 4

Further exercises · 10

Worked examples · 3