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

About the further exercises of Section 5.6

Twenty further exercises in the spirit of the problems, the worked examples and the home exercises, graded from easy to harder within two groups. Exercises 5.6.1–5.6.10 belong to the lecture: Exercises 5.6.1–5.6.2 are about phasors and the phase velocity, 5.6.3–5.6.5 about the complex permittivity and the wave number, 5.6.6–5.6.7 about the Helmholtz equation and 5.6.8–5.6.10 about power. Exercises 5.6.11–5.6.20 belong to the exercise lecture: reading time-harmonic fields, loops, mixers and receivers, power, and the medium found from measurements. They use only

u = Re {U e^{j ω t}}, sin θ = cos (θ - \frac{π}{2}), \frac{\partial}{\partial t} \to j ω, v_{p} = \frac{ω}{∣\nabla ϕ ∣}, λ = \frac{2 π}{β}, ε_{c} = ε - j \frac{σ}{ω} = ε^{'} - j ε^{''}, tan δ = \frac{ε ^{''}}{ε ^{'}}, k = ω μ ε_{c} = β - j α, \nabla^{2} E + k^{2} E = 0, \nabla \cdot E = 0 (source-free), \nabla \times E = - j ω μ H, \nabla \times H = j ω ε_{c} E, η = \frac{ω μ}{k} = \frac{μ}{ε _{c}}, emf = - \frac{d Φ}{d t},

and, from Section 6.1, the instantaneous and the complex Poynting vectors $P (t) = E (t) \times H (t)$ and $P_{c} = \frac{1}{2} E \times H^{*}$ , with $P_{a v} = Re {P_{c}}$ , and the time-harmonic Poynting relation

- \oint_{S} P_{c} \cdot d S - \int_{V} \frac{1}{2} E \cdot J_{i}^{*} d V = \int_{V} [\frac{1}{2} σ ∣ E ∣^{2} + j 2 ω (w_{m} - w_{e})] d V,

with the time-average energy densities $w_{e} = \frac{1}{4} ε ∣ E ∣^{2}$ and $w_{m} = \frac{1}{4} μ ∣ H ∣^{2}$ (real $ε$ , $μ$ ). For a wave $E_{0} e^{- j k \cdot r}$ every $\nabla$ becomes $- j k$ , so Faraday's law gives $H = k \times E / ω μ$ . A small loop of $N$ turns and area $A$ in a field $H$ of angular frequency $ω$ picks up an emf of amplitude $N ω μ ∣ H \cdot n ∣ A$ , with $n$ its normal.

Exercise 5.6.17

The power of an AM wave

An AM broadcast wave in air is

E (z; t) = E_{0} [1 + m cos ω_{m} τ] cos ω_{c} τ a_{x}, τ = t - \frac{z}{c},

with $E_{0} = 0.5 V/m$ , modulation index $m = 0.8$ , $f_{c} = 1 MHz$ and $f_{m} = 5 kHz$ . (a) Write $E$ as a sum of time-harmonic waves and give the phasor of each. Can one phasor describe $E$ ? (b) Find $H (z; t)$ . (c) Find the time-average power density, and the share carried by the sidebands. (d) Find the largest field and the largest instantaneous power density.

Drawn with f_c=15f_m. — Drawn with $f_{c} = 15 f_{m}$ .

Try it yourself first.

cf. Problem 5.6.6

Exercise 5.6.17: The power of an AM wave