Exercise 5.6.10: Power delivered by a radiating current sheet

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 5.6.10

Power delivered by a radiating current sheet

The plane $z = 0$ in free space carries a time-harmonic surface current $J_{s} = J_{s 0} a_{x}$ with $J_{s 0} = 1 A/m$ (real). It radiates plane waves away from the sheet on both sides: $E = E_{1} e^{- j k_{0} ∣ z ∣} a_{x}$ on either side (symmetry), with $H = \pm (E_{1} / η_{0}) e^{- j k_{0} ∣ z ∣} a_{y}$ for $z ≷ 0$ . (a) Use the boundary condition $a_{z} \times (H_{2} - H_{1}) = J_{s}$ to find $E_{1}$ . (b) Find $P_{a v}$ on both sides. (c) Find the power the sheet delivers per unit area, $- \frac{1}{2} Re {E \cdot J_{s}^{*}}$ , and check the Poynting relation for a thin box enclosing $1 m^{2}$ of the sheet. (d) What changes if the space on both sides is a lossless dielectric with $ε_{r} = 4$ ?

Try it yourself first.

cf. Problem 5.6.5