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 5.6.8

How much the power pulsates

A wave in free space has the phasor $E (z) = (6 a_{x} + 8 e^{- j ψ} a_{y}) e^{- j β z} V/m$ , $β = ω / c$ . (a) Find $H (z)$ . (b) Find the instantaneous Poynting vector $P (z; t)$ and write it as a constant plus terms at $2 ω$ . (c) Find $P_{a v}$ from (b) and from $\frac{1}{2} Re {E \times H^{*}}$ . (d) Find the range of $P_{z} (t)$ for $ψ = 0$ , $4 5^{\circ}$ and $9 0^{\circ}$ . For which amplitudes and phases would $P$ not pulsate at all?

Try it yourself first.

Strategy

For a wave along $+ z$ in free space Faraday's law gives $H = \frac{1}{η _{0}} a_{z} \times E$ . Then multiply the real fields and use $cos^{2} θ = \frac{1}{2} (1 + cos 2 θ)$ .

$\nabla \times E = - j β a_{z} \times E$ and $H = - \nabla \times E / j ω μ_{0} = (β / ω μ_{0}) a_{z} \times E$ with $β / ω μ_{0} = 1/ η_{0}$ :

H (z) = \frac{1}{η _{0}} (6 a_{y} - 8 e^{- j ψ} a_{x}) e^{- j β z} A/m .

With $θ = ω t - β z$ : $E = 6 cos θ a_{x} + 8 cos (θ - ψ) a_{y}$ and $η_{0} H = 6 cos θ a_{y} - 8 cos (θ - ψ) a_{x}$ , so

P = E \times H = \frac{36 cos ^{2} θ + 64 cos ^{2} ( θ - ψ )}{η _{0}} a_{z} = \frac{50 + 18 cos 2 θ + 32 cos ( 2 θ - 2 ψ )}{η _{0}} a_{z} .

The $2 ω$ terms average to zero: $P_{a v} = (50/ η_{0}) a_{z} = 0.1326 a_{z} W/ m^{2}$ for every $ψ$ . From the phasors,

\frac{1}{2} E \times H^{*} = \frac{1}{2 η _{0}} [36 (a_{x} \times a_{y}) - 64 (a_{y} \times a_{x})] = \frac{100}{2 η _{0}} a_{z},

the same.

The $2 ω$ part has the amplitude $∣18 + 32 e^{- j 2 ψ} ∣/ η_{0}$ :

$ψ$	ripple amplitude [W/m²]	range of $P_{z}$ [W/m²]
$0$	$50/ η_{0} = 0.133$	$0$ to $0.265$
$4 5^{\circ}$	$36.7/ η_{0} = 0.097$	$0.035$ to $0.230$
$9 0^{\circ}$	$14/ η_{0} = 0.037$	$0.095$ to $0.170$

The ripple is smallest for $ψ = \pm 9 0^{\circ}$ , and it vanishes only if, in addition, the two amplitudes are equal: $E_{1}^{2} cos 2 θ + E_{2}^{2} cos (2 θ \mp 18 0^{\circ}) = (E_{1}^{2} - E_{2}^{2}) cos 2 θ = 0$ .

Answer

P_{a v} = \frac{50}{η _{0}} a_{z} = 0.133 a_{z} W/ m^{2} for all ψ, no pulsation only for E_{1} = E_{2}, ψ = \pm 9 0^{\circ}

Check

For $ψ = 0$ the field is $10 cos θ$ along a fixed direction (at $53. 1^{\circ}$ from $a_{x}$ towards $a_{y}$ ), and $P_{z} = 100 cos^{2} θ / η_{0}$ swings between $0$ and twice its mean, as for any single linearly polarised wave.

Remark

The average power depends only on $∣ E_{x} ∣^{2} + ∣ E_{y} ∣^{2}$ , never on the phase between the components. The phase decides how steadily it is delivered: with $ψ = \pm 9 0^{\circ}$ and equal amplitudes (circular polarisation, Section 6.1) the field vector rotates at constant length and the power flows like a d.c. current.

cf. Home exercise 5.6.3

Exercise 5.6.8: How much the power pulsates