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 5.6.2

Phase velocity measured along a line

A small source at the origin radiates a $100 MHz$ wave in free space whose field, far from the source, has the form $E (r; t) = (A / r) cos (ω t - k_{0} r)$ . (a) Find the phase velocity from $ω /∣\nabla ϕ ∣$ . (b) A probe moves along the line $x = 3 m$ , $y = 0$ , parallel to the $z$ -axis, and measures the speed with which the crests pass along this line, $v_{z} = ω / (\partial ϕ / \partial z)$ . Find $v_{z}$ at $z = 1$ , $4$ and $30 m$ . (c) What is the spacing of the crests along the line near $z = 4 m$ ? Is anything moving faster than light?

Try it yourself first.

Strategy

The crests are the spheres $k_{0} r = ω t +$ const, and they move along $\nabla ϕ$ at $ω /∣\nabla ϕ ∣$ . Along any other line the crests cross at a different apparent speed, $ω$ divided by the component of $\nabla ϕ$ along that line.

$ϕ = k_{0} r$ , $\nabla ϕ = k_{0} a_{r}$ and $∣\nabla ϕ ∣ = k_{0} = ω / c$ , so $v_{p} = c = 3 \times 1 0^{8} m/s$ , along $a_{r}$ . Here $k_{0} = 2 π /3 rad/m$ and $λ_{0} = 3 m$ .

On the line $r = 9 + z^{2}$ , so

\frac{\partial ϕ}{\partial z} v_{z} = k_{0} \frac{z}{r} = k_{0} cos θ, = \frac{ω}{k _{0} cos θ} = c \frac{r}{z},

where $θ$ is the angle between the line and $a_{r}$ .

$z$ [m]	1	4	30
$r$ [m]	3.16	5.00	30.15
$v_{z}$ [m/s]	$9.49 \times 1 0^{8}$	$3.75 \times 1 0^{8}$	$3.01 \times 1 0^{8}$

Near $z = 4 m$ the phase advances by $\partial ϕ / \partial z = k_{0} \times 4/5$ per metre, so successive crests are

λ_{z} = \frac{2 π}{\partial ϕ / \partial z} = λ_{0} \frac{r}{z} = 3 \times \frac{5}{4} = 3.75 m

apart, longer than $λ_{0}$ .

Answer

v_{p} = c, v_{z} = c \frac{r}{z} = 9.49 \times 1 0^{8}, 3.75 \times 1 0^{8}, 3.01 \times 1 0^{8} m/s λ_{z} = 3.75 m near z = 4 m

Check

$v_{z}$ is always $\geq c$ . It tends to $c$ far along the line ( $θ \to 0$ ) and to infinity at $z = 0$ , where the line is tangent to the crests and they cross it all at once.

Remark

Nothing travels faster than light: the point where a crest cuts the line is not a thing that moves, just as the point where a wave breaking obliquely on a beach meets the shoreline runs along it faster than the water. Energy flows radially at $c$ . This is why Eq. (5.6.2) defines $v_{p}$ with $∣\nabla ϕ ∣$ , which gives the smallest of the apparent speeds: $v_{p} = ω /∣\nabla ϕ ∣ \leq ω / (\partial ϕ / \partial s)$ along any direction $s$ .

cf. Problem 5.6.13

Exercise 5.6.2: Phase velocity measured along a line