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

Most problems on time-harmonic waves begin with a field written either in the time domain or as a phasor, and ask what it tells: the frequency, the direction of travel, the wavelength, the medium, the other field, the power. Problems 5.6.6–5.6.14 practise exactly this, and the procedure is always the same.

Recipe · Reading a time-harmonic field

Bring it to the reference form $A e^{- α ℓ} cos (ω t - β ℓ + φ)$ , where $ℓ$ is the distance along the direction of travel, with $sin θ = cos (θ - π /2)$ and $cos (- u) = cos u$ (Problems 5.6.8, 5.6.9, 5.6.14).
Read the numbers: $ω$ , $β = ∣\nabla ϕ ∣$ and the direction of $\nabla ϕ$ , $α$ , the amplitude $A$ and the phase $φ$ . A minus sign between $ω t$ and $β ℓ$ means travel towards $+ ℓ$ (Problems 5.6.11, 5.6.12, 5.6.13).
Phasor and time domain: $U = A e^{j φ} e^{- α ℓ} e^{- j β ℓ}$ , and back with $Re {U e^{j ω t}}$ (Problems 5.6.8, 5.6.10, 5.6.13, 5.6.14).
The medium: $k^{2} = ω^{2} μ ε - j ω μ σ$ gives $ε$ and $σ$ , and in a lossless medium $v_{p} = c / μ_{r} ε_{r}$ (Problems 5.6.7, 5.6.8, 5.6.9, 5.6.11, 5.6.12).
The other field: $H = \frac{j}{ω μ} \nabla \times E$ and $E = \frac{1}{j ω ε _{c}} \nabla \times H$ (Problems 5.6.8, 5.6.10, 5.6.11).
Power and induction: $P_{a v} = \frac{1}{2} Re {E \times H^{*}}$ and $emf = - d Φ/ d t$ (Problems 5.6.6, 5.6.8).

Checks that catch most slips

$∣ E ∣/∣ H ∣ = ∣ η ∣$ , the intrinsic impedance of the medium ( $η_{0} = 120 π Ω$ in free space), and $E \times H$ points the way the wave travels.
$E ⊥ \nabla ϕ$ where there is no charge, since $\nabla \cdot E = 0$ .
$ε_{r} \geq 1$ and $v_{p} \leq c$ in a material.
A decaying wave written as $e^{- j k x}$ has $Im k < 0$ .
Set $t = 0$ in the answer and compare with the data.
Units: $E$ in $V/m$ , $H$ in $A/m$ , $α$ in $Np/m$ , $β$ in $rad/m$ , power density in $W/ m^{2}$ .

Table 5.6.5. The applications of this section.

Application	Physics used	Worked out in
The radio spectrum	$λ_{0} = c / f$ , half-wave dipole $\approx λ_{0} /2$ , $v_{p} = c / ε_{r}$	Worked example 5.6.2
The microwave oven	$ε_{c} = ε^{'} - j ε^{''}$ , $k = β - j α$ , $p_{a v} = \frac{1}{2} σ_{eff} E_{0}^{2}$	Worked example 5.6.1

5.6.9 Reading a time-harmonic field