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

Recap

Monochromatic wave, Eq. (5.6.1): $u = A (r) cos [ω t - ϕ (r)]$ with amplitude $A$ , angular frequency $ω$ , period $T = 2 π / ω$ and frequency $f = 1/ T$ . One frequency is enough because Fourier synthesis and linearity give every other signal.
Phase velocity, Eq. (5.6.2): $v_{p} = ω /∣\nabla ϕ ∣$ , $v_{p} = ω / β$ for $ϕ = β z$ , and $v_{p} = 1/ μ ε = c / ε_{r} μ_{r}$ in a lossless medium. Wavelength $λ = 2 π / β = v_{p} / f$ , Eq. (5.6.5).
Spectrum: ITU band $N$ from $0.3 \times 1 0^{N}$ to $3 \times 1 0^{N} Hz$ . The hardware scales with $λ_{0} = c / f$ , a half-wave dipole is $λ_{0} /2$ long.
Phasors, Eq. (5.6.7): $u = Re {U e^{j ω t}}$ , $U = A e^{- j ϕ}$ , with a cosine reference. $\partial / \partial t \to j ω$ , a sine carries the factor $- j$ , and phasors do not multiply.
Maxwell's equations in phasor form, Eq. (5.6.10): $\nabla \times E = - j ω B$ , $\nabla \times H = J + j ω D$ , and the divergence equations follow from the curl equations.
Complex permittivity, Eq. (5.6.13): $ε_{c} = ε - j σ / ω = ε^{'} - j ε^{''}$ , $tan δ = ε^{''} / ε^{'} = σ / ω ε$ , $σ_{eff} = ω ε^{''}$ . Good dielectric for $tan δ ≪ 1$ , good conductor for $tan δ ≫ 1$ , and the regime depends on the frequency.
Helmholtz equations, Eq. (5.6.17): $\nabla^{2} E + k^{2} E = 0$ together with $\nabla \cdot E = 0$ , with $k = ω μ ε_{c} = β - j α$ , Eq. (5.6.16), $k_{0} = ω / c$ in free space.
Wave number, Eq. (5.6.20): $E_{0} e^{- α z} cos (ω t - β z)$ , $β$ in $rad/m$ , $α$ in $Np/m$ , $ar g k = - δ /2$ , $γ = j k = α + j β$ . Low loss: $α \approx \frac{1}{2} β tan δ$ .
Applications: antenna sizes $λ_{0} /2$ from $150 m$ (AM) to $1.95 mm$ ( $77 GHz$ radar). The microwave oven heats water through $ε^{''}$ , $p_{a v} = \frac{1}{2} σ_{eff} E_{0}^{2}$ , at $2.45 GHz$ because the reach $1/2 α = 1.9 cm$ is still useful.

5.6.10 Recap