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

Recap

Source-free equations, Eq. (5.5.3): $\nabla \times E = - μ \partial H / \partial t$ , $\nabla \times H = ε \partial E / \partial t$ , $\nabla \cdot E = \nabla \cdot H = 0$ . The divergence equations follow from the curl equations, which carry all the dynamics.
Wave equations, Eq. (5.5.13): curl of one curl equation, substitute the other, expand with $\nabla \times \nabla \times A = \nabla (\nabla \cdot A) - \nabla^{2} A$ , use the zero divergence. Result $\nabla^{2} E - μ ε \partial^{2} E / \partial t^{2} = 0$ and the same for $H$ . Every Cartesian component obeys the scalar equation Eq. (5.5.14), cylindrical and spherical components do not. The wave equation is necessary, not sufficient: $\nabla \cdot E = 0$ and Faraday's law must still hold.
d'Alembert, Eq. (5.5.20): in one dimension $E_{x} = f (z - v t) + g (z + v t) = F (t - z / v) + G (t + z / v)$ , shapes moving rigidly in $\pm z$ .
Speed, Eq. (5.5.22): $v = 1/ μ ε = c / n$ , $n = μ_{r} ε_{r}$ , $c = 1/ μ_{0} ε_{0} = 3 \times 1 0^{8} m/s$ .
Sources and conduction, Eqs. (5.5.28) and (5.5.29): the term $μ σ \partial / \partial t$ is a loss. For $σ ≪ ω ε$ a pulse travels unchanged in shape and decays as $e^{- α z}$ , $α = \frac{σ}{2} μ / ε$ , Eq. (5.5.33). For $σ ≫ ω ε$ the equation is a diffusion equation.
Potentials, Eq. (5.5.37): $B = \nabla \times A$ , $E = - \nabla V - \partial A / \partial t$ . In the Lorenz gauge $\nabla \cdot A + μ ε \partial V / \partial t = 0$ both obey wave equations, Eq. (5.5.44), driven by $J$ and by $ρ_{v}$ . Gauge transformations $A + \nabla ψ$ , $V - \partial ψ / \partial t$ leave $E$ and $B$ unchanged.
Retarded potentials, Eq. (5.5.49): the static integrals with the sources taken at $t - R / v$ . Signals need $R / v$ to arrive, a phase lag $2 π R / λ$ for sinusoids, the amplitude falls as $1/ R$ . Quasi-static (circuit theory) when the region is much smaller than $λ /2 π$ .

5.5.11 Recap