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

5.1.5 Recap

Recap

Maxwell's equations, Eq. (5.1.1): $\nabla \times E = - \partial B / \partial t$ , $\nabla \times H = J + \partial D / \partial t$ , $\nabla \cdot D = ρ_{v}$ , $\nabla \cdot B = 0$ , with $D = ε E$ , $B = μ H$ and, in conductors, $J = σ E$ . The divergence equations are those of statics, each curl equation has one new time-derivative term. The equations are local and universal.
Counting: twelve field components, closed by the constitutive relations, Eq. (5.1.2). The divergence equations follow from the curl equations and the conservation of charge, so they act as initial conditions.
The wave: $E = E_{0} cos (ω t - k z) a_{x}$ , $H = (E_{0} / η_{0}) cos (ω t - k z) a_{y}$ solves the source-free equations with $ω / k = c = 1/ μ_{0} ε_{0} = 3 \times 1 0^{8} m/s$ and $η_{0} = μ_{0} / ε_{0} = 120 π \approx 377 Ω$ .