5.1.1 The Wave Equation for the Electric Field

Differentiation of (4.3) with respect to time and substituting $\vec{J }$ from (4.8) and $\vec{H}$ from (4.7) gives

$$\vec{\nabla}\times\frac{1}{\mu}\partial_t\vec{B} = \gamma \partial_t\vec{E} + \epsilon \partial_{tt}\vec{E}.$$ (5.1)

With $\vec{B}$ from (4.1) the wave equation for $\vec{E}$ in the frequency domain is given by the expression

$$\vec{\nabla}\times(\frac{1}{\mu}\vec{\nabla}\times\vec{E}) + (\jmath\omega\gamma - \omega^2\epsilon)\vec{E} = 0,$$ (5.2)

where $\omega$ is the angular frequency. The time convention $e^{\jmath\omega{}t}$ is used and suppressed.