Maxwell's Equations

\begin{matrix} \nabla \cdot \vec{D} = ρ \\ \nabla \cdot \vec{B} = 0 \\ \nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t} \\ \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \end{matrix}

Maxwell's Equations in Source Free, Homogeneous Medium

\begin{matrix} \nabla \cdot \vec{E} = 0 \\ \nabla \cdot \vec{H} = 0 \\ \nabla \times \vec{E} = μ \frac{\partial \vec{H}}{\partial t} \\ \nabla \times \vec{H} = ϵ \frac{\partial \vec{E}}{\partial t} \end{matrix}

\begin{matrix} \nabla \cdot \vec{E} (x, y, z) = 0 \\ \nabla \cdot \vec{H} (x, y, z) = 0 \\ \nabla \times \vec{E} (x, y, z) = - j ω μ \vec{H} (x, y, z) \\ \nabla \times \vec{H} (x, y, z) = j ω ϵ \vec{E} (x, y, z) \end{matrix}

\begin{matrix} \nabla^{2} \vec{E} (x, y, z) + ω^{2} μ ϵ \vec{E} (x, y, z) = 0 \\ \nabla^{2} \vec{H} (x, y, z) + ω^{2} μ ϵ \vec{H} (x, y, z) = 0 \end{matrix}