Generated by the Physics Derivation Graph. Eq. \ref{eq:4757562} is an initial equation. \begin{equation} \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t} \label{eq:4757562} \end{equation} Eq. \ref{eq:1199299} is an initial equation. \begin{equation} \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E} \label{eq:1199299} \end{equation} Eq. \ref{eq:4857731} is an initial equation. \begin{equation} \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0 \label{eq:4857731} \end{equation} Partially differentiate Eq. \ref{eq:1199299} with respect to \(t\); yields Eq. \ref{eq:4642245}. \begin{equation} \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:4642245} \end{equation} Apply curl to both sides of Eq. \ref{eq:4757562}; yields Eq. \ref{eq:2392932}. \begin{equation} \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} \label{eq:2392932} \end{equation} Substitute LHS of Eq. \ref{eq:2392932} into Eq. \ref{eq:4642245}; yields Eq. \ref{eq:2962831}. \begin{equation} \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:2962831} \end{equation} Eq. \ref{eq:3984852} is an assumption. \begin{equation} \rho = 0 \label{eq:3984852} \end{equation} Substitute LHS of Eq. \ref{eq:4857731} into Eq. \ref{eq:3984852}; yields Eq. \ref{eq:2837471}. \begin{equation} \vec{ \nabla} \cdot \vec{E} = 0 \label{eq:2837471} \end{equation} Eq. \ref{eq:1939485} is an identity. \begin{equation} \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:1939485} \end{equation} Substitute LHS of Eq. \ref{eq:1939485} into Eq. \ref{eq:2837471}; yields Eq. \ref{eq:3738373}. \begin{equation} \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E} \label{eq:3738373} \end{equation} Substitute LHS of Eq. \ref{eq:3738373} into Eq. \ref{eq:2962831}; yields Eq. \ref{eq:4758592}. \begin{equation} \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:4758592} \end{equation} Eq. \ref{eq:4758592} is one of the final equations.