$${\frac{\partial H_y}{ \partial z} = - \varepsilon \varepsilon_0 \frac{\partial E_x}{\partial t} \\ \frac{\partial H_x}{ \partial z} = - \varepsilon \varepsilon_0 \frac{\partial E_y}{\partial t} \\ 0 = - \varepsilon \varepsilon_0 \frac{\partial E_z}{\partial t}}$$