\[\mathbf{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon}\]
\[\mathbf{\nabla} \cdot \mathbf{H}=0\]
\[\mathbf{\nabla} \times \mathbf{E}=- \mu \frac{\partial H}{\partial t}\]
\[\mathbf{\nabla} \times \mathbf{H}=- \epsilon \frac{\partial E}{\partial t}+ \mathbf{J}\]
In the case of static fields,
\[\mathbf{E}, \mathbf{H}\]
are constant, so that \[\frac{\partial \mathbf{E}}{\partial t} =\frac{\partial \mathbf{H}}{\partial t}=0\]
Hence
\[\mathbf{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon}\]
\[\mathbf{\nabla} \cdot \mathbf{H}=0\]
\[\mathbf{\nabla} \times \mathbf{E}=0\]
\[\mathbf{\nabla} \times \mathbf{H}= \mathbf{J}\]