Pre-Quantum Electrodynamics

Our two fundamental equations for the electrostatic field are \[ {\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0} \hspace{2cm} {\boldsymbol \nabla} \times {\bf E} = 0. \] For the electrostatic potential, Gauss' law becomes

When the charge density vanishes, it becomes more simply

Since the curl of a gradient is always zero, we by construction have \({\boldsymbol \nabla} \times {\bf E} = - {\boldsymbol \nabla} \times ({\boldsymbol \nabla} V) = 0\).