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\)
Section 16.11 Second derivatives and Maxwell’s Equations
Since the electric field is (minus) the gradient of the scalar potential, it is conservative. But since
\begin{gather*}
\grad\times\grad V = \zero
\end{gather*}
for any function \(V\text{,}\) we can rewrite this as
\begin{gather*}
\grad\times\EE = \zero
\end{gather*}
which is the differential form of (the electrostatic version of) Faraday’s Law, and another of Maxwell’s Equations.
Similarly, we can apply the identity
\begin{gather*}
\grad\cdot(\grad\times\FF) = 0
\end{gather*}
for any vector field \(\FF\text{,}\) to the magnetic vector potential, which yields the fourth and final of Maxwell’s Equations, namely
\begin{gather*}
\grad\cdot\BB = \grad\cdot(\grad\times\AA) = 0 .
\end{gather*}
In summary, Maxwell’s equations for electro- and magnetostatics are:
\begin{align*}
\grad\cdot\EE \amp= \frac{\rho}{\epsilon_0} ,\\
\grad\times\EE \amp= \zero ,\\
\grad\cdot\BB \amp= 0 ,\\
\grad\times\BB \amp= \mu_0 \,\JJ .
\end{align*}
