5.1.1 The four static equations

Let us collect what the two static chapters established. Each of them produced two field equations, one for the curl of its field and one for the divergence.

Equation	Name	Built in
① $\nabla \times E=0$	first fundamental equation of electrostatics	Section 3.3
② $\nabla \times H=J$	Ampère's law, first fundamental equation of magnetostatics	Section 4.1
③ $\nabla \cdot D={\rho }_v$	Gauss's law, second fundamental equation of electrostatics	Section 3.2
④ $\nabla \cdot B=0$	conservation of magnetic flux, second fundamental equation of magnetostatics	Section 4.1

Equation ① says that the electrostatic field does no net work around a closed path, which is what allowed us to define the potential $V$. Equation ③ says that electric flux begins and ends on charges. Equation ② says that magnetic field lines circulate around currents, and ④ that they never begin or end at all: there are no magnetic charges, and every line of $B$ closes on itself.