Gauss's law for magnetism

Remarks

The law is one of Maxwell's equations; as a physical law it expresses that no magnetic monopoles have been observed. The two forms are equivalent by the divergence theorem. For a field given by the Biot–Savart law it is a theorem: such a field is a curl, and the divergence of a curl is zero. Example: the field of a long straight wire, $B = \frac{\mu_0 I}{2 \pi} \frac{(-x_2, x_1, 0)}{x_1^2 + x_2^2}$, has divergence $0$ away from the wire. Non-example: the field $q x / \abs{x}^3$ of a point source has flux $4 \pi q$ through every sphere around $0$, so it cannot be a magnetic field if $q \ne 0$.

Going deeper

For a student

Since $\nabla \cdot B = 0$ on $\R^3$, the field is the curl of a vector potential, $B = \nabla \times A$. This is the reason for the magnetic vector potential and, in the relativistic theory, for the four-potential. The law also gives the continuity of the normal component of $B$ across any interface. If magnetic monopoles existed, the right side would be a magnetic charge density.