Magnetostatics in Vacuum

A Field With No Sources

Magnetostatics mirrors electrostatics with the two roles swapped. Where $\mathbf{E}$ has divergence but no curl, $\mathbf{B}$ has curl but no divergence:

$$\nabla \cdot \mathbf{B} = 0, \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$

with $\mu_0 = 4\pi \times 10^{-7}$ T m/A. The first says there are no magnetic monopoles: every field line closes on itself, so the flux through any closed surface is exactly zero. The second is Ampère's law, whose integral form is

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$$

Because $\mathbf{B}$ is divergence-free it cannot be the gradient of a scalar, but it can be the curl of something, the vector potential $\mathbf{A}$, with $\mathbf{B} = \nabla \times \mathbf{A}$. In the Coulomb gauge this satisfies $\nabla^2\mathbf{A} = -\mu_0\mathbf{J}$, one Poisson equation per component.

Ampère's law earns its keep on symmetric geometries:

The Biot-Savart law handles the rest, integrating contributions $d\mathbf{B} \propto I\,d\mathbf{l}\times\hat{\mathbf{r}}/r^2$ along the wire.

At large distance a current loop looks like a magnetic dipole of moment $\mathbf{m} = I\mathbf{A}$, with a field of the same $1/r^3$ shape as the electric dipole's. In a uniform field it feels a torque $\boldsymbol{\tau} = \mathbf{m}\times\mathbf{B}$ but no net force, which is why a compass needle turns rather than being dragged.

Common pitfall: expecting the field to be zero wherever the enclosed current is zero. Ampère's law constrains the circulation, not the field. An Amperian loop drawn outside a solenoid encloses equal and opposite currents and gives zero circulation, while the field just inside is large.