Magnetostatics and vector potential

Magnetostatics means that currents and charge densities have settled into time-independent configurations. Charge conservation then requires zero current divergence. A wire cannot continuously deliver charge into an isolated endpoint while maintaining the magnetostatic assumption. In a laboratory, return paths and sources complete the circuit. An infinitely long straight wire is a useful local idealization when the observation point is far from those geometric complications.

Ampère's circulation law holds for steady currents in vacuum, but extracting a field magnitude requires symmetry. For an ideal straight wire, cylindrical symmetry makes the field azimuthal and constant along a centered circle. The right-hand rule fixes its direction from the current direction. An irregular loop still obeys the law, but generally requires a Biot–Savart integral rather than a single division by circumference.

Writing $\mathbf B=\nabla\times\mathbf A$ automatically satisfies the absence of magnetic charge. In a contractible region with suitable regularity, a divergence-free field admits this global representation; more complicated topology can require patches or additional flux conditions. Its units are tesla metres, equivalently webers per metre. A potential component is not a magnetic field component, and differentiating it is essential before comparing with a magnetometer.

Adding $\nabla\chi$ changes $\mathbf A$ without changing its curl, because mixed derivatives cancel. Choosing the Coulomb gauge $\nabla\cdot\mathbf A=0$ converts the static field equations into a vector Poisson equation. Boundary and decay conditions remain necessary, just as for an electrostatic potential. A gauge choice organizes the redundant description; it does not add a new measured force.

Stokes's theorem relates the line integral of $\mathbf A$ around a closed loop to magnetic flux through a spanning surface. For a single-valued smooth gauge function, the added gradient integrates to zero around that loop. By contrast, an open-path integral changes by the difference of gauge-function values at its endpoints. Closing the path is therefore a physical and mathematical distinction, not a drawing convention.

For the ideal wire, one convenient potential has only a longitudinal component proportional to minus the logarithm of radius. The arbitrary reference radius changes that component by a constant and leaves the field unchanged. The logarithm cannot be forced to vanish at infinity for an infinite wire: the source itself is nonlocalized. This limitation is another reminder to match boundary assumptions to the source idealization.