 Solution

Since $\nabla \cdot \mathbf{B}=0$, a magnetic field can locally be represented by a magnetic vector potential,It is not unique: $\mathbf{A}\to \mathbf{A}+\nabla \chi$ leaves its curl unchanged. In magnetostatics, choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. Ampère's law then gives ${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J}$ in vacuum. For a localized steady current distribution with the usual decay condition,Its curl gives the Biot–Savart field. For a uniform field, one convenient choice is $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$. Stokes's theorem relates ${\oint }_C\mathbf{A}\cdot d\mathbf{l}$ to magnetic flux through a spanning surface. In time-dependent electrodynamics, $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$; changing $\mathbf{A}$ by $\nabla \chi$ requires $\varphi \to \varphi -{\partial }_t\chi$ to preserve the electric field. Gauge freedom simplifies calculation without changing the electromagnetic observables.