= Coulomb-gauge vector potential of a long solenoid
{c}
For an ideal infinitely long <solenoid> with interior <magnetic field> $B_0\hat{\mathbf z}$ and radius $R$, an axisymmetric <Coulomb gauge> representative of the <magnetic vector potential> is
$$
\mathbf A=A_\phi(r)\hat{\boldsymbol\phi},\qquad A_\phi(r)=\begin{cases}B_0r/2,&r<R,\\B_0R^2/(2r),&r>R.\end{cases}
$$
This follows from $2\pi rA_\phi=\Phi(r)$ by <Stokes theorem>. The exterior has zero <curl> but nonzero circulation around the enclosed <magnetic flux>. The <Coulomb gauge> condition alone does not eliminate every residual <gauge transformation>; symmetry and regularity select this customary representative.
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