Setting $I=0$ in <cylindrical magnetostatic pressure balance> gives $p+B_z^2/(8\pi)=\mathrm{constant}$. The axial boundary value sets this constant to $p_0$. Thus the <magnetic pressure support with vanishing axial field> solution is
$$
\boxed{B_z(R)=\sigma\sqrt{8\pi p_0}\sqrt{1-e^{-R^2/a^2}},\qquad \sigma=\pm1.}
$$
The azimuthal component of the magnetostatic <Ampère-Maxwell equation> gives
$$
\boxed{j_\phi(R)=-\sigma\frac{c\sqrt{8\pi p_0}}{4\pi a^2}
\frac{R e^{-R^2/a^2}}{\sqrt{1-e^{-R^2/a^2}}}.}
$$
Since $1-e^{-R^2/a^2}=R^2/a^2+O(R^4)$, the radial limit is
$$
\boxed{\lim_{R\downarrow0}j_\phi(R)=-\sigma\frac{c\sqrt{8\pi p_0}}{4\pi a}.}
$$
There is a regularity subtlety: $B_z\sim\sigma\sqrt{8\pi p_0}R/a$ is not a <smooth function> of Cartesian position at the axis, and the nonzero limiting $j_\phi$ multiplies an azimuthal unit vector with no unique direction there. These formulas solve the radial problem for $R>0$ and give the requested scalar limit, but the data in this part do not admit the fully smooth vector-field regularity assumed in part (a).

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 3 / d / Solution

With no axial <electric current>, <cylindrical magnetostatic pressure balance> reduces to constant $p+B_z^2/(8\pi)$. For $p=p_0e^{-R^2/a^2}$ and $B_z(0)=0$, its radial solutions are
$$
B_z=\sigma\sqrt{8\pi p_0}\sqrt{1-e^{-R^2/a^2}},
\qquad
j_\phi=-\sigma\frac{c\sqrt{8\pi p_0}}{4\pi a^2}
\frac{R e^{-R^2/a^2}}{\sqrt{1-e^{-R^2/a^2}}},
\qquad \sigma\in\{-1,1\}.
$$
As $R\downarrow0$, $B_z\sim\sigma\sqrt{8\pi p_0}R/a$ and $j_\phi\to-\sigma c\sqrt{8\pi p_0}/(4\pi a)$. These finite scalar limits do not give a <smooth function> of Cartesian position: $R=\sqrt{x^2+y^2}$ has a cusp at the axis, and the azimuthal unit vector has no unique limit there. Thus these data describe radial equilibrium away from the axis, but cannot satisfy smooth vector-field regularity on the axis without modifying the data.

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 Incoming links: Magnetic pressure support with vanishing axial field