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

For constant $B_z$, cylindrical magnetostatic pressure balance givesRegularity gives $I(0)=0$. With $x=R^2/a^2$, integration yields the power-law pressure-supported axial currentThe sign of $I$ may be chosen either way and fixes the sense of the toroidal magnetic field. For $p_0>0$ and nontrivial decreasing pressure ($k>0$), the integral converges at infinity precisely whenAt $k=1$ it diverges logarithmically; for $0<k<1$ it diverges as $R^{2(1-k)}$. The degenerate case $k=0$ has constant pressure and zero electric current.