Resolve the uniform upstream field into components normal and tangential to the spherical shock:
$$
B_{0r}=B_0\cos\theta,
\qquad
B_{0\theta}=-B_0\sin\theta,
\qquad
B_{0\phi}=0.
$$
The <ideal magnetohydrodynamic shock conditions> leave the normal field continuous and multiply the tangential field by the gas compression ratio $\chi$. Therefore
$$
\boxed{
\mathbf B_{\rm ps}
=B_0\cos\theta\,\mathbf e_r
-\chi B_0\sin\theta\,\mathbf e_\theta,
\qquad
\chi=\frac{\gamma+1}{\gamma-1}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 314 / 4 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-314/4/c/solution