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

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- Solution c

Solution

Resolve the uniform upstream field into components normal and tangential to the spherical shock:

B_{0 r} = B_{0} cos θ, B_{0 θ} = - B_{0} sin θ, B_{0 ϕ} = 0.

The ideal magnetohydrodynamic shock conditions leave the normal field continuous and multiply the tangential field by the gas compression ratio

χ

. Therefore

B_{ps} = B_{0} cos θ e_{r} - χ B_{0} sin θ e_{θ}, χ = \frac{γ + 1}{γ - 1} .

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