Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 4 / c

Table of contents
- Solution c

Solution

Resolve the upstream uniform field

B_{0} = B_{0} e_{z}

into components normal and tangential to the spherical shock:

B_{r, 1} = B_{0} cos θ, B_{θ, 1} = - B_{0} sin θ, B_{ϕ, 1} = 0.

The ideal magnetohydrodynamic shock conditions preserve the normal magnetic component and multiply the tangential component of a weak passive field by the gas compression ratio. Part (b) therefore gives

B_{r} (R, θ, t) = B_{0} cos θ, B_{θ} (R, θ, t) = - 4 B_{0} sin θ, B_{ϕ} = 0 .

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