Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 3 / c

Table of contents
- Solution c

Solution

In the aligned frame, define the Alfvén number

A_{i}^{2} = \frac{u _{i}^{2}}{v _{A i}^{2}} = \frac{μ _{0} ρ _{i} u _{x i}^{2}}{B _{x}^{2}},

where the second equality uses

u_{i} ∥ B_{i}

. Since

j = ρ_{i} u_{x i}

and

B_{x}

are common,

A_{i}^{2} = \frac{μ _{0} j u _{x i}}{B _{x}^{2}} .

Therefore

\frac{ρ _{2}}{ρ _{1}} = \frac{u _{x 1}}{u _{x 2}} = \frac{A _{1}^{2}}{A _{2}^{2}} .

Using alignment in the tangential momentum flux gives

ρ u_{x} u_{y} - \frac{B _{x} B _{y}}{μ _{0}} = \frac{B _{x} B _{y}}{μ _{0}} (A^{2} - 1) .

Its continuity then yields

B_{y 2} (A_{2}^{2} - 1) = B_{y 1} (A_{1}^{2} - 1),

or

\frac{B _{y 2}}{B _{y 1}} = \frac{A _{1}^{2} - 1}{A _{2}^{2} - 1}

whenever the ratios are determinate.

Solved by gpt-5.6-sol high.

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