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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-60/3/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 60 / 3 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Put

x = α^{2}

and

p = π^{2}

. Then

R_{c} = \frac{( x + p ) ^{3} + Qp ( x + p )}{x}, \frac{d R _{c}}{d x} = \frac{( x + p ) ^{2} ( 2 x - p ) - Q p ^{2}}{x ^{2}} .

The minimizing condition is

2 x^{3} + 3 p x^{2} = p^{3} + Q p^{2} .

Its left side increases strictly for

x > 0

and the neutral curve diverges at both ends, so there is a unique global minimum. For large

Q

,

x \to \infty

and the leading balance is

2 x^{3} \sim Q p^{2}

. Therefore

α \sim (π^{4} /2)^{1/6} Q^{1/6} .

This is the strong-field wavenumber selection in magnetoconvection scaling. In a box of fixed width one compares the two admissible horizontal wavenumbers bracketing the continuum optimum.

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