Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 3 / d

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- Solution d

Solution

For

B = B e_{x}

, the vertical magnetic perturbation is

δ B_{z} = i k_{x} B ξ_{z} .

The

z

component of the Fourier-transformed equation of motion is therefore

- ρ ω^{2} ξ_{z} = - i k_{z} δ Π - ρ v_{A}^{2} k_{x}^{2} ξ_{z} .

Using

ω = v k_{x}

gives

\frac{δ Π}{ξ _{z}} = - i ρ \frac{k _{x}^{2}}{k _{z}} (v^{2} - v_{A}^{2}) .

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