 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-75/2/1/solution

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

For the theta pinch, the electric current is azimuthal:

j_{0} = \frac{c}{4 π} \nabla \times (B_{0} \hat{z}) = - \frac{c B _{0}^{'}}{4 π} \hat{θ} .

j_{0} \times B_{0} / c = - B_{0} B_{0}^{'} \hat{r} / (4 π)

. Static radial balance with the pressure gradient therefore gives

p_{0}^{'} + \frac{B _{0} B _{0}^{'}}{4 π} = 0, p_{0} (r) + \frac{B _{0} ( r ) ^{2}}{8 π} = constant .

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