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Past exam of the mathematics course of the University of Cambridge

2021

Paper 314

For the stated <magnetic vector potential>, part a gives
$$
\boxed{\mathbf B=\nabla\beta\times\nabla\phi
+(L\alpha)\nabla\phi}.
$$
Applying the same identity once more and using the <Ampère-Maxwell equation> in the magnetostatic limit gives
$$
\boxed{\mathbf J=\frac1{\mu_0}
\left[\nabla(L\alpha)\times\nabla\phi
+(L\beta)\nabla\phi\right]}.
$$
Expanding the <Lorentz force density> $\mathbf F_m=\mathbf J\times\mathbf B$, using $\nabla\phi=\mathbf e_\phi/r$ and the vector triple-product identity, separates its poloidal and azimuthal parts:
$$
\boxed{r^2\mu_0\mathbf F_m
=(L\beta)\nabla\beta-(L\alpha)\nabla(L\alpha)
+f\mathbf e_\phi},
$$
where
$$
\boxed{f=[\nabla(L\alpha)\times\nabla\beta]\cdot\mathbf e_\phi}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 3 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 3 / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 3 / b / Solution

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