The representation
$$
\mathbf B_p=\nabla\psi\times\nabla\phi
$$
gives
$$
B_r=-\frac1r\frac{\partial\psi}{\partial z},
\qquad
B_z=\frac1r\frac{\partial\psi}{\partial r}.
$$
Hence $\mathbf B\mathbin\cdot\nabla\psi=0$: the level sets of the <poloidal magnetic flux function> are magnetic surfaces. Moreover, the magnetic flux through a circular disc is
$$
\int_0^r2\pi r'B_z(r',z)\,dr'
=2\pi[\psi(r,z)-\psi(0,z)].
$$
Thus $2\pi\psi$ is the enclosed poloidal magnetic flux up to an additive axial reference.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 4 / a / Solution

{title2=$\psi$}

For $\mathbf B_p=\nabla\psi\times\nabla\phi$, surfaces $\psi=\text{constant}$ are poloidal magnetic surfaces and $2\pi\psi$ is magnetic flux up to the choice of axial reference value.


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 Incoming links: Poloidal magnetic flux function