Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 4 / c

Solution

Axisymmetry makes the azimuthal direction geometrically distinct, so the field splits into a meridional poloidal part and an azimuthal toroidal part:

B = B_{p} + B_{ϕ} e_{ϕ} .

B_{p} = \nabla ψ \times \nabla ϕ,

or in cylindrical coordinates

B_{r} = - \frac{1}{r} \frac{\partial ψ}{\partial z}, B_{z} = \frac{1}{r} \frac{\partial ψ}{\partial r} .

r

at fixed

z

Φ_{p} (r, z) = 2 π [ψ (r, z) - ψ (0, z)] .

Thus, after choosing

ψ = 0

on the axis,

2 π ψ

is the enclosed poloidal flux and surfaces of constant

ψ

are magnetic surfaces.

There are no discussions about this article yet.

There are currently no matching articles.