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

Table of contents
- Solution b

Solution

For the stated magnetic vector potential, part a gives

B = \nabla β \times \nabla ϕ + (Lα) \nabla ϕ .

Applying the same identity once more and using the Ampère-Maxwell equation in the magnetostatic limit gives

J = \frac{1}{μ _{0}} [\nabla (Lα) \times \nabla ϕ + (L β) \nabla ϕ] .

Expanding the Lorentz force density

F_{m} = J \times B

, using

\nabla ϕ = e_{ϕ} / r

and the vector triple-product identity, separates its poloidal and azimuthal parts:

r^{2} μ_{0} F_{m} = (L β) \nabla β - (Lα) \nabla (Lα) + f e_{ϕ},

where

f = [\nabla (Lα) \times \nabla β] \cdot e_{ϕ} .

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