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

Table of contents
- Solution d

Solution

Define the poloidal Alfvén number by

A^{2} = \frac{μ _{0} ρ u _{p}^{2}}{B _{p}^{2}} = \frac{μ _{0} k ^{2}}{ρ},

where

u_{p} = k B_{p} / ρ

. Thus, along a given field line,

ρ = \frac{μ _{0} k ^{2}}{A ^{2}};

density decreases as the Alfvén number grows.

The azimuthal velocity representation gives

u_{ϕ} = r ω + \frac{k B _{ϕ}}{ρ} .

Substitution into

ℓ = r u_{ϕ} - r B_{ϕ} / (μ_{0} k)

yields

ℓ = r^{2} ω + \frac{r B _{ϕ}}{μ _{0} k} (A^{2} - 1) .

At the Alfvén surface,

A^{2} = 1

. Smoothness requires the numerator of the solution for

B_{ϕ}

to vanish there, so

ℓ = ω r_{a}^{2} .

Solving away from the surface then gives

B_{ϕ} = \frac{μ _{0} kω}{r} \frac{r ^{2} - r _{a}^{2}}{1 - A ^{2}} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version