Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 1 / c

Table of contents
- Solution c

Solution

Define the accumulated rotation angle

θ (x) = \int_{x_{*}}^{x} q (s) d s .

The coupled first-order equations describe a rotation of

(B_{y}, B_{z})

and have the general solution

B_{z} = C_{1} cos θ + C_{2} sin θ, B_{y} = C_{1} sin θ - C_{2} cos θ, B_{x} = 0 .

Direct differentiation verifies both force-free equations, and

B_{y}^{2} + B_{z}^{2} = C_{1}^{2} + C_{2}^{2}

is constant. Thus a one-dimensional force-free magnetic field rotates without changing its magnitude.

 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