Define the accumulated rotation angle
$$
\theta(x)=\int_{x_*}^{x}q(s)\,ds.
$$
The coupled first-order equations describe a rotation of $(B_y,B_z)$ and have the general solution
$$
\boxed{
B_z=C_1\cos\theta+C_2\sin\theta,\qquad
B_y=C_1\sin\theta-C_2\cos\theta,\qquad
B_x=0}.
$$
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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-314/1/c/solution