Problem 3 – Hint 5

As the length of a flux tube is being increased while its volume remains constant, its average radius must decrease. Furthermore, the hydrostatic pressure along the surface of each flux tube is constant, because the fluid can flow freely along the field lines. Notice also that where the magnetic field strength changes in a direction perpendicular to the field lines, there is a current in the fluid, and hence an Ampère force. If the curvature of the magnetic field lines can be neglected, this Ampère force can be said to be caused by a gradient of the magnetic pressure $B^2/2\mu_0$. Taken together, these considerations mean that over a major part of the surface of a flux tube, the magnetic field strength is constant (why exactly?). This also means that the radius of a flux tube is constant over most of its length, (why?). Thus, one can estimate the magnetic energy inside a flux tube as a function of its length.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. There will be no more hints for the Problem 3, but it will remain open at least until 19th March 2023. Thee next updates of the intermediate results will be published at 13:00 GMT, 12th March 2023.