# Physics Cup – TalTech 2023 – Problem 3

By Jaan Kalda (TalTech).

Two spherical permanent magnets of radius $R$ and of magnetic dipole moment $m$ are initially at a distance $3R$ from each other (this is the distance between the centers of the spheres); the dipole moments are parallel to each other and to the axis connecting the centers of the spheres. The magnets are immersed into an incompressible liquid which is brought into a highly conductive state: in your calculations, you can consider it as superconductive, except that the existing magnetic field is not expelled from it. The liquid fills an infinite (i.e. very large) volume of space.

Now, the magnets are pulled from each other along the axis connecting their centers so that the new distance $L$ between the magnets is much larger than $R$. Estimate the magnitude of the force $F$ needed to keep the magnets at these new positions.

Remark: “estimate” means that the numerical prefactor of your answer does not need to be correct, but all the physical parameters need to enter the expression in a correct way.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The first intermediate results for Problem 2 will be published on 29th January, 13:00 GMT. The first hints will appear here on 5th February 2023, 13:00 GMT. After the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab.