Magnetic Dipole in a Cuboidal Superconducting Trap

We derive the exact image-dipole potential of a point dipole inside a closed cuboidal superconducting trap.

The construction generalises the parallel-plate result to a geometry that confines every translational degree of freedom, and we prove that the image lattice satisfies the Meissner boundary condition on all six walls.

For a centred dipole the orientational energy reduces to a diagonal quadratic form whose three coefficients are Epstein-zeta-type lattice sums.

We show that in both the infinite and finite rectangular traps the dipole orientation aligns with the \emph{short} cross-sectional axis over a finite range of aspect ratios.

The equilibrium orientation in both cases is described by a phase diagram whose degeneracies we classify.

Every prediction is verified against finite-element solutions of the same boundary-value problem, with agreement better than $0.16\%$.