106MT2Pr-solutions

Question 2

(a) The normal component of ${\vec {B}}$ must be zero.

(b) To find the field outside the superconductor, we place an "image dipole" inside the superconductor a depth $d$ below the surface, oriented at an angle ($\pi -\theta$) from vertical. This automatically satisfies the boundary condition. The field outside the superconductor is the field due to the original dipole and the image dipole.

(c) Taking the $y$-axis normal to $\mu$, parallel to the superconducting sheet, and into the page:

${\vec {\tau }}=-{\frac {\mu ^{2}\sin 2\theta }{64\pi \epsilon _{0}c^{2}d^{3}}}{\hat {y}}$

(d) Stable equilibria: $\theta =\pi /2$, $\theta =3\pi /2$. Unstable equilibria: $\theta =0$, $\theta =\pi$

(e) $F=-{\frac {3\mu ^{2}\sin 2\theta }{64\pi \epsilon _{0}c^{2}d^{4}}}(3+\cos 2\theta )$ away from the superconductor.