Problem $2$: Meissner effect $[40$ points$]$

The magnetic scalar potential outside a spinning shell $($radius $R,$ angular velocity $,$ total charge

q$)$ of charge is:

${}_{\text{o}\text{u}\text{t}}=-\frac{{}_{0}q{R}^2}{12r^2}{P}_1(cos)$

a$.$ Show that the magnetic field outside a spinning spherical shell of charge is

vec$(B)(\frac{?}{\text{b}\text{a}\text{r}}(r))=-$grad${}_M(vec(r))=\frac{{}_0}{4}(\frac{3\text{v}\text{e}\text{c}(m)*vec(r)}{r^5}$vec$(r)-\frac{\text{v}\text{e}\text{c}(m)}{r^3})$

where the magnetic dipole moment of the spinning sphere is

vec$(m)=\frac{1}{3}q{R}^2$vec$().$

b$.$ Below the critical magnetic field $H_c,$ type I superconductors exhibit the Meissner effect, in

which all magnetic fields are excluded from the volume of the superconductor. Consider a

superconducting sphere of radius a placed in an otherwise uniform magnetic induction $B_0.$

Since the magnetic field is excluded from the sphere, the magnetic induction $B$ is tangent to

the sphere at its surface as shown in Figure below.

i$.$ Show that the total magnetic induction outside the sphere is

vec$(B)(vec(r))=$vec$(B{)}_0+\frac{{}_0}{4}(\frac{3\text{v}\text{e}\text{c}(m)*vec(r)}{r^5}$vec$(r)-\frac{\text{v}\text{e}\text{c}(m)}{r^3})$

where the induced dipole moment is

vec$(m)=-\frac{2a^3}{{}_0}$vec$(B{)}_0$

ii$.$ Calculate the energy required to place the sphere in the magnetic field.

$[$hint: the magnetic field induction exerts a pressure on the surface of the sphere. Compute $W$

done on the field first when the radius of the sphere expands from $r$ to $r+r.]$

$[10$ points$]$