In a perfect conductor, the conductivity is infinite, so $\mathbf{E}\mathbf{=}\mathbf{0}$(Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics).

(a) Show that the magnetic field is constant $\left(\frac{\partial B}{\partial t}=0\right)$, inside a perfect conductor.

(b) Show that the magnetic flux through a perfectly conducting loop is constant.

A superconductor is a perfect conductor with the additional property that the (constant) B inside is in fact zero. (This "flux exclusion" is known as the Meissner effect.)

(c) Show that the current in a superconductor is confined to the surface.

(d) Superconductivity is lost above a certain critical temperature $\mathbf{\left(}{\mathbf{T}}_{\mathbf{c}}\mathbf{\right)}$, which varies from one material to another. Suppose you had a sphere (radius ) above its critical temperature, and you held it in a uniform magnetic field ${\mathbf{B}}_{\mathbf{0}}\hat{\mathbf{z}}$while cooling it below ${\mathbf{T}}_{\mathbf{c}}$. Find the induced surface current density K, as a function of the polar angle$\mathit{\theta }$.

Based on this law whenever a conductor is kept inside a varying magnetic field then it experiences a force known as ‘electro motive force (emf)’ as well as a certain current is induced.

The value of emf generated on a conducting coil relies upon the change of magnetic flux as well as the number of turns of the coil.

Applying Faraday’s law, the expression for the magnetic field inside a perfect conductor is given by,

$\nabla \times E=-\frac{\partial B}{\partial t}$

Here, E is the electric field and B is the magnetic fieldinside a perfect conductor.

Putting E=0 in the expression,

$$\begin{gathered} \nabla \times E=-\frac{\partial B}{\partial t} \\ \nabla \times 0=-\frac{\partial B}{\partial t} \\ \frac{\partial B}{\partial t}=0 \end{gathered}$$

Hence, the magnetic field is constant inside a perfect conductor.

Using Faraday’s law, the integral formula for themagnetic flux through a perfectly conducting loop is given by,

$\int E.dl=-\frac{d\Phi }{dt}$

Here, E is the electric field and$\Phi$ is the magnetic fluxthrough aperfectly conducting loop.

Putting$E=0$in the expression,

$$\begin{gathered} \int 0.dl=-\frac{d\Phi }{dt} \\ -\frac{d\Phi }{dt}=0 \\ \frac{d\Phi }{dt}=0 \end{gathered}$$

Hence, the magnetic flux through a perfectly conducting loop is constant.

The generalized form of Ampere-Maxwell formula is given by,

$\nabla \times B={\mu }_{0}J+{\mu }_0{\in }_0\frac{\partial E}{\partial t}$

Here, E represents the electric field,${\mu }_0$ is the permeability of free space, J is the current in the superconductor and$\frac{\partial E}{\partial t}$ is the change in electric field.

Putting$E=0$and$B=0$ in expression,

$$\begin{gathered} \nabla \times 0={\mu }_{0}J+{\mu }_0{\in }_0\times 0 \\ {\mu }_{0}J=0 \\ J=0 \end{gathered}$$

Hence, the current in a superconductor is confined to the surface.

The expression for the uniform magnetic field generated inside a rotating shell in polar form is given by,

$$\begin{gathered} B=\nabla \times A \\ B=\frac{2{\mu }_{0}R\omega \delta }{3}(\cos \theta \hat{r}-\sin \theta \hat{\theta }) \\ B=\frac{2}{3}{\mu }_0\delta R\omega \hat{z} \\ B=\frac{2}{3}{\mu }_0\delta R\omega \end{gathered}$$

Putting the value of radius R=a in the expression,

$$\begin{gathered} B=\frac{2}{3}{\mu }_0\delta \omega a\hat{Z} \\ \delta \omega a=-\frac{2B_0}{3{\mu }_0} \end{gathered}$$

The formula for the induced surface current density of the sphere is given by,

$K=\delta \nu$

Here,$\delta$ is the surface charge density and$\nu$ is the velocity of the charge.

Putting the value of charge velocity$\nu =\omega \times a\sin \theta \hat{\varphi }$ in the expression,

$$\begin{gathered} K=\delta \omega a\sin \theta \hat{\varphi } \\ K=-\frac{3B_0}{2{\mu }_0}\sin \theta \hat{\varphi } \end{gathered}$$

Hence, the induced surface current density is$k=-\frac{3B_0}{2{\mu }_0}\sin \theta \hat{\varphi }$.