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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,

$∇×E=-\frac{∂B}{∂t}$

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

Putting E=0 in the expression,

$$\begin{gathered} ∇×E=-\frac{∂B}{∂t} \\ ∇×0=-\frac{∂B}{∂t} \\ \frac{∂B}{∂t}=0 \end{gathered}$$

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,

$∫E.dl=-\frac{d\mathrm{Φ}}{dt}$

Here, E is the electric field and$\mathrm{Φ}$ is the magnetic fluxthrough aperfectly conducting loop.

Putting$E=0$in the expression,

$$\begin{gathered} ∫0.dl=-\frac{d\mathrm{Φ}}{dt} \\ -\frac{d\mathrm{Φ}}{dt}=0 \\ \frac{d\mathrm{Φ}}{dt}=0 \end{gathered}$$

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

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

$∇×B={\mathrm{μ}}_{0}J+{\mathrm{μ}}_0{∈}_0\frac{∂E}{∂t}$

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

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

$$\begin{gathered} ∇×0={\mathrm{μ}}_{0}J+{\mathrm{μ}}_0{∈}_0×0 \\ {\mathrm{μ}}_{0}J=0 \\ J=0 \end{gathered}$$

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,

$$\begin{gathered} B=∇×A \\ B=\frac{2{\mathrm{μ}}_{0}R\mathrm{Ó\neg }\mathrm{δ}}{3}(\cos \mathrm{θ}\hat{r}-\sin \mathrm{θ}\widehat{\mathrm{θ}}) \\ B=\frac{2}{3}{\mathrm{μ}}_0\mathrm{δ}R\mathrm{Ó\neg }\hat{z} \\ B=\frac{2}{3}{\mathrm{μ}}_0\mathrm{δ}R\mathrm{Ó\neg } \end{gathered}$$

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

$$\begin{gathered} B=\frac{2}{3}{\mathrm{μ}}_0\mathrm{δ}\mathrm{Ó\neg }a\hat{Z} \\ \mathrm{δ}\mathrm{Ó\neg }a=-\frac{2B_0}{3{\mathrm{μ}}_0} \end{gathered}$$

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

$K=\mathrm{δ}\mathrm{ν}$

Here,$\mathrm{δ}$ is the surface charge density and$\mathrm{ν}$ is the velocity of the charge.

Putting the value of charge velocity$\mathrm{ν}=\mathrm{Ó\neg }×a\sin \mathrm{θ}\widehat{\mathrm{Ï•}}$ in the expression,

$$\begin{gathered} K=\mathrm{δ}\mathrm{Ó\neg }a\sin \mathrm{θ}\widehat{\mathrm{Ï•}} \\ K=-\frac{3B_0}{2{\mathrm{μ}}_0}\sin \mathrm{θ}\widehat{\mathrm{Ï•}} \end{gathered}$$

Hence, the induced surface current density is$k=-\frac{3B_0}{2{\mathrm{μ}}_0}\sin \mathrm{θ}\widehat{\mathrm{ϕ}}$.