91影视

Based on this law whenever a conductor is kept inside a varying magnetic field then it experiences a force known as 鈥榚lectro motive force (emf)鈥� 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鈥檚 law, the expression for the magnetic field inside a perfect conductor is given by,

$鈭�脳E=-\frac{鈭�B}{鈭�t}$

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

Putting E=0 in the expression,

$$\begin{gathered} 鈭�脳E=-\frac{鈭�B}{鈭�t} \\ 鈭�脳0=-\frac{鈭�B}{鈭�t} \\ \frac{鈭�B}{鈭�t}=0 \end{gathered}$$

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

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

$鈭�E.dl=-\frac{d桅}{dt}$

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

Putting$E=0$in the expression,

$$\begin{gathered} 鈭�0.dl=-\frac{d桅}{dt} \\ -\frac{d桅}{dt}=0 \\ \frac{d桅}{dt}=0 \end{gathered}$$

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

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

$鈭�脳B={渭}_{0}J+{渭}_0{鈭�}_0\frac{鈭�E}{鈭�t}$

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

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

$$\begin{gathered} 鈭�脳0={渭}_{0}J+{渭}_0{鈭�}_0脳0 \\ {渭}_{0}J=0 \\ J=0 \end{gathered}$$

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,

$$\begin{gathered} B=鈭�脳A \\ B=\frac{2{渭}_{0}R蝇未}{3}(\cos 胃\hat{r}-\sin 胃\widehat{胃}) \\ B=\frac{2}{3}{渭}_0未R蝇\hat{z} \\ B=\frac{2}{3}{渭}_0未R蝇 \end{gathered}$$

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

$$\begin{gathered} B=\frac{2}{3}{渭}_0未蝇a\hat{Z} \\ 未蝇a=-\frac{2B_0}{3{渭}_0} \end{gathered}$$

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

$K=未谓$

Here,$未$ is the surface charge density and$谓$ is the velocity of the charge.

Putting the value of charge velocity$谓=蝇脳a\sin 胃\widehat{蠒}$ in the expression,

$$\begin{gathered} K=未蝇a\sin 胃\widehat{蠒} \\ K=-\frac{3B_0}{2{渭}_0}\sin 胃\widehat{蠒} \end{gathered}$$

Hence, the induced surface current density is$k=-\frac{3B_0}{2{渭}_0}\sin 胃\widehat{蠒}$.