A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R. Thickness d(dltltR) and length L. A variable current `i=i_(0) sin omega t` flows through the coil. If the resistivity of the material of cylindrical shell is `(rho)`, find the induced current in the shell.

Answer

A, B, C, D

Explanation

The megnetic field in the solenoid is given by
`B=(mu_0)ni`

implies B=(mu_0)ni_(0) sin omega t [:' I=(I_0)=sin omega t (given)]`
The magnetic flux linked with the soleniod
`(phi)=vec(B).vec(A) =B A cos 90^(@)=(mu_(0)ni_(0)sin omega t)(pi a^2)`
`:. The rate of change of magnetic flux through the solenoid `
(d phi)/(dt_=pi (mu_0)n(a^2)i_(0)(omega)t`
The same rate of change of flux is linked with the cylindrical shell. By the priciple of electromagnetic inductio, the induced emf produced in the cylindrical shell is

`e=-(d phi)/(dt)=-(pi)(mu_0)n(a^2)(i_0)(omega) cos (omega)t` ...(i) The resistance offered by the cylinder shell to the flow of induced current I will be
`R=(rho)(l)/(A)`
here, `l=2 pi R and A=Lxxd`
`:. R=(rho)(2 pi R)/(Ld)` ...(ii)
The induced current I will be
`I=(|e|)/(R ) =([pi mu_(0)n(a^2)I_(0)omega cos omega t]xxLd)/(rho xx 2pi R)`
`implies I=(mu_(0)na^(2)Ld(i_0)omega cos omega t)/(2 rho R)`.