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.

Explanation

Here, `I = I_(0) sin omega t`
Magnetic field produced due to solenoid,
`B = mu_(0) n I`
`B = mu_(0) n I_(0) sin omega t`
As no magnetic field exists outside the solenoid, therefore magnetic flux linked with the cylindrical vessel is
`phi = B xx` area of cross-section of soleniod
`= (mu_(0) n I_(0) sin omega t) pi a^(2)`
`= mu_(0) n pi a^(2) I_(0) sin omega t`.
The e.m.f. induced in cylindrical vessel is
`e = (dphi)/(dt) = mu_(0) n pi a^(2) I_(0) omega cos omega t`
Resistance of cylinder vessel is
`R' = (rho L)/(A)`, where `A = 2 pi R xx d`
`= (rho L)/(2 pi R xx d)`
Induced current in outer cylindrical vessel
`I = (e)/(R') = (mu_(0) n pi a^(2) omega cos omega t)/(rho L//2 pi Rd)`
`I = (2 mu_(0) n pi^(2) a^(2) RdI_(0) omega cos omega t)/(rho L)`