Question

A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R, thickness d (d << R) and length L. A variable current $i = i_0$ sin $\omega t$ flows through the solenoid. If the resistivity of the material of cylindrical shell is $\rho$, find the induced current in the shell.

Answer 1

Out side the solenoid net magnetic field is zero. It can be assumed only inside the solenoid and equal to $\mu_0 nI$.
Induced $e = - \frac{d \phi}{dt} = - \frac{d}{dt} (\mu_0 nI \pi a^2 )$
or $|e| = (\mu_0n\pi a^2)(I_0 \omega \, cos \, \omega t)$
Resistance of the cylindrical vessel $R = \frac{\rho l}{s} = \frac{\rho (2 \pi R)}{Ld} $
$\therefore $ Induced current $i = \frac{|e|}{R} = \frac{\mu_0 L dna^2I_0 \omega \, cos \, \omega t}{2 \rho R}$