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 $e=-\frac{d\varphi }{dt}=-\frac{d}{dt}({\mu }_{0}nI\pi a^2)$
or $|e|=({\mu }_{0}n\pi a^2)(I_0\omega cos\omega t)$
Resistance of the cylindrical vessel $R=\frac{\rho l}{s}=\frac{\rho (2\pi R)}{Ld}$
$\therefore$ Induced current $i=\frac{|e|}{R}=\frac{{\mu }_{0}Ldn{a}^2{I}_0\omega cos\omega t}{2\rho R}$