Question

A long solenoid has $n$ turns per unit length and radius $a$. A current $I=I_o\sin \omega t$ flows through it. A cylindrical vessel of radius $R$, length $L$, thickness $d(d<<R)$ and resistivity $p$ is kept coaxially as shown in the figure. Find the induced current in the outer cylindrical vessel.

• A. $I=\frac{\left({\mu }_{o}n{i}_o\omega cos\omega t\right)\pi a^2\left(Ld\right)}{\rho 2\pi R}$
• B. $I=\frac{\left({\mu }_{o}n{i}_o\omega cos\omega t\right)\pi a^2}{\rho 2\pi R}$
• C. $I=\frac{\left({\mu }_{o}n{i}_o\omega cos\omega t\right)}{\rho 2\pi R}$
• D. None of these

Answer 1

The correct option is A $I=\frac{\left({\mu }_{o}n{i}_o\omega cos\omega t\right)\pi a^2\left(Ld\right)}{\rho 2\pi R}$

The current flowing through the solenoid is $I=I_{o}sinwt$

$\therefore$ Magnetic field due to solenoid $B={\mu }_{o}nI={\mu }_{o}n{I}_{o}sinwt$

Flux through the solenoid $\varphi =B\times \pi a^2={\mu }_{o}n{I}_o\left(\pi a^2\right)sinwt$

$\therefore$ Emf induced $|E|=\frac{d\varphi }{dt}={\mu }_{o}n{I}_{o}w\left(\pi a^2\right)coswt$ .............(1)

As $B$ increases with time, thus induced current $\left(i\right)$ flows in clockwise direction to prevent the increase in flux (Lenz Law).

Given : Resistivity of the cylindrical vessel is $\rho$

Cross-sectional area through which the induced current flows $A=Ld$

Thus resistance of the cylindrical vessel $r=\rho \frac{2\pi R}{Ld}$

$\therefore$ Induced current $i=\frac{E}{r}=\frac{{\mu }_{o}n{I}_{o}w\left(\pi a^2\right)Ld}{\rho \left(2\pi R\right)}coswt$