Question

An insulating thin rod of length $\ell$ has a x linear charge density $p(x)={\rho }_0\frac{x}{\ell }$ on it . The rod is rotated about an axis passing through the origin $(x=0)$ and perpendicular to the rod. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is :

• A. $\frac{\pi }{4}n\rho {\ell }^3$
• B. $n\rho {\ell }^3$
• C. $\pi n\rho {\ell }^3$
• D. $\frac{\pi }{3}n\rho {\ell }^3$

Answer 1

Answer: (A)

$\because M=NIA$
$dq=\lambda dx\&A=\pi x^2$
$\int dm=\int \left(x\right)\frac{{\rho }_{0}x}{\ell }dx.\pi x^2$
$M=\frac{n{\rho }_0\pi }{\ell }.{\int }_0^{\ell }{x}^3.dx=\frac{n{\rho }_0\pi }{\ell }.\left[\frac{L^4}{4}\right]$
$M=\frac{n{\rho }_0\pi {\ell }^3}{4}$ or $\frac{\pi }{4}n\rho {\ell }^3$