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

The Correct Option is 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^{\ell}_{0} 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} $