Q. 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 :

Solution:

$\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} $

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