An insulating rod of length $l$ carries a charge $q$ distributed uniformly on it. The rod is pivoted at an end and is rotated at a frequency $f$ about a fixed perpendicular $t$ axis. The magnetic moment of the system is
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$\lambda=\frac{Q}{\ell}$
$\mathrm{dQ}=\ell \mathrm{d} \mathrm{x}$
$\mathrm{I}=(\mathrm{dQ}) \mathrm{f}$
$\mathrm{dm}=(\mathrm{I})(\mathrm{A})$
$\int \mathrm{d} \mathrm{m}=(\mathrm{d} \mathrm{Q} \mathrm{f})\left(\pi \mathrm{x}^{2}\right)$
$\int \mathrm{d} \mathrm{m}=\int_{0}^{\mathrm{Q}} \lambda \mathrm{f} \pi \mathrm{x}^{2} \mathrm{d} \mathrm{x}$
$=\frac{Q}{\ell} \times f \times \pi \times \frac{\ell^{3}}{3}$
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