A thin circular disk of radius $R$ is uniformly charged with density $\sigma>0$ per unit area. The disk rotates about its axis with a uniform angular speed $\omega$. The magnetic moment of die disk is
AIEEE 2011, Diffcult
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$dq =\sigma \times 2 \pi x \times dx =2 \pi \sigma xdx$
$dI =\frac{ dq }{ dt }$ $=\frac{2 \pi \cdot \sigma \cdot x \cdot dx }{ dt }$
$\frac{2 \pi}{ dt }=\omega$
$\therefore dI =\omega \cdot \sigma \cdot x \cdot dx$
$dM = dI \times \pi \times x ^{2}$
$=\omega \sigma x d x \cdot \pi \cdot x^{2}$
$=\omega \sigma x ^{3} dx$
$M=\int_{0}^{R} d M$
$=\int_{0}^{R} \omega \sigma \pi x^{3} d x$
$=\omega \sigma \pi \int_{0}^{ R } x ^{3} \cdot dx$
$=\omega \cdot \sigma \cdot \pi\left(\frac{ x ^{4}}{4}\right)_{0}^{ R }$
$=\omega \cdot \sigma \cdot \pi\left(\frac{ R ^{4}}{4}\right)$
$=\frac{1}{4} \pi R ^{4} \cdot \sigma w$
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