An insulating thin rod of length \(l\) has a linear charge density \(\rho \left( x \right) = {\rho _0}\,\frac{x}{l}\) 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

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Answer & Solution

Correct Answer

(C) \(\frac{\pi }{4}n\rho {l^3}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{dM} =\mathrm{di} \mathrm{A} \)

\(=\left(\frac{\mathrm{d} \mathrm{q} \omega}{2 \pi}\right) \pi \mathrm{x}^{2} \)

\(=(\rho \mathrm{d} \mathrm{x}) \frac{\omega}{2 \pi} \pi \mathrm{x}^{2} \)

\(\mathrm{M} =\int_{0}^{\mathrm{L}} \mathrm{dM}\)

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