Q 12-04-201JEE MainJEE Main 2019 (10 Jan, Shift 1)Medium
An insulating thin rod of length $l$ has a linear charge density $\rho(x) = \rho_0\dfrac{x}{l}$ 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:
Answer: (A) $\dfrac\pi4n\rho_0l^3$
The element $dx$ at distance $x$ carries $dq = \rho_0\dfrac xl\,dx$. Rotating $n$ times a second it is a current $n\,dq$ around a circle of area $\pi x^2$:
$$d\mu = n\,\rho_0\frac xl\,dx\cdot\pi x^2$$
$$\mu = \frac{\pi n\rho_0}{l}\int_0^l x^3\,dx = \frac\pi4n\rho_0l^3$$
Solution by Sreeraj P, M.Sc Physics