An insulting thin rod of length `l` has a linear charge density `rho(x)=rho_(0)(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 :

Options

A. `n rho_0 l^3`
B. `pi/3 rho_0 l^3`
C. `pi/2 n rho_0 l^3`
D. `pi n rho_0 l^3`

Answer

C

Explanation

`dq = rho dx = rho_0 (2x)/(L) dx`
`di = (dq)/(T) = (rho_0 (2x)/(L) dx )/( 1/n) = (2 rho_0 n)/(L) xdx`
`dm = di.pi x^2 = (2 rho_0 n)/(L) xdx. Pi x^2`
`m_("net") = int dm (2 rho_0 n pi)/(L) int_(0)^(L) x^3dx = 2 rho_0 n pi (L^4)/(4L) = (2 rho_0 n pi L^3)/(4) = pi/2 rho_0 nL^3`