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 1^(3)`
B. `(pi)/3 n rho 1^(3)`
C. `(pi)/4 n rho 1^(3)`
D. `pi n rho 1^(3)`

Answer

C

Explanation

`dq= rho dx =rho_(0)x/L dx`
`di=(dq)/T=(rho_(0)x/Ldx)/(1/n)=(rho_(0)n)/L x dx, dm=dipix^(2)`
`=(rho_(0)n)/L xdx pi x^(2)`
`m_("net")=int dm=(rho_(0)npi)/L int_(0)^(L) x^(3) dx =rho_(0) n pi (L^(4))/(4L)=(rho_(0)n pi L^(3))/r=(pi)/4 rho_(0)nL^(3)`