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. `npl^(3)`
B. `pi/3npl^(3)`
C. `pi/4npl^(3)`
D. `pinpl^(3)`

Answer

C

Explanation

`dq=pdx=p_(0)x/Ldx`
`di=(dq)/T=(p_(0)x/Ldx)/(1/n)=(p_(0)n)/Lxdx`
`dm=di.pix^(2)=(p_(0)n)/Lxdx.pix^(2)`
`m_("net")=intdm=(p_(0)npi)/Lint_(0)^(L)x^(3)dx=p_(0)npi(L^(4))/(4L)=(p_(0)npiL^(3))/4=pi/4p_(0)nL^(3)`