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An insulting thin rod of length l has a ...

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 :

A

`npl^(3)`

B

`pi/3npl^(3)`

C

`pi/4npl^(3)`

D

`pinpl^(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
`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)`
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Knowledge Check

  • The radius of gyration of an uniform rod of length l about an axis passing through one of its ends and perpendicular to its length is.

    A
    `l/sqrt 2`
    B
    `l/3`
    C
    `l/sqrt 3`
    D
    `l/2`

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