In Fig. , AB is a non-conducting rod. Equal charges of magnitude q are fixed at various points on the rod as shown. The rod is rotated uniformly about an axis passing through O and perpendicular to its length such that linear speed at the end A or B of the rod is `3 ms^(-1)`. Magnetic field at O is

Options

A. `(11mu_(0)q)/(12 pi)`
B. `(3mu_(0)q)/(7 pi)`
C. `(mu_(0)q)/(2 pi)`
D. `(6mu_(0)q)/(13 pi)`

Answer

A

Explanation

Since the rod is a rigid a rigid body, every point has same angular velocity `W=1 rad//s`.
For points at a distance of `1m, v_(1)=1m//s`
For points at a distance of `2m, v_(2)=2m//s`
If a charge rotating in circle constitues a current `I`,
`B=(mu_(0)i)/(2r)`, where `iq/t=(qv)/(2pir)`
Since field due to rotation of all charges are in same direction
`B=Sigma (mu_(0)qv)/(4pir^(2)) =(mu_(0)q)/(4pi) (1+1+1/2+1/2+1/3+1/3)`
`implies B=(11 mu_(0)q)/(12pi)`