Study the current distribution given in the figure and calculate the magnetic field at the centre O .

`OP = r, OQ = 2 r`

Options

A. `( mu _(0) I )/( 24 r )`
B. `(3 mu _(0) I )/( 24 r )`
C. `(13 mu _(0) I )/( 24 r )`
D. `( 9 mu _(0) I )/( 24 r )`

Answer

D

Explanation

The magnetic field at a point due to a circle arc is given by,
`B = ( mu _(0) i theta)/( 4pi r)`

`OP = r, OQ = 2 r`
The magnetic field at the centre will be due to the three circular acros of radius r and radius 2 r.
The magnetic field due to the arcs of radius r will be equal to
`B _(1) = 3 (( mu _(0) i )/( 4 pi r ) xx (pi )/(3))`
The magnetic field due to the arcs of radius 2 r will be equal to,
`B _(2) = 3 (( mu _(0) i )/( 4 pi 2 r) xx (pi)/(3))`
`vecB _(n et) = vecB _(1) + vecB _(2)`
we get the direction of magnetic by Right-Hand Screw Rule, The direction of `B _(1) and B _(2)` are outside the plane
`B _(n et) = ({:(mu_(0I) , mu _(0I) ), (,+), (4r, 8r):})= {:(3 mu _(0I)),(8r):}`