A current `I` is flowing thorugh the loop. The direction of the current and the shpae of the loop are as the shown in the figure. The magnetic fild at the centre of the loop is `(mu_(0) I)/(R)` times
`(MA = R, MB = 2R, /_DMA = 90^(@))`

Options

A. `5/16`, out of the plane of the paper.
B. `5/16`, into the plane of the paper.
C. `7/16`, out of the plane of the paper.
D. `7/16`, out the plane of the paper.

Answer

D

Explanation

Magnetic field at the centre M due to current through the curved portion DA is

`B_(1)=(mu_(0)I)/(2piR)xx((3pi)/(2))`
into the plane of the paper
`=(3mu_(0)I)/(8R)` into the plane of the paper
Magnetic field at the centre M due to current through the straight protion AB is `B_(2)=0`
(`because` point M lies on the axis of the straight portion AB)
Magnetic field at the centre M due to current through the curved protion BC is
`B_(3)=(mu_(0)l)/(4pi(2r))xx(pi)/(2)` into the plane of the paper
`=(mu_(0)I)/(4pi(2R))xx(pi)/(2)` into the plane of the paper
`=(mu_(0)I)/(16R)` into the plane of the paper
Magnetic field at tge cebtre M due to current through the straight portion CD is `B_(4)=0`
(`because` point M lies on the axis of the straight portion CD)
The resultant magnetic field at M is
`B=B_(1)+B_(2)+B_(3)+B_(4)`
`=(3mu_(0)I)/(8R)+0+(mu_(0)I)/(16R)+0=(3mu_(0)I)/(8R)+(mu_(0)I)/(16R)`
`=(7mu_(0)I)/(16R)` into the plane of the paper