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

Options

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

Answer

D

Explanation

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

`B=(mu_(0)I)/(4piR)xx((3pi)/(2))` into the plane of the paper
Since point M lies on the axis of the straight portion AB `:.` Magnetic field at the centre M due to current through the curved portion AB is `B_(2)=0`
Magnetic field at the centre M due to current through the curved portion BC is
`B_(3)=(mu_(0)I)/(4pi2R)xx(pi)/(2)`
into the plane of the paper
`=(mu_(0)I)/(4pi2R)` into the plane of the paper
Since point M lies on the axis of the straight portion CD. `:.` Magnetic field at the centre M due to current through the straight portion CD is `B_(4)=0`
`B=B_(1)+B_(2)+B_(3)+B_(4)`
`=(mu_(0)I)/(R)+0+(mu_(0)I)/(16R)+0=(3mu_(0)I)/(8R)+(mu_(0)I)/(16R)=(7mu_(0)I)/(16R)`
`B=(7mu_(0)I)/(16R)` into the plane of the paper