The figure shows a circular loop of radius `a` with two long parallel wires `(numbered 1 and 2)` all in the plane of the paper . The distance of each wire from the centre of the loop is `d`. The loop and the wire are carrying the same current `I`. The current in the loop is in the counterclockwise direction if seen from above.
(q) The magnetic fields(B) at `P` due to the currents in the wires are in opposite directions.
(r) There is no magnetic field at `P`.
(s) The wires repel each other.

(4) When `d~~a` but wires are not touching the loop , it is found that the net magnetic field on the axis of the loop . In that case

Options

A. current in wire 1 and wire 2 is in the direction PQ and RS, respectively and `h=a`
B. current in wire 1 and wire 2 is in the direction PQ and SR, respectively and `h=a`
C. current in wire 1 and wire 2 is in the direction PQ and SR, respectively and `h=1*2a`
D. current in wire 1 and wire 2 is in the direction PQ and RS, respectively and `h=1*2a`

Answer

C

Explanation

Let O be a point at a height h above the loop. The magnetic field induction at O due to current loop

`B_1=(mu_0)/(4pi)(2piIa^2)/((a^2+h^2)^(3//2))=(mu_0Ia^2)/(2(a^2+h^2)^(3//2))`
Magnetic field induction at O due to currents in the wire 1 and 2.
`B_2=2xx(mu_0)/(4pi)(2I)/(sqrt(d^2+h^2))xx(d)/(sqrt(d^2+h^2))`
`=(mu_0Id)/(pi(d^2+h^2))=(mu_0Ia)/(pi(a^2+h^2)) ( :' d=a)`
If net magnetic field at O is zero, then
`B_1=B_2` or `(mu_0Ia^2)/(2(a^2+h^2)^(3//2))=(mu_0Ia)/(pi(a^2+h^2))`
or `(a)/(2sqrt(a^2+h^2))=1/pi`
or `(a^2)/(4(a^2+h^2))=(1)/(pi^2)=1/10` `[:' pi^2=10]`
or `10a^2=4a^2+4h^2` or `h^2=3a^2//2`
or `h=1*2a`