Infinite wires, each carrying a current 0.5 A, are kept parallel to y-axis and intersecting x-axis at `x=+-1` m, `+02 m`, `+-4 m`, `+-8m` etc. Wires kept at positive values of x carry current in the same direction while wires kept at negative values of x carry current successively in opposite directions as shown in Fig.

A wire is kept parallel to x-axis and intersecting y-axis at y=2 m. It carries a current i. Assuming this wire to be insulated from others then the current i in it such that magnetic induction at O is zero is
(assuming each wire to be infinitely long)

Options

A. `8/3 A` along positive x-axis
B. `4/3 A` along negative x-axis
C. `5/3 A` along negative x-axis
D. `7/3 A` along positive x-axis

Answer

A

Explanation

Given that wires kept at negative x value carry current successively in opposite directions. The field due to wire kept `x= +- 2m, +-8m` etc, cancel each other. Due to wires kept at `x= +-1m, +-4m,` Field are in upwards direction and net field is given by:
`B=2xx(mu_(90)i)/(2 pi) (1+1/4+1/13 +....)`
`implies B=(4 mu_(0)i)/(3 pi)=(2 mu_(0)i)/(3 pi) =(2mu_(0)i)/(3 pi) ("since" i=0.5A)`
This field is negative by field due to wire placed at y=2m. This wire must carry current I along positive x-axis such that
`(mu_(0)i)/(2 pi xx 2)=(2 mu_(0)i)/(3 pi)`
Thus, `I=8/3 A` along +x-axis.