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` carry current successively in opposite direction. The fields due to wires kept at `X=+-2m,+-8m` etc. cance each other. Due to wires kept at `X=+1m,+-4m`, fields are in upward direction and net field is given by:
`B=2xx(mu_(0)i)/(2pi)(1+1/4+1/16+......)`
` implies B=(4mu_(0)i)/(3pi) =(2mu_(0))/(3pi)` Since `=0.5 A`
This field is negated by field due to wire placed at `y=2m`. This wire must carry current `I` along positve `x`-axis such that
`(mu_(0)I)/(2pixx2)=(2mu_(0))/(3pi)` hence, `I=8/3 A` along `+x`-axis.