Infinite wires, each carrying a current 0.5 A, are kept parallel to y-axis and intersecting x-axis at x = ±1 m, ±2 m, ±4 m, ±8 m, 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 figure.
A wire is kept parallel to x-axis and intersecting y-axis at 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:
(Assume each wire to be infinitely long)

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8/3 A along positive x-axis

4/3 A along negative x-axis

5/3 A along negative x-axis

7/3 A along positive x-axis

answer is A.

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Detailed Solution

$G i v e n t h a t w i r e s k e p t a t n e g a t i v e x v a l u e s c a r r y c u r r e n t s u c c e s s i v e l y i n o p p o s i t e d i r e c t i o n s . T h e f i e l d s d u e t o w i r e s k e p t a t x = \pm 2 m, \pm 8 m e t c . c a n c e l e a c h o t h e r . D u e t o w i r e s k e p t a t x = \pm 1 m, \pm 4 m, f i e l d s a r e i n u p w a r d d i r e c t i o n a n d n e t f i e l d i s g i v e n b y : B = 2 \times \frac{μ_{o} i}{2 π} (1 + \frac{1}{4} + \frac{1}{16} + \dots) \Rightarrow B = \frac{4 μ_{o} l}{3 π} = \frac{2 μ_{o} l}{3 π} (Since i = 0.5 A) T h i s f i e l d i s n e g a t e d b y f i e l d d u e t o w i r e p l a c e d a t y - 2 m . T h i s w i r e m u s t c a r r y c u r r e n t l a l o n g p o s i t i v e x - a x i s s u c h t h a t \frac{μ_{o} I}{2 π \times 2} = \frac{2 μ_{o} i}{3 π} Thus, I = \frac{8}{3} A along + x -axis.$