Infinite wires, each carrying a current 0.5 A, are kept parallel to y-axis and intersecting x-axis at $x = \pm 1 m, \pm 2 m, \pm 4 m, \pm 8 m$ ,etc. Wires kept at positive values of x axis carry current in the same direction while wires kept at negative values of x axis carry current successively in opposite directions as shown in figure. 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: (assume each wire to be infinitely long)

Difficult

Solution

Given that wires kept at negative x values carry current successively in opposite directions. The fields due to wires kept at $x = \pm 2 m, \pm 8 m$ etc. cancel each other. Due to wires kept at $x = \pm 1 m, \pm 4 m$ , fields are out of the page and net field is given by:
$B = 2 \times \frac{μ_{0} i}{2 π} (1 + \frac{1}{4} + \frac{1}{16} + ....)$
$\Rightarrow B = \frac{4 μ_{0} i}{3 π} = \frac{2 μ_{0}}{3 π}$ (since $i = 0.5 A$ )
This field is negated by field due to wire placed at $y = 2 m$ . This wire must carry current I along positive x-axis such that $\frac{μ_{0} I}{2 π \times 2} = \frac{2 μ_{0}}{3 π}$
Thus, $I = \frac{8}{3} A$ along + x-axis.
= 0002.67

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