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Infinitely long wires parallel to the $y$ - axis are present in the $x-y$ plane at $x = 1, x = 2, x = 4$, and so on. The currents in all the wires with odd serial no is $1A $ in the $+y$ direction while it is $1A$ in the $–y$ direction for all the wires with even serial no. The magnetic field at the origin would be

$\begin {array} {1 1} (a)\;2.67 \times 10^{-7} \hat k & \quad (b)\;1.33 \times 10^{-7} \hat k \\ (c)\;-2.67 \times 10^{-7} \hat k & \quad (d)\;-1.33 \times 10^{-7} \hat k \end {array}$

 

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As may be clear from the situation, the magnetic field at the origin can be represented as a difference of two geometric series
$ \overrightarrow B = \large\frac{\mu_0}{2 \pi } $$\hat k \bigg[ \bigg( \large\frac{1}{1} + \large\frac{1}{4}+ \large\frac{1}{16} +....\bigg) - \bigg( \large\frac{1}{2} + \large\frac{1}{8}+....\bigg) \bigg]$$ \large\frac{2}{3}$$ \large\frac{\mu_0}{2 \pi}$
$ = 1.33 \times 10^{-7} \hat k$
Ans : (b)
answered Feb 25, 2014 by thanvigandhi_1
 

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