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A Considerably large number of very long wires are placed parallel to the Y-axis at location with X- values given by 1m,2m,4m,8m and so on. Each wires having current of 1A in opposite direction .The magnetic field at Origin (in tesla) is

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a

$- 2.67 \times 10^{- 7} \hat{k}$

b

$- 1.33 \times 10^{- 7} \hat{k}$

c

$1.33 \times 10^{- 7} \hat{k}$

d

$2.67 \times 10^{- 7} \hat{k}$

answer is C.

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Detailed Solution

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$\bar{B} = \frac{μ_{0}}{2 π} [(\frac{1}{1} + \frac{1}{4} + \frac{1}{16} + .......) \bar{K} - (\frac{1}{2} + \frac{1}{8} + ........) \bar{K}] = \frac{μ_{0}}{2 π} [\frac{1}{(1 - \frac{1}{4})} \bar{K} - \frac{1}{2} [\frac{1}{(1 - \frac{1}{4})}] \bar{K}]$

$\bar{B} = \frac{2}{3} \frac{μ_{0}}{2 π} \bar{K} = \frac{2}{3} \times 2 \times 10^{- 7} \bar{K} = \frac{4 \times 10^{- 7}}{3} \bar{K} = 1.33 \times 10^{- 7} \bar{K}$

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A Considerably large number of very long wires are placed parallel to the Y-axis at location with X- values given by 1m,2m,4m,8m and so on. Each wires having current of 1A in opposite direction .The magnetic field at Origin (in tesla) is