Two very long straight parallel wires, parallel to y- axis,carryc… - Vidyadip

Question

Two very long straight parallel wires, parallel to $y-$ axis,carrycurrents $4I$ and $I,$ along $+y$ directionand$-y$ direction, respectively. The wires are passes through the $x-$axis at the points $(d, 0, 0)$ and $(- d, 0, 0)$ respectively.The graph of magnetic field $z-$component as one moves along the $x-$axis from $x=- d$ to $x= +d,$ is best given by

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Answer

$B=\left[\frac{\mu_{0}}{4 \pi} \frac{2 I}{x}+\frac{\mu_{0}}{4 \pi} \frac{2(4 I)}{4 \pi(2 d-x)}\right] \odot$

$=\frac{\mu_{0}}{4 \pi} 2 I\left[\frac{1}{x}+\frac{4}{(2 d-x)}\right]$

For $B$ to be minimum $\frac{d B}{d x}=0 \Rightarrow x=\frac{2 d}{3}$

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