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An infinite uniform current carrying wire is kept along z-axis, carrying current I₀ in the direction of the positive z-axis. OABCDEFG represents a circle (where all the points are equally spaced), whose centre at point (4m, 0m) and radius 4 m as shown in the figure. If $\int_{D E F}^{} \vec{B} . \vec{d} l = \frac{μ_{0} I_{0}}{k}$ in S.I. unit, then the value of k is

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answer is 8.

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Detailed Solution

Draw a circle assuming OF as radius and M is point where the circle cuts the x-axis.

$\oint_{M D E F M} \vec{B} . \vec{d} l = \int_{M}^{D} \vec{B} . \vec{d} l + \int_{D E F} \vec{B} . \vec{d} l + \int_{F}^{M} \vec{B} . \vec{d} l = 0$

$\Rightarrow 0 + x - μ_{0} l_{0} \frac{45^{o}}{360^{o}} = 0$

$\Rightarrow x = \frac{μ_{0} l_{0}}{8}$

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