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A circular loop of radius $a = 8 c m$ and carrying a current $I_{2} = 1 A$ is placed near an infinitely long straight wire carrying a current $I_{1} = 1 A$ as shown in the figure. The straight wire and circular loop are coplanar. The distance of the centre of the loop from the wire is $b = 10 c m$ . The force of interaction between the straight wire and the loop is $\frac{nπ}{3} \times 10^{- 7} N$ . Find the value of n.

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Detailed Solution

Consider an element on the circular loop at an angle $θ$ with the horizontal. From the symmetry of the figure, only the horizontal component will contribute in net fore. Therefore

$d F = 2 (d F_{1}) \cos θ = 2 (\frac{μ_{0} I_{1}}{2 π (b - a \cos θ)}) (I_{2} a d θ) \cos θ$

$F = \frac{μ_{0} I_{1} I_{2} a}{π} \int_{0}^{π} \frac{\cos θ d θ}{(b - a \cos θ)}$

$\Rightarrow F = μ_{0} I_{1} I_{2} (\frac{b}{\sqrt{b^{2} - a^{2}}} - 1) = 1 \times 4_{π} \times 10^{- 7} (\frac{10}{6} - 1)$

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