Question 217 of 276

A square loop of side $2a,$ and carrying current $I$ is kept in $XZ$ plane with its centre at origin. A long wire carrying the same current $I$ is placed parallel to the $z$-axis and passing through the point $(0,b,0),(b\gt \gt a)$. The magnitude of the torque on the loop about $z$-axis is given by.

A$\dfrac{{\mu }_{0}{I}^{2}{a}^{2}}{2\pi b}$B$\dfrac{{\mu }_{0}{I}^{2}{a}^{3}}{2\pi {b}^{2}}$C$\dfrac{2{\mu }_{0}Ia22}{\pi b}$D$\dfrac{2{\mu }_{0}Ia32}{\pi {b}^{2}}$

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Answer: (C) $\dfrac{2{\mu }_{0}Ia22}{\pi b}$

Net force acting on wire $AB$ & $CD$ are zero

${F}_{1}=ILB=I\times \left(2a\right)\times \left(\dfrac{{\mu }_{0}I}{2\pi \sqrt{{a}^{2}+{b}^{2}}}\right)=\dfrac{{\mu }_{0}{I}^{2}a}{\pi \sqrt{{a}^{2}+{b}^{2}}}$
$\tau ={F}_{1}cos\theta \times 2a=\dfrac{{\mu }_{0}{I}^{2}a}{\pi \sqrt{{a}^{2}+{b}^{2}}}\times \dfrac{b}{\sqrt{{a}^{2}+{b}^{2}}}\times 2a$
$\tau =\dfrac{2{\mu }_{0}{I}^{2}{a}^{2}b}{\pi \left({a}^{2}+{b}^{2}\right)}\approx \dfrac{2{\mu }_{0}{I}^{2}{a}^{2}}{\pi b},\left(a≪b\right)$