Question 216 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 $z$-axis and passing through point $\left(0,b,0\right),\left(b\gt \gt a\right)$. The magnitude of torque on the loop about $z$-axis will be :

A$\dfrac{2{\mu }_{0}{I}^{2}{a}^{2}}{\pi b}$B$\dfrac{2{\mu }_{0}{I}^{2}{a}^{2}b}{\pi \left({a}^{2}+{b}^{2}\right)}$C$\dfrac{{\mu }_{0}{I}^{2}{a}^{2}b}{2\pi \left({a}^{2}+{b}^{2}\right)}$D$\dfrac{{\mu }_{0}{I}^{2}{a}^{2}}{2\pi b}$

Show answer and solution▾

Answer: (B) $\dfrac{2{\mu }_{0}{I}^{2}{a}^{2}b}{\pi \left({a}^{2}+{b}^{2}\right)}$

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

$\tau ={F}_{1}cos\theta \times 2a=\dfrac{{\mu }_{0}}{2\pi }\dfrac{{i}^{2}}{\sqrt{{a}^{2}+{b}^{2}}}(2a)\times \dfrac{b}{\sqrt{{a}^{2}+{b}^{2}}}\times 2a$
$\tau =\dfrac{2{\mu }_{0}.{i}^{2}{a}^{2}b}{\pi \left({a}^{2}+{b}^{2}\right)}$