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Q.

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 $(0, b, 0), (b > > a)$ . The magnitude of torque on the loop about z-axis will be :

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a

$\frac{2 μ_{0} I^{2} a^{2} b}{π (a^{2} + b^{2})}$

b

$\frac{2 μ_{0} I^{2} a^{2}}{π b}$

c

$\frac{μ_{0} I^{2} a^{2}}{2 π b}$

d

$\frac{μ_{0} I^{2} a^{2} b}{2 π (a^{2} + b^{2})}$

answer is C.

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Detailed Solution

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Magnetic moment of loop is $4 a^{2} I \hat{j}$ and magnetic filed induction at origin due to long wire $\vec{B} = \frac{μ_{o} I}{2 π b} \hat{i}$ .

So as b>>a, $|\vec{τ}| = |\vec{M} \times \vec{B}| = \frac{2 μ_{o} I^{2} a^{2}}{π b}$

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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 (0,b,0) , b>>a. The magnitude of torque on the loop about z-axis will be :