Write the expression for the magnetic moment vector (m) due to a planar square loop of side ‘l’ carrying a steady current

Question

Write the expression for the magnetic moment \((\overrightarrow{m})\) due to a planar square loop of side ‘l’ carrying a steady current I in a vector form.

In the given figure this loop is placed in a horizontal plane near a long straight conductor carrying a steady current I₁ at a distance l as shown. Give reasons to explain that the loop will experience a net force but no torque. Write the expression for this force acting on the loop.

Answer 1

Magnetic moment due to a planar square loop of side l carrying current I is

\(\overrightarrow{m}=I\,\overrightarrow{A}\)

For square loop A = I²

Where ̂ is unit vector normal to loop. 

Magnetic field due to current carrying wire at the location of loop is directed downward perpendicular to plane of loop. Since the magnetic moment is parallel to area vector hence torque is zero. 

Force on QR and SP are equal and opposite, so net force on these sides is zero.

That is loop experiences a repulsive force but no torque.