Question

A long current carrying wire, carrying current such that it is flowing out from the plane of paper, is placed at O. A steady state current is flowing in the loop ABCD. Then,

• A. the net force is zero
• B. the net torque is zero
• C. as seen from O, the loop will rotate in clockwise direction along axis OO'
• D. as seen from O, the loop will rotate in anticlockwise direction along axis OO'

Answer 1

Answer: (A), (C)

the net force is zero
as seen from O, the loop will rotate in clockwise direction along axis OO'

Magnetic field produced by the long current carrying conductor O on the arc AB is tangential to the arc AB.
$\therefore \overrightarrow{d{l}_{AB}}\parallel \overrightarrow{B}$
$\therefore \overrightarrow{F}=i_2\left(\overrightarrow{d{l}_{AB}\times \overrightarrow{B}}\right)=0$
Similarly, Magnetic field produced by the long current carrying conductor O on the arc CD is tangential to the arc CD.
$\therefore \overrightarrow{d{l}_{CD}}:\parallel (-\overrightarrow{B})$
$\therefore \overrightarrow{F}=i_2\left(\overrightarrow{d{l}_{CD}\times \overrightarrow{B}}\right)=0$
Force on segments BC and DA will be zero since force at any point on BC will be equal and opposite to that for a corresponding point on arc DA.
$\therefore$ net force on the loop ABCD is equal to zero.
${\overrightarrow{F}}_{ABCD}=0$
Torque on arc AB is zero since force on AB is zero.
Torque on arc CD is zero since force on CD is zero.
Force on segment BC and DA are equal and opposite but the perpendicular distance between the forces is not zero.
Hence, there will be a net torque on the loop.
${\overrightarrow{F}}_{BC}\parallel \hat{k}$
${\overrightarrow{\tau }}_{BC}\perp \overrightarrow{BC}$ and is towards OO'.
${\overrightarrow{F}}_{DA}\parallel (-\hat{k})$
${\overrightarrow{\tau }}_{DA}\perp \overrightarrow{DA}$ and is away from OO'.
$\therefore$ as seen from O the net torque is clock-wise.