Question

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

• A. $\frac{2{\mu }_0{I}^2{a}^3}{\pi b^2}$
• B. $\frac{{\mu }_0{I}^2{a}^3}{2\pi b^2}$
• C. $\left[\frac{2{\mathrm{\mu }}_0{\mathrm{I}}^2{\mathrm{a}}^2}{\mathrm{\pi b}}\right]$
• D. $\frac{{\mu }_0{I}^2{a}^3}{2\pi b}$

Answer 1

The correct option is C

$\left[\frac{2{\mathrm{\mu }}_0{\mathrm{I}}^2{\mathrm{a}}^2}{\mathrm{\pi b}}\right]$

Step 1: Given data

Side of square$=2a$

Current $=I$

Step 2: Find the magnitude of the torque

The magnetic moment $M$ of the square is,

$M=$Current $I\times$Area of Square $A$

Area of Square $A=($side of square ${)}^2$,

$\Rightarrow$ $A={\left(2a\right)}^2$

$\Rightarrow$ $\begin{array}{rcl}M& =& I\times {\left(2a\right)}^2\\ & =& 4a^{2}I\end{array}$

Step 3: Calculation

Magnetic Flux $B$, and Torque $\tau$ is given by,

$$\begin{gathered} B=\frac{{\mu }_{0}I}{2\pi b} \\ \tau =MB\sin \theta \end{gathered}$$

Where ${\mu }_0$ is the permeability of free space, $b$ is the distance.

$\theta$ is the angle between $B$ and $M$ $[\theta =90°]$

Substituting the value of the magnetic moment and magnetic field in the torque formula,

$\tau =4\left(a^{2}I\right)\left[\frac{{\mu }_{0}I}{2\pi b}\right]$

$\Rightarrow$$\tau =\left[\frac{4{\mu }_0{I}^2{a}^2}{2\pi b}\right]$

$\Rightarrow$$\tau =\left[\frac{2{\mu }_0{I}^2{a}^2}{\pi b}\right]$

Thus, the magnitude of the torque on the loop about the $z-axis$ is $\tau =\left[\frac{2{\mathrm{\mu }}_0{\mathrm{I}}^2{\mathrm{a}}^2}{\mathrm{\pi b}}\right]$.

Hence, option (C) is correct.