Question

$A$ current $I$ flows in a rectangularly shaped wire whose center lies at $(x_{0}, 0,0)$ and whose vertices are located at the points $A(x_{0} + d, - a, -b)$, $B(x_{0} - d, a, -b)$, $C(x_{0} - d, a, b)$, and $D(x_{0} + d, -a, b)$ respectively. Assume that $a,b, d <\,< x_{0}$ Find the magnitude of magnetic dipole moment vector of the rectangular wire frame. (Given: $b = 10 \,m$, $d = 4 \,m$, $a = 3 \,m$, $I= 0.01 \,A)$

• A. $2\,J\,T^{-1}$
• B. $4\,J\,T^{-1}$
• C. $3\,J\,T^{-1}$
• D. $9\,J\,T^{-1}$

Answer 1

Answer: (A)

Magnetic moment of a current carrying loop,
$\vec{\mu}=I\,\vec{S} $
Area of the loop, $\vec{S}=\overrightarrow{AB}\times\overrightarrow{BC}$
Here, $\overrightarrow{AB}=-2d\, \hat{i}+2a\,\hat{j}, \overrightarrow{BC}=2b\,\hat{k}$
$\therefore \vec{S}=\left(-2d\,\hat{i}+2a\,\hat{j}\right)\times\left(2b\,\hat{k}\right)$
$=4bd \hat{j}+4ab \hat{i} $
$\therefore \left|\vec{\mu}\right|=I\left|\vec{S}\right|=4Ib \sqrt{a^{2}+d^{2}}$
$=4\times0.01\times10\times\sqrt{3^{2}+4^{2}}$
$=0.4\times5=2\,J\,T^{-1}$