Question

A rectangular loop carrying a current $i$ is placed in a uniform magnetic field $B$. The area enclosed by the loop is $A$. If there are $n$ turns in the loop, the torque acting on the loop is given by
A. $ni\left( {\bar A \times \bar B} \right)$
B. $ni\left( {\bar A \cdot \bar B} \right)$
C. $\dfrac{{i\left( {\bar A \times \bar B} \right)}}{n}$
D. $\dfrac{{i\left( {\bar A \cdot \bar B} \right)}}{n}$

Answer 1

Use the relation between the torque acting on a rectangular current carrying loop, magnetic dipole moment of the current carrying loop and the uniform magnetic field in which the rectangular loop is placed. Also use the relation between the dipole moment of the current carrying loop, number of turns in the loop and the current.

Formulae used:
The torque acting on a rectangular current carrying loop is given by
$\tau = m \times B$
Here, $\tau$ is the torque on the rectangular loop, $m$ is the magnetic dipole moment of the loop and $B$ is the uniform magnetic field.
The magnetic dipole moment $m$ of the loop is given by
$m = niA$
Here, $n$ is the number of turns in the loop, $i$ is the current in the loop and $A$ is the area enclosed by the loop.

Complete step by step answer:
Rewrite the equation for the torque $\tau$ acting on a current carrying rectangular loop in a uniform magnetic field.
$\tau = m \times B$
Substitute $niA$ for $m$ in the above equation.
$\tau = \left( {niA} \right) \times B$
$\Rightarrow \tau = ni\left( {\bar A \times \bar B} \right)$
Hence, the correct option is A..

Note: If there are two vectors $\bar A$ and $\bar B$ and $c$ is a scalar in the cross product, then the scalar $c$ can be drawn out of the cross product as the cross product can be done of two vectors only.
$\Rightarrow c\bar A \times \bar B = c\left( {\bar A \times \bar B} \right)$