Question

A charge $q$ is placed at (1,2,1) and another charge $-q$ is placed at (0,1,0) such that they form an electric dipole . There exists a uniform electric field $\overrightarrow E = 2\hat i$. Calculate the torque experienced by the dipole.

Answer 1

Torque is the measurement of the force that allows an object to rotate around an axis. Torque is a vector quantity whose direction is determined by the force acting on the axis.

Complete step by step answer:
The torque vector's magnitude is determined as follows:
$T = Fr\sin \theta$
Where,$r$ is the moment arm's length and $\theta$ the angle formed by the moment arm and the force vector.

An electric dipole is a pair of electric charges of equal magnitude but opposite charges divided by a distance d. The product of the magnitude of these charges and the difference between them is the electric dipole moment for this. The electric dipole moment is a vector that has a direction from negative to positive charge,
$\overrightarrow p = q\overrightarrow d$
Given that,
Position of charge at $q$ , ${r_1} = (1\hat i + 2\hat j + 1\hat k)$
Position of charge at $-q$ , ${r_2} = - (0\hat i + 1\hat j + 0\hat k)$
We know that dipole moment of the dipole is,
$\overrightarrow p = q\overrightarrow d$
Here,
$\overrightarrow d = {r_1} + {r_2}$
$\Rightarrow \overrightarrow d = (1\hat i + 2\hat j + 1\hat k) - (0\hat i + 1\hat j + 0\hat k)$
$\Rightarrow \overrightarrow d = (1 - 0)\hat i + (2 - 1)\hat j + (1 - 0)\hat k$
$\Rightarrow \overrightarrow d = 1\hat i + 1\hat j + 1\hat k$
$\Rightarrow \overrightarrow d = \hat i + \hat j + \hat k$
$\Rightarrow \overrightarrow p = q\overrightarrow d$
$\Rightarrow \overrightarrow p = q(\hat i + \hat j + \hat k)$
We know that, torque due to electric field on dipole is,
$\tau = \overrightarrow P \times \overrightarrow E$
$\Rightarrow \tau = q(\hat i + \hat j + \hat k) \times (2\hat i)$
$\Rightarrow \tau = (2\hat j + 2\hat k)q$
Hence, the magnitude of torque is,
$\left| \tau \right| = \sqrt {({2^2} + {2^2})q}$
$\Rightarrow \left| \tau \right| = \sqrt {8q}$
$\therefore \left| \tau \right| = 2\sqrt {2q}$

Hence the torque is $2\sqrt {2q}$.

Note: Torque is the measure of force that causes an object to spin around an axis. Since the force magnitudes are equal and divided by a distance d, the torque on the dipole is given by: Torque $\left( \tau \right)$ = Force $\times$ Distance between the forces.
Remember the equation, $\tau = \overrightarrow P \times \overrightarrow E$.