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

Hint: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, \[\tau = \overrightarrow P \times \overrightarrow E \].