Question

An electric dipole of length $2\,cm$ , when placed with its axis making an angle of ${60^ \circ }$ with a uniform electric field, experiences a torque of $8\sqrt 3 \,Nm$ . Calculate the potential energy of the dipole, if it has a charge of $\pm 4\,nC$ .

Answer 1

The separation between two opposite charges is known as dipole. A pair of equal and opposite charges separated by a distance is known as electric dipole. Dipole moment is defined as the total polarity of the system. Potential energy of a dipole moment is the dot product of the dipole moment and the electric field.

Formula used:
The formula to calculate the dipole moment is given as:
$P = q \times (2a)$
Here, $P$ is the dipole moment and the distance between the charges is $2a$ .
$q$ is the charge.
The torque on the dipole moment of uniform electric field is given by,
$\tau = P.E\sin \theta$
Here, $E$ is the uniform electric field and $\theta$ is the angle between the dipole and the electric field.
The formula for potential energy $U$ of the dipole is:
$U = - P.E\cos \theta$

Complete step by step answer:
The given values are:
$a = 2\,cm = 2 \times {10^{ - 2}}\,m$
$\theta = {60^ \circ }$
Charge $q = 4 \times {10^{ - 9}}\,C$
The dipole moment will be
$P = q \times (2a)$
$\Rightarrow P = 4 \times {10^{ - 9}} \times 2 \times 2 \times {10^{ - 2}}$
$\Rightarrow P = 16 \times {10^{ - 11}}\,Cm$
The torque is given to be $8\sqrt 3 \,Nm$
$\tau = P.E\sin \theta$
From here we can find the electric field as:
$E = \dfrac{\tau }{{P\sin \theta }}$
We need to find the potential energy; potential energy is given as:
$U = - P.E\cos \theta$
Substituting the value of $E$ we get
$U = - P\left( {\dfrac{\tau }{{P\sin \theta }}} \right)\cos \theta$
$\Rightarrow U = - \left( {\dfrac{\tau }{{\tan \theta }}} \right)$
Substituting the values of torque and angle, we get
$U = - \left( {\dfrac{{8\sqrt 3 }}{{\tan {{60}^ \circ }}}} \right)$
$\therefore U = - 8\,Nm$

The potential energy is $- 8\,Nm$.

Note: Dipole polarity is the total dipole moment in the system while dipole is the separation between two opposite charges. The energy acquired by the dipole under the influence of an electric field is known as potential energy. The two charges in the dipole are equal in magnitude but opposite in charge.Dipole moment is used in the application of dipole antennas which is an important form of RF antenna, widely used for radio transmission.