Question

An electric dipole of moment P is placed in the position of stable equilibrium in the uniform electric field of intensity E. It is rotated through an angle θ from the initial position. Determine the position of the potential energy of the electric dipole.
A. \[PE\cos \theta \]
B. \[PEsin\theta \]
C. \[PE\left( {1 - \cos \theta } \right)\]
D. \[ - PE\cos \theta \]

Answer 1

Hint: Before we start addressing the problem, we need to know about the electric dipole moment. It measures the separation of positive and negative electrical charges within a system, in other words, it is a measure of the system's overall polarity and its SI unit is a coulomb-meter (Cm).

Formula Used:
To find the potential energy of the dipole, we have
\[U = - PE\cos \theta \]
Where, P is electric dipole of dipole moment, E is the electric field and \[\theta \]is angle of rotation.

Complete step by step solution:
Suppose if an electric dipole of moment P is placed in the position of stable equilibrium in uniform electric field E. It is rotated through an angle \[\theta \] from the initial position. We need to find the position of the potential energy of the electric dipole. We know the formula to find the potential energy of dipole,
\[U = - PE\cos \theta \]
\[\Rightarrow \Delta U = - PE\left( {\cos {\theta _2} - \cos {\theta _1}} \right)\]

Since it is rotated through an angle \[\theta \] from the initial position then, \[{\theta _1} = 0\] and \[{\theta _2} = \theta \]
Then, the above equation will become,
\[\Delta U = - PE\left( {\cos \theta - 1} \right)\]
\[ \therefore \Delta U = PE\left( {1 - \cos \theta } \right)\]
Therefore, the potential energy of an electric dipole in the position \[PE\left( {1 - \cos \theta } \right)\].

Hence, option C is the correct answer.

Note: We have an alternate method to this question which is as follows. We know that the dipole moment is denoted by P. now the electric field E will be produced a torque on the dipole that is,
\[\tau = PE\sin \theta \]
Now work done from rotating it from an equilibrium position by an angle \[\theta \] is,
\[W = \int\limits_0^\theta {\tau d\theta } \]
\[\Rightarrow W = PE\int\limits_0^\theta {\sin \theta } d\theta \]
\[\Rightarrow W = - PE\left( {\cos \theta } \right)_0^\theta \]
\[\Rightarrow W = - PE\left( {\cos \theta - \cos 0} \right)\]
\[\Rightarrow W = - PE\left( {\cos \theta - 1} \right)\]
\[\therefore W = PE\left( {1 - \cos \theta } \right)\]
Therefore, the potential energy of an electric dipole in the position \[PE\left( {1 - \cos \theta } \right)\].