An electric dipole of moment `p` is placed in the positive of stable equilibrium in uniform electric field of intensity `E`. It is rotated through an angle `theta` from the initial position. The potential energy of electric dipole in the final position is

Options

A. `pE cos theta`
B. `pE sin theta`
C. `pE (1 - cos theta)`
D. `-pE cos theta`

Explanation

To find the potential energy of an electric dipole in a uniform electric field after it has been rotated through an angle \( \theta \) from its stable equilibrium position, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Electric Dipole**:
An electric dipole consists of two equal and opposite charges separated by a distance \( 2L \). The dipole moment \( \mathbf{p} \) is defined as:
\[ \mathbf{p} = q \cdot 2L \]
where \( q \) is the magnitude of one of the charges.

2. **Initial Position in the Electric Field**:
The dipole is initially in a stable equilibrium position, which means it is aligned with the electric field \( \mathbf{E} \). In this position, the angle \( \theta \) is \( 0^\circ \).

3. **Torque on the Dipole**:
When the dipole is rotated through an angle \( \theta \), it experiences a torque \( \tau \) given by:
\[ \tau = \mathbf{p} \times \mathbf{E} = pE \sin \theta \]
This torque tends to restore the dipole to its equilibrium position.

4. **Work Done and Potential Energy**:
The work done \( W \) in rotating the dipole from \( 0^\circ \) to \( \theta \) is equal to the change in potential energy \( U \) of the dipole:
\[ U = -\int_0^\theta \tau \, d\theta = -\int_0^\theta pE \sin \theta \, d\theta \]

5. **Calculating the Integral**:
The integral can be calculated as follows:
\[ U = -pE \int_0^\theta \sin \theta \, d\theta \]
The integral of \( \sin \theta \) is \( -\cos \theta \):
\[ U = -pE \left[-\cos \theta \right]_0^\theta = -pE \left[-\cos \theta + \cos 0\right] \]
Since \( \cos 0 = 1 \):
\[ U = -pE \left(-\cos \theta + 1\right) = -pE (1 - \cos \theta) \]

6. **Final Expression for Potential Energy**:
The potential energy of the electric dipole in the final position after being rotated through an angle \( \theta \) is:
\[ U = -pE \cos \theta \]

### Final Answer:
The potential energy of the electric dipole in the final position is:
\[ U = -pE \cos \theta \]