An electric dipole of moment `p` is palced 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 `(1 - cos theta)`
C. pE cos `theta`
D. pE sin `theta`

Answer

To find the potential energy of an electric dipole in a uniform electric field after it has been rotated through an angle θ, we can follow these steps:

### Step 1: Understand the Initial Position
Initially, the electric dipole (with dipole moment \( p \)) is aligned parallel to the electric field \( E \). In this position, the potential energy \( U_i \) of the dipole is at its minimum value, which is given by:
\[ U_i = -p \cdot E = -pE \cos(0) = -pE \]
Here, the angle between the dipole moment and the electric field is \( 0^\circ \).

### Step 2: Determine the Final Position
After the dipole is rotated through an angle \( \theta \), the angle between the dipole moment and the electric field becomes \( \theta \).

### Step 3: Calculate the Potential Energy in the Final Position
The potential energy \( U_f \) of the dipole in the electric field when it makes an angle \( \theta \) with the field is given by the formula:
\[ U_f = -p \cdot E = -pE \cos(\theta) \]
This formula arises from the dot product of the dipole moment vector and the electric field vector.

### Step 4: Conclusion
Thus, the potential energy of the electric dipole in the final position after being rotated through an angle \( \theta \) is:
\[ U_f = -pE \cos(\theta) \]

### Final Answer
The potential energy of the electric dipole in the final position is \( -pE \cos(\theta) \).

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