An electric dipole of dipole moment p is aligned parallel to a uniform electric field E. the energy required to rotate the dipole by `90^(@)` is

Options

A. `pE^(2)`
B. `p^(2)`E
C. pE
D. Infinity

Answer

To find the energy required to rotate an electric dipole from an initial position aligned parallel to a uniform electric field to a final position perpendicular to the field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial and Final Positions**:
- The dipole moment \( \mathbf{p} \) is initially aligned parallel to the electric field \( \mathbf{E} \). This means the angle \( \theta \) between \( \mathbf{p} \) and \( \mathbf{E} \) is \( 0^\circ \).
- After rotation, the dipole is perpendicular to the electric field, which means \( \theta = 90^\circ \).

2. **Potential Energy of the Dipole**:
- The potential energy \( U \) of an electric dipole in an electric field is given by the formula:
\[ U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos \theta \]
- Here, \( p \) is the magnitude of the dipole moment, \( E \) is the magnitude of the electric field, and \( \theta \) is the angle between \( \mathbf{p} \) and \( \mathbf{E} \).

3. **Calculate Initial Potential Energy**:
- For the initial position where \( \theta = 0^\circ \):
\[ U_{\text{initial}} = -pE \cos(0^\circ) = -pE \cdot 1 = -pE \]

4. **Calculate Final Potential Energy**:
- For the final position where \( \theta = 90^\circ \):
\[ U_{\text{final}} = -pE \cos(90^\circ) = -pE \cdot 0 = 0 \]

5. **Calculate the Work Done (Energy Required)**:
- The work done (or energy required) to rotate the dipole is the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} = 0 - (-pE) = pE \]

### Final Answer:
The energy required to rotate the dipole by \( 90^\circ \) is \( pE \).

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