An electric dipole of moment p is placed parallel to the uniform electric field. The amount of work done in rotating the dipole by `90^@` is :

Options

A. 2pE
B. PE
C. `(pE)/2`
D. Zero

Answer

To find the amount of work done in rotating an electric dipole of moment \( p \) from a position parallel to a uniform electric field \( E \) to a position perpendicular to it (90 degrees), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial and Final Positions**:
- Initially, the dipole moment \( \vec{p} \) is parallel to the electric field \( \vec{E} \). This means the angle \( \theta_1 = 0^\circ \).
- After rotation, the dipole moment \( \vec{p} \) is perpendicular to the electric field. This means the angle \( \theta_2 = 90^\circ \).

2. **Use the Work Done Formula**:
The work done \( W \) in rotating a dipole in an electric field can be calculated using the formula:
\[ W = pE \left( \cos \theta_1 - \cos \theta_2 \right) \]

3. **Substitute the Angles**:
- Substitute \( \theta_1 = 0^\circ \) and \( \theta_2 = 90^\circ \) into the formula.
- We know that:
- \( \cos 0^\circ = 1 \)
- \( \cos 90^\circ = 0 \)

Thus, we have:
\[ W = pE \left( \cos 0^\circ - \cos 90^\circ \right) = pE \left( 1 - 0 \right) = pE \]

4. **Final Result**:
The amount of work done in rotating the dipole by \( 90^\circ \) is:
\[ W = pE \]