A molecule with a dipole moment `p` is placed in an electric field of strength `E`. Initially the dipole is aligned parallel to the field. If the dipole is to be rotated to be anti-parallel to the field, the work required to be done by an external agency is

Options

A. `-2pE`
B. `-pE`
C. `pE`
D. 2pE

Answer

To find the work required to rotate a dipole from being parallel to an electric field to being anti-parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial and Final Positions**:
- The dipole moment \( \vec{p} \) is initially aligned parallel to the electric field \( \vec{E} \). This means the angle \( \theta_i = 0^\circ \).
- The dipole is to be rotated to an anti-parallel position, which means the angle \( \theta_f = 180^\circ \).

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

3. **Calculating Initial Potential Energy**:
- For the initial position where \( \theta_i = 0^\circ \):
\[ U_i = -pE \cos(0) = -pE \cdot 1 = -pE \]

4. **Calculating Final Potential Energy**:
- For the final position where \( \theta_f = 180^\circ \):
\[ U_f = -pE \cos(180^\circ) = -pE \cdot (-1) = pE \]

5. **Calculating the Work Done**:
- The work done \( W \) by the external agency in moving the dipole from the initial to the final position is equal to the change in potential energy:
\[ W = U_f - U_i \]
- Substituting the values:
\[ W = pE - (-pE) = pE + pE = 2pE \]

6. **Conclusion**:
- The work required to rotate the dipole from parallel to anti-parallel to the electric field is:
\[ W = 2pE \]

### Final Answer:
The work required to be done by an external agency is \( 2pE \).