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

Answer

To find the work required to rotate a dipole moment \( \mathbf{p} \) from being aligned parallel to an electric field \( \mathbf{E} \) to being aligned anti-parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial and Final Positions**:
- Initially, the dipole moment \( \mathbf{p} \) is aligned parallel to the electric field \( \mathbf{E} \). This means the angle \( \theta \) between \( \mathbf{p} \) and \( \mathbf{E} \) is \( 0^\circ \).
- Finally, we want to rotate the dipole to be anti-parallel to the electric field, which means the angle \( \theta \) will be \( 180^\circ \).

2. **Potential Energy in an Electric Field**:
- The potential energy \( U \) of a dipole in an electric field is given by the formula:
\[ U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos(\theta) \]
- For the initial position (parallel), where \( \theta = 0^\circ \):
\[ U_{\text{initial}} = -pE \cos(0^\circ) = -pE \cdot 1 = -pE \]

3. **Calculating Potential Energy in the Final Position**:
- For the final position (anti-parallel), where \( \theta = 180^\circ \):
\[ U_{\text{final}} = -pE \cos(180^\circ) = -pE \cdot (-1) = pE \]

4. **Calculating the Work Done**:
- The work \( W \) done by an external agency to rotate the dipole from the initial to the final position is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} \]
- Substituting the values we calculated:
\[ W = pE - (-pE) = pE + pE = 2pE \]

5. **Conclusion**:
- Therefore, the work required to rotate the dipole from parallel to anti-parallel alignment in the electric field is:
\[ W = 2pE \]

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

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