An electric dipole of dipole moment `vecP` is placed in a uniform electric field `vecE` such that `vecP` is perpendicular to `vecE` . The work done to turn the dipole through an angle of `180^(@)` is -

Options

A. zero
B. pE
C. 2pE
D. `sqrt(2)` pE

Explanation

To find the work done to turn an electric dipole through an angle of 180 degrees when it is initially placed perpendicular to a uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- The dipole moment vector \(\vec{P}\) is perpendicular to the electric field vector \(\vec{E}\). This means the initial angle \(\theta_1\) between \(\vec{P}\) and \(\vec{E}\) is \(90^\circ\).

2. **Determine the Final Conditions**:
- We need to rotate the dipole through an angle of \(180^\circ\). Therefore, the final angle \(\theta_2\) will be \(180^\circ\).

3. **Use the Work Done Formula**:
- The work done \(W\) in rotating a dipole in an electric field is given by the formula:
\[ W = -P E (\cos \theta_2 - \cos \theta_1) \]
- Here, \(P\) is the dipole moment, \(E\) is the electric field strength, \(\theta_1\) is the initial angle, and \(\theta_2\) is the final angle.

4. **Substitute the Angles**:
- Substitute \(\theta_1 = 90^\circ\) and \(\theta_2 = 180^\circ\) into the formula:
\[ W = -P E (\cos 180^\circ - \cos 90^\circ) \]

5. **Calculate the Cosine Values**:
- We know that:
- \(\cos 180^\circ = -1\)
- \(\cos 90^\circ = 0\)

6. **Plug in the Values**:
- Now substituting these values into the equation:
\[ W = -P E (-1 - 0) = -P E (-1) = P E \]

7. **Final Result**:
- Therefore, the work done to turn the dipole through an angle of \(180^\circ\) is:
\[ W = P E \]

### Conclusion:
The work done to turn the dipole through an angle of \(180^\circ\) is \(P E\).

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