An electric dipole of dipole moment p is placed in a uniform external electric field E. Then, the

Options

A. torque experienced by the dipole is `E xxp`
B. torque is zero, if p is perpendicular to E
C. torque is maximum, if p is perpendicular to E
D. potential energy is maximum, if p is parallel to E

Answer

To solve the problem of an electric dipole of dipole moment \( \mathbf{p} \) placed in a uniform external electric field \( \mathbf{E} \), we need to analyze the torque experienced by the dipole and the potential energy associated with its orientation in the field.

### Step-by-Step Solution:

1. **Understanding Torque on a Dipole**:
The torque \( \tau \) experienced by an electric dipole in an electric field is given by the formula:
\[ \tau = \mathbf{p} \times \mathbf{E} \]
where \( \times \) denotes the cross product.

2. **Magnitude of Torque**:
The magnitude of the torque can be expressed as:
\[ |\tau| = |\mathbf{p}| |\mathbf{E}| \sin \theta \]
where \( \theta \) is the angle between the dipole moment \( \mathbf{p} \) and the electric field \( \mathbf{E} \).

3. **Analyzing Torque Conditions**:
- **Torque is Maximum**: The torque is maximum when \( \theta = 90^\circ \) (i.e., when \( \mathbf{p} \) is perpendicular to \( \mathbf{E} \)). In this case, \( |\tau| = |\mathbf{p}| |\mathbf{E}| \).
- **Torque is Zero**: The torque is zero when \( \theta = 0^\circ \) or \( \theta = 180^\circ \) (i.e., when \( \mathbf{p} \) is parallel or anti-parallel to \( \mathbf{E} \)).

4. **Potential Energy of the Dipole**:
The potential energy \( U \) of the dipole in the electric field is given by:
\[ U = -\mathbf{p} \cdot \mathbf{E} = -|\mathbf{p}| |\mathbf{E}| \cos \theta \]
- **Potential Energy is Maximum**: The potential energy is maximum when \( \theta = 180^\circ \) (i.e., when \( \mathbf{p} \) is anti-parallel to \( \mathbf{E} \)). In this case, \( U = |\mathbf{p}| |\mathbf{E}| \).
- **Potential Energy is Minimum**: The potential energy is minimum when \( \theta = 0^\circ \) (i.e., when \( \mathbf{p} \) is parallel to \( \mathbf{E} \)). In this case, \( U = -|\mathbf{p}| |\mathbf{E}| \).

5. **Conclusion**:
- The torque is maximum when \( \mathbf{p} \) is perpendicular to \( \mathbf{E} \).
- The potential energy is maximum when \( \mathbf{p} \) is anti-parallel to \( \mathbf{E} \).

### Summary of Results:
- Torque is maximum if \( \mathbf{p} \) is perpendicular to \( \mathbf{E} \).
- Potential energy is maximum if \( \mathbf{p} \) is anti-parallel to \( \mathbf{E} \).