An electric dipole when placed in a uniform electric field `E` will have minimum potential energy, if the positive direction of dipole moment makes the following angle with `E`

Options

A. zero
B. `(pi)/(2)`
C. `pi`
D. `(3pi)/(2)`

Answer

To solve the problem of determining the angle at which an electric dipole has minimum potential energy in a uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Potential Energy of a Dipole**:
The potential energy \( U \) of an electric dipole in a uniform electric field \( E \) is given by the formula:
\[ U = -\vec{P} \cdot \vec{E} = -PE \cos \theta \]
where \( P \) is the dipole moment, \( E \) is the magnitude of the electric field, and \( \theta \) is the angle between the dipole moment \( \vec{P} \) and the electric field \( \vec{E} \).

2. **Minimizing the Potential Energy**:
To find the angle for minimum potential energy, we need to minimize the expression \( U = -PE \cos \theta \). Since \( P \) and \( E \) are constants, we focus on minimizing \( -\cos \theta \).

3. **Analyzing the Cosine Function**:
The cosine function \( \cos \theta \) achieves its maximum value of 1 when \( \theta = 0^\circ \). Therefore, the potential energy \( U \) will be minimized when:
\[ U_{\text{min}} = -PE \cdot 1 = -PE \]

4. **Conclusion**:
The angle \( \theta \) that gives the minimum potential energy is:
\[ \theta = 0^\circ \]
This corresponds to the dipole moment being aligned in the same direction as the electric field.

### Final Answer:
The positive direction of the dipole moment makes an angle of **0 degrees** with the electric field \( E \) for minimum potential energy.

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