An electric dipole of moment `vecp` placed in a uniform electric field `vecE` has minimum potential energy when the angle between `vecp` and `vecE` is

Options

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

Answer

To determine the angle at which the potential energy of an electric dipole in a uniform electric field is minimized, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Electric Dipole Moment and Electric Field**:
An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment, denoted as **p**, is a vector quantity that points from the negative charge to the positive charge. The electric field, denoted as **E**, is a uniform field in which the dipole is placed.

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

3. **Analyzing the Potential Energy**:
The potential energy is dependent on the cosine of the angle θ. The cosine function varies between -1 and 1:
- When **θ = 0°** (or 0 radians), **cos θ = 1**:
\[ U = -pE \cdot 1 = -pE \quad (\text{minimum potential energy}) \]
- When **θ = 90°** (or π/2 radians), **cos θ = 0**:
\[ U = -pE \cdot 0 = 0 \quad (\text{potential energy is zero}) \]
- When **θ = 180°** (or π radians), **cos θ = -1**:
\[ U = -pE \cdot (-1) = +pE \quad (\text{maximum potential energy}) \]

4. **Finding the Minimum Potential Energy**:
From the analysis above, we see that the potential energy is minimized when **θ = 0°**. This is because at this angle, the potential energy reaches its most negative value, which is the minimum.

5. **Conclusion**:
Therefore, the angle between the dipole moment **p** and the electric field **E** at which the potential energy is minimized is:
\[ \theta = 0° \]

### Final Answer:
The angle between the dipole moment **p** and the electric field **E** for minimum potential energy is **0°**.