A dipole of electric dipole moment p is placed in a uniform electric field of strength E. If `theta` is the angle between positive direction of p and E, then the potential energy of the electric dipole is largest when `theta` is

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. zero

Answer

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

### Step-by-Step Solution:

1. **Understanding the Electric Dipole Moment**:
An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment \( \mathbf{p} \) is defined as \( \mathbf{p} = q \cdot \mathbf{d} \), where \( q \) is the charge and \( \mathbf{d} \) is the vector pointing from the negative charge to the positive charge.

2. **Potential Energy of the Dipole**:
The potential energy \( U \) of an electric dipole in a uniform electric field \( \mathbf{E} \) is given by the formula:
\[ U = -\mathbf{p} \cdot \mathbf{E} \]
This can also be expressed in terms of the angle \( \theta \) between the dipole moment \( \mathbf{p} \) and the electric field \( \mathbf{E} \):
\[ U = -pE \cos \theta \]

3. **Maximizing Potential Energy**:
To find the angle \( \theta \) that maximizes the potential energy \( U \), we need to minimize \( \cos \theta \) since \( U \) is negative of \( pE \cos \theta \).

4. **Finding the Minimum Value of Cosine**:
The cosine function \( \cos \theta \) has a minimum value of -1. This occurs when:
\[ \cos \theta = -1 \]

5. **Determining the Corresponding Angle**:
The angle \( \theta \) for which \( \cos \theta = -1 \) is:
\[ \theta = 180^\circ \quad \text{or} \quad \theta = \pi \text{ radians} \]

6. **Conclusion**:
Therefore, the potential energy of the electric dipole is largest when \( \theta \) is \( 180^\circ \) or \( \pi \) radians.

### Final Answer:
The potential energy of the electric dipole is largest when \( \theta = 180^\circ \) or \( \theta = \pi \) radians.

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