When dipole moment `vec(p)` of a dipole is parallel to electric field intensity `vec(E )` (stable equilibrium) the potential energy of dipole is

Options

A. positive, maximum
B. positve, minimum
C. negative, maximum
D. negative, minimum

Answer

To find the potential energy of a dipole when its dipole moment \( \vec{p} \) is parallel to the electric field intensity \( \vec{E} \), we can follow these steps:

### Step 1: Understand the relationship between potential energy and dipole moment
The potential energy \( U \) of an electric dipole in a uniform electric field is given by the formula:
\[ U = -\vec{p} \cdot \vec{E} \]
This means that the potential energy depends on the angle \( \theta \) between the dipole moment \( \vec{p} \) and the electric field \( \vec{E} \).

### Step 2: Analyze the case when \( \vec{p} \) is parallel to \( \vec{E} \)
When the dipole moment \( \vec{p} \) is parallel to the electric field \( \vec{E} \), the angle \( \theta \) between them is \( 0^\circ \). Therefore, we can substitute \( \theta = 0 \) into the potential energy formula.

### Step 3: Substitute the angle into the potential energy formula
Using the formula for potential energy:
\[ U = -\vec{p} \cdot \vec{E} = -pE \cos(0^\circ) \]
Since \( \cos(0^\circ) = 1 \), we have:
\[ U = -pE \]

### Step 4: Interpret the result
The negative sign indicates that the potential energy is minimized when the dipole is aligned with the electric field, which corresponds to a stable equilibrium. Thus, the potential energy of the dipole in this configuration is:
\[ U = -pE \]

### Conclusion
When the dipole moment \( \vec{p} \) is parallel to the electric field intensity \( \vec{E} \), the potential energy of the dipole is given by:
\[ U = -pE \]