In which of the states shown in figure, is potential energy of a electric dipole maximum ?

Options

A.
B.
C.
D.

Explanation

To determine in which state the potential energy of an electric dipole is maximum, we can follow these steps:

### Step 1: Understand the formula for potential energy
The potential energy (U) of an electric dipole in an electric field is given by the formula:
\[ U = -\mathbf{P} \cdot \mathbf{E} = -PE \cos \theta \]
where:
- \( P \) is the magnitude of the dipole moment,
- \( E \) is the magnitude of the electric field,
- \( \theta \) is the angle between the dipole moment and the electric field direction.

### Step 2: Analyze the formula
From the formula \( U = -PE \cos \theta \), we can see that:
- The potential energy is maximum when \( \cos \theta \) is at its minimum value.
- The minimum value of \( \cos \theta \) occurs when \( \theta = 180^\circ \) (or \( \pi \) radians), which gives \( \cos 180^\circ = -1 \).

### Step 3: Determine the maximum potential energy condition
When \( \theta = 180^\circ \):
\[ U = -PE \cos 180^\circ = -PE \times (-1) = PE \]
This indicates that the potential energy is maximum when the dipole is aligned opposite to the electric field.

### Step 4: Evaluate the given options
Now, we need to evaluate the orientations of the dipole in the provided figure (not shown here) to identify which orientation corresponds to \( \theta = 180^\circ \).

1. **If the dipole is aligned with the electric field (0 degrees)**: \( U = -PE \)
2. **If the dipole is perpendicular to the electric field (90 degrees)**: \( U = 0 \)
3. **If the dipole is opposite to the electric field (180 degrees)**: \( U = PE \) (maximum)

### Step 5: Conclusion
The orientation of the dipole that corresponds to maximum potential energy is when it is aligned opposite to the direction of the electric field.

### Final Answer
The potential energy of the electric dipole is maximum in the orientation where it is opposite to the electric field direction (typically represented as option A in the figure).

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