The electric potential at a point on the axis of an electric dipole depends on the distance `r` of the point from the dipole as

Options

A. `prop (1)/(r )`
B. `prop (1)/(r ^(2))`
C. `prop r`
D. `prop (1)/(r^(3))`

Explanation

To solve the question regarding the electric potential at a point on the axis of an electric dipole and its dependence on the distance `r`, we can follow these steps:

### Step 1: Understand the Concept of Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a small distance. The electric potential due to a dipole at a point in space varies based on the position of that point relative to the dipole.

### Step 2: Identify the Relevant Formula
The electric potential \( V \) at a point on the axis of an electric dipole is given by the formula:
\[ V = \frac{k \cdot p \cdot \cos(\theta)}{r^2} \]
where:
- \( k \) is the electrostatic constant,
- \( p \) is the dipole moment,
- \( \theta \) is the angle between the dipole moment and the line connecting the dipole to the point,
- \( r \) is the distance from the dipole to the point.

### Step 3: Analyze the Situation on the Axis
For points along the axis of the dipole, \( \theta = 0^\circ \) (since the angle is measured from the dipole moment direction). Thus, \( \cos(0) = 1 \).

### Step 4: Simplify the Formula
Substituting \( \cos(0) \) into the formula, we get:
\[ V = \frac{k \cdot p}{r^2} \]
This shows that the electric potential \( V \) is inversely proportional to the square of the distance \( r \).

### Step 5: Conclusion
Therefore, we can conclude that the electric potential at a point on the axis of an electric dipole depends on the distance \( r \) as:
\[ V \propto \frac{1}{r^2} \]

### Final Answer
The electric potential at a point on the axis of an electric dipole depends on the distance \( r \) as \( V \propto \frac{1}{r^2} \).

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