Electric potential V due to a short dipole is related to the distance r of the observation point as

Options

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

Answer

To solve the problem of how the electric potential \( V \) due to a short dipole relates to the distance \( r \) of the observation point, we can follow these steps:

### Step 1: Understand the Dipole Moment
A short dipole consists of two equal and opposite charges separated by a small distance. The dipole moment \( p \) is defined as:
\[ p = q \cdot d \]
where \( q \) is the magnitude of one of the charges and \( d \) is the separation distance between the charges.

### Step 2: Electric Potential Due to a Dipole
The electric potential \( V \) at a point in space due to a dipole is given by the formula:
\[ V = \frac{k \cdot p \cdot \cos \theta}{r^2} \]
where:
- \( k \) is the Coulomb's constant,
- \( p \) is the dipole moment,
- \( \theta \) is the angle between the dipole moment and the line connecting the dipole to the observation point,
- \( r \) is the distance from the dipole to the observation point.

### Step 3: Analyze the Relationship Between \( V \) and \( r \)
From the formula \( V = \frac{k \cdot p \cdot \cos \theta}{r^2} \), we can see that the potential \( V \) is inversely proportional to the square of the distance \( r \):
\[ V \propto \frac{1}{r^2} \]
This means that as the distance \( r \) increases, the electric potential \( V \) decreases.

### Step 4: Express the Relationship Mathematically
We can express the relationship mathematically as:
\[ V \propto r^{-2} \]
This indicates that the potential decreases with the square of the distance from the dipole.

### Conclusion
Thus, the electric potential \( V \) due to a short dipole is inversely proportional to the square of the distance \( r \) from the dipole.

### Final Answer
The relationship is:
\[ V \propto \frac{1}{r^2} \quad \text{or} \quad V \propto r^{-2} \]

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