This problem examines how electrostatic potential varies with distance \( r \) for an electric dipole. The solution proceeds as follows:
Electric Dipole Definition: An electric dipole comprises two equal and opposite charges separated by a small distance. It is characterized by a dipole moment \( \mathbf{p} \), defined as \( q \cdot d \), where \( q \) is the magnitude of each charge and \( d \) is the separation distance.
Electrostatic Potential Formula: The electrostatic potential \( V \) at a point located at a distance \( r \) from the dipole's center, along the axial line (the line connecting the charges), is given by:
\( \varepsilon_0 \) represents the permittivity of free space.
\( \mathbf{p} \cdot \mathbf{r} \) is the dot product of the dipole moment vector and the position vector.
This formula indicates that the potential generated by a dipole decreases with distance \( r \) at a rate proportional to \( \frac{1}{r^2} \) for points on the axial line.
Conclusion: The potential formula clearly shows that the potential diminishes with distance \( r \) as \( \frac{1}{r^2} \). Consequently, the correct answer is \( \frac{1}{r^2} \).
Incorrect Option Analysis:
A \( \frac{1}{r} \) relationship describes a monopole (a single charge), not a dipole.
A \( \frac{1}{r^3} \) dependence might relate to electric field variations, but not the potential.
A \( r \) dependence implies a linearly increasing potential, which is not relevant in this scenario.
Therefore, the correct answer is \( \frac{1}{r^2} \), aligning with the electrostatic potential formula for an electric dipole.
