Question

The electrostatic potential due to an electric dipole at a distance \( r \) varies as:

• A. \(\frac{1}{r}\)
• B. \(\frac{1}{r^3}\)
• C. \(\frac{1}{r^2}\)
• D. \(r\)

Answer 1

The Correct Option is C

The problem involves understanding the variation of the electrostatic potential due to an electric dipole at a distance \( r \). Let's solve this problem step-by-step:

1. Understanding Electric Dipole: An electric dipole consists of two equal and opposite charges separated by a small distance. It possesses a dipole moment \(\mathbf{p}\), which can be given as \(q \cdot d\), where \(q\) is the magnitude of one of the charges and \(d\) is the separation between the charges.
2. Electrostatic Potential Due to a Dipole: The electrostatic potential \(V\) at a point at a distance \(r\) from the center of the dipole, on the axial line (line joining the charges) is given by the formula:

V = \(\frac{1}{4 \pi \varepsilon_0} \cdot \frac{\mathbf{p} \cdot \mathbf{r}}{r^3}\)

• Here, \(\varepsilon_0\) is the permittivity of free space.
• \(\mathbf{p} \cdot \mathbf{r}\) denotes the dot product of the dipole moment vector and the position vector.
• The formula shows that the potential due to a dipole falls off as \(\frac{1}{r^2}\) at points on the axial line.

1. Conclusion and Correct Option: From the potential formula, we see that the potential decreases with distance \(r\) in a manner proportional to \(\frac{1}{r^2}\). Therefore, the correct answer to the question is \(\frac{1}{r^2}\).
2. Explanation of Incorrect Options:

• \(\frac{1}{r}\) indicates a monopole (like a single charge), not a dipole.
• \(\frac{1}{r^3}\) might represent variations such as the electric field, but not the potential.
• \(r\) suggests a linearly increasing potential, which is not applicable here.

Thus, the correct answer is \(\frac{1}{r^2}\), consistent with the electrostatic potential formula for an electric dipole.

Answer 2

The electrostatic potential V at a point along the axial line of an electric dipole (aligned along the x-axis) is given by:

\( V = \frac{kp \cos \theta}{r^2} \)

where: k is Coulomb's constant, p is the dipole moment (\( p = q \times d \), where q is the charge and d is the separation distance), r is the distance from the dipole to the point where the potential is being calculated, and θ is the angle between the dipole axis and the line connecting the dipole to the point.

Since the potential V is inversely proportional to r², we conclude that the electrostatic potential due to a dipole varies as:

\( V \propto \frac{1}{r^2} \)