Electric Dipole Potential Understanding

An electric dipole consists of two equal and opposite charges separated by a small distance. The electric potential at any point in space due to an electric dipole depends on several factors related to the dipole's properties and the observation point's location.

Dipole Potential Formula Explanation

The electric potential ($V$) at a point P, located at a distance $r$ from the center of the dipole, is given by the formula:

$ V = \frac{1}{4\pi\epsilon_0} \frac{\vec{p} \cdot \vec{r}}{r^3} $

This can also be expressed using the angle $\theta$ between the dipole moment vector ($\vec{p}$) and the position vector ($\vec{r}$):

$ V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2} $

Here,

• $\epsilon_0$ is the permittivity of free space.
• $p$ is the magnitude of the dipole moment ($p = qd$, where $q$ is the charge magnitude and $d$ is the separation distance).
• $r$ is the magnitude of the position vector from the dipole's center to the point P.
• $\theta$ is the angle between $\vec{p}$ and $\vec{r}$.

Analyzing Statement (A): Dependence on r

Statement (A) says the potential depends on $r$, the magnitude of the position vector. Looking at the formula $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$, it's clear that $V$ is inversely proportional to $r^2$. Therefore, the electric potential due to a dipole **depends on $r$**. Statement (A) is correct.

Analyzing Statement (B): Dependence on Angle

Statement (B) states that the potential depends on the angle between the position vector $\vec{r}$ and the dipole moment vector $\vec{p}$. The term $\cos\theta$ in the formula $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$ explicitly shows this dependence. The potential varies with the angle $\theta$; for example, it's maximum along the axis ($\theta=0^\circ$) and zero on the perpendicular bisector ($\theta=90^\circ$). Statement (B) is correct.

Analyzing Statement (C): Fall-off Rate at Long Distances

Statement (C) claims the potential falls off as $1/r^2$ at long distances. As observed from the formula $V \propto \frac{1}{r^2}$, the potential decreases with the square of the distance from the dipole. This is characteristic of a dipole field at large distances. Statement (C) is correct.

Analyzing Statement (D): Independence of Charge Separation

Statement (D) claims the potential does not depend on the distance separating the charges. The dipole moment $p$ is defined as the product of the magnitude of one charge ($q$) and the distance ($d$) separating the two charges ($p = qd$). Since the potential $V$ is directly proportional to $p$, it inherently **depends on the distance separating the charges** ($d$). Therefore, statement (D) is incorrect.

Conclusion on Dipole Potential Factors

Based on the analysis of the formula and the behavior of electric potential due to a dipole:

• Statement (A) is true (depends on $r$).
• Statement (B) is true (depends on the angle $\theta$).
• Statement (C) is true (falls off as $1/r^2$ at long distances).
• Statement (D) is false (it does depend on the charge separation distance).

Therefore, the correct combination of statements is (A), (B), and (C).