The electric potential due to an electric dipole

$\text{(A) depends on r, where r is the magnitude of position vector}$  r →

$\text{(B) depends on the angle between the position vector}\overrightarrow{r}\text{and the dipole moment vector}\overrightarrow{\mathrm{ p}}$

$\text{(C) falls off at long distances, as }\frac{1}{r^2}$

$\text{(D) does not depend upon the distance separating the charges}$

$\text{Choose the correct answer from the options given below:}$

The correct answer is (A), (B) and (C) only.

To understand why this is the correct choice, let us derive and analyze the formula for the electric potential due to an electric dipole at a point far away from it.

An electric dipole consists of two equal and opposite charges, $-q$ and $+q$, separated by a small distance $d$ (often represented as $2a$). The dipole moment vector $\overrightarrow{p}$ is directed from the negative charge to the positive charge, and its magnitude is given by:
$p=q·d$

For a point located at a position vector $\overrightarrow{r}$ relative to the center of the dipole, where the distance $r$ is much larger than the separation distance $d$ ($r>>d$), the electric potential $V$ is given by the formula:
$V=\frac{1}{4\pi {\epsilon }_0}\frac{\overrightarrow{p}·\hat{r}}{r^2}=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{r^2}$
where $\theta$ is the angle between the dipole moment vector $\overrightarrow{p}$ and the position vector $\overrightarrow{r}$, and ${\epsilon }_0$ is the permittivity of free space.

Let us evaluate each of the statements based on this formula:

Statement (A): depends on r, where r is the magnitude of position vector $\overrightarrow{r}$
Looking at the formula, $V$ is inversely proportional to $r^2$. Thus, the potential directly depends on the magnitude of the position vector $r$. Therefore, statement (A) is correct.

Statement (B): depends on the angle between the position vector $\overrightarrow{r}$ and the dipole moment vector $\overrightarrow{p}$
The formula includes the term $\cos \theta$, where $\theta$ is the angle between the position vector and the dipole moment. Hence, the electric potential depends on this angle. Therefore, statement (B) is correct.

Statement (C): falls off at long distances, as $\frac{1}{r^2}$
Since $V\propto \frac{1}{r^2}$, the potential falls off with distance as $\frac{1}{r^2}$ (unlike the potential of a single point charge which falls off as $\frac{1}{r}$). Therefore, statement (C) is correct.

Statement (D): does not depend upon the distance separating the charges
The dipole moment $p$ is equal to $q·d$, where $d$ is the separation between the charges. Since the potential $V$ depends on $p$, it directly depends on the separation distance $d$. Therefore, statement (D) is incorrect.

Conclusion: Statements (A), (B), and (C) are the only correct statements, making (A), (B) and (C) only the correct answer.