Question

The electric dipole potential at a large distance $r$ from the dipole falls off. The falling off the magnitude of the electric potential depends on,

• A. $\frac{1}{r}$
• B. $\frac{1}{r^4}$
• C. $\frac{1}{r^3}$
• D. $\frac{1}{r^2}$

Answer 1

The correct option is D $\frac{1}{r^2}$

Given that $r>>a$, where $2a$ is the distance between the two charges of the dipole.

The dipole moment is
$\Rightarrow \overrightarrow{p}=(q\times 2a)\hat{p}=\left(2aq\right)\hat{p}$

Potential due to a dipole with dipole moment $\overrightarrow{p}$ at a point $P$ having position vector $\overrightarrow{r}$ with respect to the dipole moment orientation is given by (if $r>>a$)

$\Rightarrow V=\frac{\overrightarrow{p}\cdot \overrightarrow{p}}{4\pi {ϵ}_0{r}^3}=\frac{(2aq\hat{p}\cdot (r\hat{r}\left)\right)}{4\pi {ϵ}_0{r}^3}=\frac{2qa\hat{p}\hat{r}}{4\pi {ϵ}_0{r}^2}$

Thus, the potential falls off by a factor of $\frac{1}{r^2}$

Therefore, option $\left(B\right)$ is correct.