Question

The electric potential due to dipole at a general point which is at distance $r$ from center of the dipole is

• A. $V=\frac{k(\overrightarrow{p}\cdot \overrightarrow{r})}{r^3}$
• B. $V=\frac{k(\overrightarrow{p}\cdot \overrightarrow{r})}{r^2}$
• C. $V=\frac{k(\overrightarrow{p}\cdot \overrightarrow{r})}{r}$
• D. $V=\frac{k(\overrightarrow{p}\cdot \overrightarrow{r})}{r^4}$

Answer 1

The correct option is A $V=\frac{k(\overrightarrow{p}\cdot \overrightarrow{r})}{r^3}$

Let the general point be $\text{A}(r,0)$.
Now, resolve the dipole moment $\overrightarrow{p}$ into components $p\cos \theta$ along the radial line and $p\sin \theta$ perpendicular to the radial line.


$\Rightarrow$ For point $\text{A}$ the $p\cos \theta$ component will yield potential at an axial point.

$V_1=\frac{k\left(p\cos \theta \right)}{r^2}$

Similarly, for $p\sin \theta$ component, point $\text{A}$ is at equatorial position. Thus, potential at a due to $p\sin \theta$

$V_2=0$

So, net potential at given point will be

$\Rightarrow V_{net}=V_1+V_2$

$\Rightarrow V_{net}=\frac{kp\cos \theta }{r^2}=\frac{kpr\cos \theta }{r^3}$

$\therefore V=\frac{k(\overrightarrow{p}\cdot \overrightarrow{r})}{r^3}$

So, option $\left(a\right)$ is correct.