Question

A point dipole $\overrightarrow{p}=-p_0\hat{x}$ kept at the origin. The potential and electric field due to this dipole on the $y-$axis at a distance $d$ are, respectively :
(Take $V=0$ at infinity)

• A. $0,\frac{|\overrightarrow{p}|}{4\pi {\epsilon }_0{d}^2}$
• B. $0,\frac{-\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$
• C. $\frac{|\overrightarrow{p}|}{4\pi {\epsilon }_0{d}^2},\frac{\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$
• D. $0,\frac{\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$

Answer 1

The correct option is B $0,\frac{-\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$

The dipole lies along negative $x-$ axis and the given point is on $y-$ axis.

So, the given point lies on the equatorial line of a point dipole.

The electric potential due to a point dipole is given by,

$V=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{d^2}$

$\text{On the equatorial line,}\theta ={180}^{\circ }$

$\Rightarrow V=0$

We know that, the electric field due to a point dipole on the equatorial line is given by,

$\overrightarrow{E}=\frac{p}{4\pi {\epsilon }_0{d}^3}(-\hat{p})$

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