Question

Prove that the electric potential at a point in broad side on position due to an electric dipole is zero.

Answer 1

Potential due to a dipole in Broad side on position: Consider an electric dipole $AB$ consisting of charges $-q$ and $+q$ coulomb at a separation $2a$ meter and placed in a medium of dielectric constant $K$ as shown in Fig. We are to find the electric potential due to dipole at a point $P$ at distance $r$ meter from its mid point $O$ on the perpendicular bisector of dipole axis $AB$. It is clear from Fig. that the distance of point $P$ from each charge is the same, i.e.,

$AP=BP=\sqrt{r^2+a^2}$ meter

Now potential at point $P$ due to charge $+q$ is

$V_1=\frac{1}{4\pi {\epsilon }_{0}K}\frac{(+q)}{\left(BP\right)}=\frac{1}{4\pi {\epsilon }_{0}K}\frac{q}{\sqrt{r^2+a^2}}$

and potential at point $P$ due to charge $-q$ is

$V_2=\frac{1}{4\pi {\epsilon }_{0}K}\frac{(-q)}{AP}=\frac{-1}{4\pi {\epsilon }_{0}K}\frac{q}{\sqrt{r^2+a^2}}$

Hence resultant potential at the point $P$ is

$V=V_1+V_2=\frac{1}{4\pi {\epsilon }_{0}K}\frac{q}{\sqrt{r^2+a^2}}-\frac{1}{4\pi {\epsilon }_{0}K}\frac{q}{\sqrt{r^2+a^2}}$

or $V=0$

Thus the electric potential due to a dipole at any point on its broad side on (or transverse) position is zero (but the electric field intensity at this point is not zero).