Question

Derive an expression for the potential at a point due to a short dipole. Hence show what will be the potential at an axial and an equatorial point :

Answer 1

Consider an electric dipole AB in which point charges +q and -q are separated by a distance 2l. Let P be a point on it's axis seperated by a distance r from the dipole's center O.
Electric dipole moment, p = q (2l)
Electric potential ($V_A$) due to +q charge
$V_A=\frac{1}{4\pi {ϵ}_0}\frac{q}{r-l}$
Electric potential ($V_B$) due to +q charge
$V_B=\frac{1}{4\pi {ϵ}_0}\frac{-q}{r+l}$
Therefore, total electric potential at P due to dipole will be
$V=\frac{q}{4\pi {ϵ}_0}[d\frac{1}{r-l}-\frac{1}{r+l}]$
$=\frac{q}{4\pi {ϵ}_0}\frac{2l}{r^2-l^2}$
$=\frac{p}{4\pi {ϵ}_0}\frac{1}{r^2-l^2}$
Now, $l<<r$ (short dipole)
$V=\frac{p}{4\pi {ϵ}_0}\frac{1}{r^2}$ (for axial position)

For equatorial position, $V=0$ as charges are equal in magnitude with opposite signs they cancel each other's effect.