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 :
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 (VA) due to +q charge
VA=14πϵ0qr−l
Electric potential (VB) due to +q charge
VB=14πϵ0−qr+l
Therefore, total electric potential at P due to dipole will be
V=q4πϵ0[d1r−l−1r+l]
=q4πϵ02lr2−l2
=p4πϵ01r2−l2
Now, l<<r (short dipole)
V=p4πϵ01r2 (for axial position)
For equatorial position, V=0 as charges are equal in magnitude with opposite signs they cancel each other's effect.