Derive an expression for the potential at a point along the axial line of a short electric dipole.

Explanation

Electric potential at an exial point of a dipole Let us consider an electric dipole consisting of charges +q and -q separated by a distance 2a.

Let P be a point on the axis of a dipole at a distance r from the centre of dipole. Electric potential at point P due to charge `-q` is
`V_(1)= (1)/(4pi epsi_(0)) .((-q))/(AP)`
`V_(1)= - (1)/(4pi epsi_(0)).(q)/(r+ a)`
Electric potential at point P due to charge +q is
`V_(2)= (1)/(4pi epsi_(0)) .(q)/(BP)`
`V_(2)= (1)/(4pi epsi_(0)).(q)/(r-a)`
Hence, electric potential at point P due to the dipole is given by
`V= V_(1) + V_(2)`
`rArr V= (q)/(4pi epsi_(0)) [(1)/(r-a)- (1)/(r+a)]= (q)/(4pi epsi_(0)).(2a)/(r^(2)-a^(2))`
`therefore V= (1)/(4pi epsi_(0)).(p)/(r^(2)-a^(2)) [ because q xx 2a= p]`
For short dipole `a lt lt r`, a can be neglected in denominator.
`therefore V= (1)/(4pi epsi_(0)).(p)/(r^(2))`