Let P be a point on the axis of a dipole of Moment \(\vec p\)and length 2a at a distance 'r' from the mid point.
Electric-field at 'P' due to +ve charge is
\(E_1\,=\frac{1}{4\pi\varepsilon_0}\times\frac{q}{(r-a)^2}\) along OP.
Electric-field at 'P' due to -ve charge is
\(E_2\,=\frac{1}{4\pi\varepsilon_0}\times\frac{q}{(r+a)^2}\) along PO.
Electric-field at 'P' due to the dipole \(=E_A=E_1-E_2.\)
\(E_A\,=\frac{1}{4\pi\varepsilon_0}\times\frac{2pr}{(r^2-a^2)^2}\) along the axis from -ve charge towards +ve charge.
when a < < d, a2 can be neglected.
\(E_A\,=\frac{1}{4\pi\varepsilon_0}\times\frac{2pr}{r^4}=\frac{1}{4\pi\varepsilon_0}\times\frac{2p}{r^3}.\)
Electric field is along the direction of dipole moment.
