Electric field at an equatorial point of a dipole-

Consider an electric dipole consisting of charges -q and +q, seperated by distance 2a and placed in vacuum.

Let P be a point on the equatorial line of the dipole at distance r from it.

OP=r

Electric field at point P due to +q charge-

E_+q= 1/4πε_o .q/(r² +a²)             along BP

Electric field at point P due to -q charge-

E-_q= 1/4πε_o .q/(r² +a²)             along PA

Thus the magnitudes of E_+q and E_-q are equal

As we can see in figure, the components of E_+q and E_-q normal to the dipole axis will cancel out.

The components parallel to the dipole axis add up.

The total electric field E_equa is opposite to p.          (p is dipole moment)

So.E_equa = -(E_-q cosθ+E_+q cosθ)p

= -(2E_-q cosθ)p

= -2.(1/4πε_o).q/(r² +a²) .a/√(r² +a²) p              (cosθ=a/√(r² +a²) )

E_equa= -(1/4πε_o).p/(r² +a²)^3/2 p

Where,p=2qa, the electric dipole moment.