Electric Field due to an Electric Dipole at any point

Consider a point R having polar R having polar coordinates r and θ. O is the centre vector p into two components, one parallel and the other perpendicular to OR. The point R is on the axis of a dipole of moment p cos θ and on the equatorial line of a dipole of moment p sin θ and on the equatorial line of a dipole of moment p sin θ.



Field due to the component p cos θ } E1 =    2p cos θ
                   __________________________4π ε₀ r³

Field due to the component p sin θ } E2 =    p sin θ
                    _________________________4π ε₀ r³

. : Magnitude of the resultant field

E = √E²₁ + E²₂ =     ___p___ √ (4 cos² θ + sin² θ)
        _____________4π ε₀ r³

= ___p___ √ 1 + 3 cos² θ
    4π ε₀ r³

The resultant field E makes an angle Ф with OR.

tan Ф =     p sin θ / (4π ε₀ r³) = ½ tan θ
    _____2p cos θ / (4π ε₀ r³)