Intensity of electric field due to electric dipole

1. Axial or Longitudinal Position

Suppose an electric dipole with dipole moment P. We want to calculate the electric field intensity at a point P located at a distance r from the centre o of the dipole.

Electric field intensity at point P due to +q charge

E₁ = [1/4π€0].q/(r+l)²]

Similarly,

Electric field intensity at point P due to -q charge

E₂ = [1/4π€0].q/(r-l)²]

Therefore,

The resultant electric field intensity at point P is

E = E₂-E₁

E = [1/4π€0].q/(r-l)²]-[1/4π€0].q/(r+l)²]

E = q/4π€0[1/(r-l)²-1/(r+l)²]

E = q/4π€0[{(r+l)²-(r-l)²}/(r²-l²)²]

E = q/4π€0[4rl/(r²-l²)²]

Therefore,

r >> l

Hence,

E = q/4π€0[4rl/r⁴]

E = 2/4π€0[2ql/r³]

E= 1/4π€0[2P/r³] (from electric dipole P=2ql)

2. Non-axis position of electric dipole

Consider an electric dipole whose dipole moment is P.

The electric field in the axisymmetric position at a point P situated at a distance r from the middle point o of the dipole is to be calculated.

Hence,

The electric field intensity at point P due to +q charge is

E₁ = [1/4π€0].q/(r²+l²)]  in the direction of PB.

Similarly,

The electric field intensity at point P due to -q charge is

E₂ = [1/4π€0].q/(r²+l²)]   in the direction of PA.

Hence,

The resultant electric field intensity at point P.

E = E₁cosθ + E₂cosθ

E₁ = E₂

E = 2E₁cosθ

E = 2.[1/4π €0].q/(r²+l²)].l/√(r²+l²)

E = [1/4π€0].P/(r²+l²)^3/2

Since l << r

E = [1/4π€0].p/r³

Note

Electric field intensity due to a point charge

E ∝ 1 /r²

Electric field intensity due to electric dipole

E ∝ 1/r³