Answer to Question #144971 in Electricity and Magnetism for Khansa

Question

: Devise a strategy to derive an expression of Electric Field at a point R due to an Electric Dipole without using the expression of Electric Scalar Potential. You can use either Coulumb’s or Gauss’s Law.

Answer 1

As point P is very far from the dipole, given dipole can be considered as a point dipole.

From Coulumb’s Law for a point dipole electric field on axial line is

$E_a = \frac{2kp}{R^3}$

p = qd

Electric field on equatorial line is

$E_e = -\frac{kp}{R^3}$

$E_2 = \frac{2kpcosθ}{R^3}$

$E_1 = -\frac{kpsinθ}{R^3}$

The net field at P is

$E = \sqrt{E_1^2 + E_2^2} = \frac{kp}{R^3}\sqrt{sin^2θ + 4cos^2θ} = \frac{kp}{R^3}\sqrt{1 + 3cos^2θ}$