Electric potential due to electric dipole in end on position or Axial position or longitudinal positions

“Electric potential due to electric dipole in end on position or Axial position or longitudinal position”:

Consider AB  is electric dipole made up by two point charges -q and +q and of length 2a. Thus its dipole moment is  P = 2a.q
 we are to find the electric potential due to this dipole in axial position at distance r from its midpoint O. let the electric constant of a medium around the dipole is K .

Electric potential due to +q charge of electric dipole at point P is ---

V1 = 1/4π£oK × +q/BP
V1 = 1/4π£oK × +q/(r-a)   volt


similarly,   Electric potential due to -q charge of electric dipole at point P is-----

V2 = 1/4π£oK × -q/AP
V2 = 1/4π£oK × -q/(r+a)     volt

Thus,  the total electric potential due to dipole at point P will be ---
V = V1 + V2

Putting value of V1 and V2

V = [ 1/4π£oK × +q/(r-a) ] +
       [ 1/4π£oK × -q/(r+a) ]

V = q/4π£oK [ 1/(r-a) - 1/(r+a) ]

V = q/4π£oK [ (r+a - r+a)/ (r+a)(r-a) ]

V = q/4π£oK [ 2a /(r² - a²) ]

V = 1/4π£oK [ 2a.q/ (r² - a²)]

(Since, Electric dipole moment, 

            P = 2a.p)

Hence, V = 1/4π£oK × P/(r² - a²)]

By the definition of electric dipole 

     a<<r  hence.    a²≈0

Hence,  V =  1/4π£oK × P/(r² - 0)]

            [ V =  1/4π£oK × P/r² ]  volt