Electric potential due to an electric dipole in broad side on position or transversal position

“Electric potential due to an electric dipole in broad side on position or transversal 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 in its broad side on position at distance r from its midpoint O. let the electric constant of a medium around the dipole is K .

In right angle ΔAOP
By Pythagoras theorem
AP² = OP² + OA²
AP² = r² + a²
AP  = (r² + a²)½
Here,  AP = BP
Hence,  BP = (r² + a²)½

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

V1 = 1/4π£oK × +q/BP
V1 = 1/4π£oK × +q/(r²+a²)½    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/(r²+a²)½    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/(r²+a²)½ ] +
       [ 1/4π£oK × -q/(r²+a²)½ ]

V = [ 1/4π£oK × q/(r²+a²)½ ] --
       [ 1/4π£oK × q/(r²+a²)½ ]

V = 0

Hence, Electric potential due to electric dipole in its broad side position is zero