Consider a point charge #q#.
We are free to chose the origin anywhere we please. In this case, we do it at the location of the charge.
Thus, the field due to the charge at a distance #r# from the charge is by Coulomb's law,
#vec E = 1/(4piepsilon_0)q/r^2hatr# where #hatr# is the radially outward unit vector.
But, we know that electric potential #V# is negative gradient of electric field,
#vec E = -nablaV#
Thus, employing spherical polar coordinates,
#vec E = -(delV)/(delr)hatr#
#implies
1/(4piepsilon_0)q/r^2hatr = -(delV)/(delr)hatr#
Integrating,
#V = -int_r^prop 1/(4piepsilon_0)q/r^2dr#
#V = int_prop^r 1/(4piepsilon_0)q/r^2dr#
#implies V = q/(4piepsilon_0)[1/r - 1/prop]#
But, #1/prop = 0#
#implies V = q/(4piepsilon_0r)#
This is the expression for electrostaic potential due to a point charge #q# at a distance #r# from it.
