Question #f7a31

1 Answer
Sep 4, 2017

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.