Derive an expression for electrostatic potential due to a point charge .

Explanation

Let us consider a positive charge q.

The electric potential at the point P is given by
`V = int_oo^r-vecE.dvecr = -int_oo^rvecE.dvecr`
Electric field due to positive point charge q is given by
`vecE = 1/(4piepsilon_0)q/r^2 hatr`
`V = (-1)/(4piepsilon_0)int_oo^rq/r^2 hatr.dr`
The infinitesimal displacement vector, `dvecr = drvecr` using `hatr.hatr=1` we get,
`V = (-1)/(4piepsilon_0)int_oo^rq/r^2 hatr.drhatr`
`= -1/(4piepsilon_0)int_oo^rq/r^2dr`
After the integration,
`V = (-1)/(4piepsilon_0)q{-1/r}_oo^r = 1/(4piepsilon_0)q/r`
Hence the electric potential due to a point charge q at a distance r is given by
`V = 1/(4piepsilon_0)q/r`