Give an expression for the electric potential at a point due to a point charge.

Explanation

Electric potential is defined as the work done in bringing a unit positive test charge from infinity to that point against the direction of the field.
Suppose a point-charge of +q coulomb is situated at a point O in a medium of dielectric constant K. Let P be a point, distant r from O, at which the electric potential is to be determined. For this, we must calculate the work done in bringing a test-charge from infinity to P. Suppose, a test-charge `+ q_0` is placed at point A, distant x from O, and away from P. By Coulomb.s law, the magnitude of the electric force `vecF` acting on `q_0` is given by
`F=1/(4piepsilon_0K)(q q_0)/x^2 ` newton ...(i)

The direction of `vecF` is away from O, If the test charge `+q_0` is displaced by dx to the point B, the work done
dW=F.(-dx)
Negative sign is due to the reason that displacement dx is opposite to the direction of force. Hence work done in bringing `+q_0` from infinity `(x=oo)` to the point P(x=r) is
`W=-int_(oo)^r F dx=-(q q_0)/(4piepsilon_0 K) int_0^r 1/x^2 dx`
`=-(qq_0)/(4piepsilon_0K)(-1/x)_oo^r = -(qq_0)/(4piepsilon_0K)(-1/r+1/oo)`
`W=1/(4piepsilon_0K) . (qq_0)/r`
Thus , the potential at P `V=W/q_0 =1/(4piepsilon_0K) q/r`
From vaccum (or air ) K=1 `V=1/(4piepsilon_0)q/r` volt.