Define electric potential due to a point charge and arrive at the expression for the electric potential at a point due to a point charge.

Explanation

The electric potential at a point due to a given point charge may be measured by finding the amount of work required to bring a unit positive test charge , from a point at infinity to that point inside a field region.
Let .P. be a point at infinity . Let A,B and C be points inside the field . Let BC =dx. Let the displacement be .dx. and .+1C. is brought closer by .dx. in a direction opposite to the direction of electric field .
Amount work done =-Fdx
For .+1C. of charges , F=E and dW=dV where .dV. is electric potential defined as `dV=(dW)/(q_0)` where `q_0=+1C` of test charge . Electric field intensity at `C=(1/(4piepsilon_0))(q/x^2)` .

Hence , dV=-Edx. Electric potential at .A.
`V_A=int_oo^r dV=-int_oo^r (1/(4piepsilon_0))(q/x^2)dx`
i.e., `V_A=-(1/(4piepsilon_0))q((-1)/x)_oo^r`
i.e., `V_A=(1/(4piepsilon_0))q[1/r-1/oo]`
i.e., `V_A=(1/(4piepsilon_0))(q/r)`
Hence, electric potential at a point is inversely proportional to the distance of that point .