How is the electric potential at a point due to a given charge measured? Obtain an expression for the electric potential at a point due to an isolated 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 positve 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=dc. Let the displacement be 'dx' and '+1C' is brought closer by 'dx' in a direction opposite to the direction Amount work done=-Fdx.
For '+1C' of charge, F=E and dW=dV where 'dV' is electric potential difined as `dV=(dW)/(q_(0))` where `q_(0)=+1C` of test charge. electric filed intesntiy at `C=((1)/(4piepsi_(0)))((q)/(x^(2)))`.

Hence dV=-Edx. electric potential at 'A'
`V_(A)=int_(oo)^(r)dV=-int_(oo)^(r)((1)/(4piepsi_(0)))((q)/(x^(2)))dx`
i.e., `V_(A)=-((1)/(4piepsi_(0)))q((-1)/(x))_(oo)^(r)`
i.e., `V_(A)=((1)/(4piepsi_(0)))q[(1)/(r)-(1)/(oo)]`
i.e., `V_(A)=((1)/(4piepsi_(0)))((q)/(r))`.
Hence electric potential at a point is inversely proportional to the distance of the point.