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 electrostatic potential at any point in an electrostatic field is defined as the work done in carrying a unit positive charge from infinity to that point against the field.

Consider a point Charge + Q at O.
Let P be a point at a distance r from O.
The magnitude of electrostatic force on unit positive charge at A is,
`F = (1)/(4pi varepsilon_(0)) (Q times 1)/(x^(2))`
`F = (Q)/(4pi varepsilon_(0) x^(2))` along OP
work done in moving a unit positive chargea from A to B through a small distance dx is,
`triangle W = vec F . d vec x`
`= F dex cos 180^(@)`
`triangle W = - Fdx`
`triangle W = (-Q)/(4 pi varepsilon_(0) x^(2)) dx`
Total workdone in moving a unit positive change from infinity to the point P is,
`W =int_(oo)^(r) (-1)/(4 pi varepsilon_(0)) (Q)/(x^(2)) dx`
`= - (Q)/(4 pi varepsilon_(0)) int_(oo)^(r) (1)/(X^(2))dx`
`W = - (Q)/(4 pi varepsilon_(0)) [-(1)/(X)dx]_(oo)^(r)`
`=(Q)/(4 pi varepsilon_(0)) [(1)/(r)-(1)/(oo)]`
`W=(Q)/(4 pi varepsilon_(0)r)`
By definition, this done is equal to the electrostatic potential (V) at that point due to Q.
`V=(1)/(4 pi varepsilon_(0)) (Q)/(r)`