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 in its vicinity.
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 electrostatic force of the field.Consider a point Charge +Q at O.Let P be a point at a distance r from O.Consider a point A at a distance x from O.The magnitude of electrostatic force on unit-positive charge at A, isF=14πε0Q×ix2F=Q4πε0x2 along OPWork Done in moving a unit positive charge from A to B through a small distance dx is,ΔW=→F.d→x=Fdxcos180oΔW=−FdxΔW=−Q4πε0x2dxTotal work done in moving a unit positive change from infinity to the point P is,W=∫r∞−14πε0Qx2dx=−Q4πε0x2∫r∞1x2dx
W=−Q4πε0[1x]r∞=Q4πε0[1r−1∞]
W=Q4πε0r
By definition, this work done is equal to the electrostatic potential (V) at that point due to Q.
V=14πε0Qr