# Electric potential and potential due to point charge

## 3. Electric Potential

• We now move towards the electric potential which is potential energy per unit charge.
• Thus electrostatic potential at any point of an electric field is defined as potential energy per unit charge at that point.
• Electric potential is represented by letter V.
V=U/q' or U=q'V                                             (6)
• Electric potential is a scalar quantity since both charge and potential energy are scalar quantities.
• S.I. unit of electric potential is Volt which is equal to Joule per Coulumb. Thus,
1 Volt = 1 JC-1
• In equation 4 if we divide both sides by q' we have

where V(r1) is the potential energy per unit charge at point R and V2) is potential energy per unit charge at point S and are known as potential at points R and S respectively.
• Again consider figure 1. If point S in figure 1 would be at infinity then from equation 7

Since potential energy at infinity is zero therefore V(∞)=0. Therefore

hence electric potential at a point in an electric field is the ratio of work done in bringing test charge from infinity to that point to the magnitude of test charge.
• Dimensions of electric potential are [ML2T-3A-1] and can be calculated easily using the concepts of dimension analysis.

## 4. Electric potential due to a point charge

• Consider a positive test charge +q is placed at point O shown below in the figure.

• We have to find the electric potential at point P at a distance r from point O.
• If we move a positive test charge q' from infinity to point P then change in electric potential energy would be

• Electric potential at point P is

• Potential V at any point due to arbitrary collection of point charges is given by

• here we see that like electric field potential at any point independent of test charge used to define it.
• For continous charge distributions summation in above expressin will be replaced by the integration

where dq is the differential element of charge distribution and r is its distance from the point at which V is to be calculated.