1.17 Electrostatic Potential Energy

        To get a clear idea about electrostatic potential energy, consider the field E due to a charge Q placed at the origin. Now, imagine that we bring a positive test chrge q from a point R to a point P against the repulsive force on it due to the charge Q.
         
             In this situation, work done by the external force(Fex) is the negative of the work done by the electric force, and gets fully stored in the form of potential energy of the charge q. Thus, work done by external forces in moving a charge q from R to P is

                   Wrp= Integral limit from R to P Fext.dr
                             (note here that this displacement is in an opposite sense to the electric force (E) and hence work done by electric field is negative,. This work done is against electrostatic repulsive force and gets stored as potential energy.

           At every point in electric field, a particle with charge q possesses a certain electrostatic potential energy, this work done increases its potential energy by an amount equal to potential energy difference between points R and P. Thus, potential energy difference.

       Therefore, we can define electric potential energy difference between two points as the work required to be done by an external force in moving (without accelerating) charge q from one point to another.

        We have discussed potential energy and potential energy difference, Which is more significant, potential energy or potential energy difference?
          
          Equation (2) defines potential energy difference in terms of the physically meaningful quantity work. The actual value of potential energy is not physically significant; it is only the difference of potential energy that is significant. To clarify this, let us add a constant value alfa to Up and Ur and find the difference

               This shows that, we can always add an arbitrary constant to potential energy at every point, since this will not change the potential energy difference:

                 The above argument gives a freedom in choosing the point where potential energy is zero without changing the value of potential energy difference. Potential energy at infinity is zero( We will learn this in next section). With this choice, if we take the point R at infinity, we get it.
       
             Since the point P is arbitrary, provides us with a definition of potential energy of a charge q at any point. Potential energy of charge q at a point ( in the presence of field due to any charge configuration) is the work done by the external force (equal and opposite to the electric force) in bringing the charge q from infinity to that point.