ELECTRIC POTENTIAL ENERGY AND POTENTIAL DIFFERENCE

A stationary electric charge at infinity has no energy ( kinetic or potential ) associated with it. If a unit positive charge is brought from infinity to an arbitrary point P in the electric field such that it has no velocity at that point then, the field being conservative, work done on it is stored with it in the form of potential energy and is called the electric potential of the point P and is given by

                P

 V (P) = - ∫ q E . dl

               ∞

If the electric charge is of magnitude q instead of unity, then the work done is called the potential energy of the charge q at point P and is given by

                               P

  U (P) = qV (P) = - ∫ q E . dl

                              ∞

The original electric field or the arrangement of charges in the field should remain unaffected by bringing the electric charge q or the unit charge from an infinite distance to the point in the electric field.

Generally, one needs to calculate the potential difference or the difference in potential energy of charge q when it is moved from P to Q, which is given by

                            Q

 U (Q) - U (P) = - ∫ q E . dl

                            P

It should be noted that this potential energy or the potential energy change is associated not only with the charge q but also with the entire charge distribution which gives rise to the electric field.