explain the concept of electric potential energy. Derive an expression for potential energy of a system of two point charges .Generalise the expression for N discrete charges.

Electrostatic potential energy of a system of point charges is the total amount of work done in bringing the various charges to their respective positions from infinitely large mutual separations.
Let us consider a point charge q1 which is held at a point P1 with position vector r1 in space. Another point charge q2 is at infinite distance from q1. This is to be brought to the position P2(r2), where P1P2=r12

​Electric potential energy at P2 due to charge q1 at P1 is


V=14π0q1r12Now,Work done in carrying charge q2 from infinity to P2W=V× q2=14π0q1r12×q2This is stored in the system in the form of electrostatic potential energy. Therefore,U=W=14π0q1r12×q2=14π0q1q2r12The electrostatic potential energy of a system of N point charges,U=14π0all pairs qjqkrjkIn the above, we should include only one term for each pair of charges, ThusU=12j=1j=N    k=1 and j not equal to kk=N 14π0qjqkrjk

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