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Determine the electric potential energy for a system of three-point charges.

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Electric Potential Energy of a System of Charges
Electric potential energy of a system of charges is equal to the amount of work done in forming the system of charges by bringing them at their particular positions from infinity without any acceleration and against the electrostatic force. It is denoted by U.U=W=qV(r)
(a) Electric potential energy of system of two charges:
Suppose two charges +q1 and +q2 are situated at a distance r. We have to find out the electrical potential energy of this system.
When the charge q1 is brought from infinity to in its position, no work is done because there is no other charge to repel or attract it. The electric potential due to this charge q1 at a distance i.e., in the position of charge q2
V1=14πε0q1r
Therefore the work done in bringing the charge q2 from infinity to its own position i.e., the electric potential energy of the system.
W=U=V1q2
or =14πε0q1q2r(1)
If both the charges are of same nature, the potential energy will be positive and for unlike charges it will be negative. Therefore during the process of evaluation of potential energy, the value of charge should be used with its sign.
(b) Electric potential energy of a system of three charges:
Let us first of all consider a system of three point charges q1,q2 and q3. Such that q2 and q3 are initially at infinite distance from the charge (q1) [In figure 3.18(a)], Work done to bringing the charge q2 from infinity to point B
W12=14πε0q1q2r12(2)
Work done bringing charge q3 (In the presence of charge q1 and q2 ) from infinity to point C
W123=V1q3+V2q3=14πε0q1q3r31+14πε0q2q3r23(3)
Now the total work done to bringing all the three charges from infinity to their respective position is given by
W=W12+W123=14πε0q1q2r12+14πε0+q1q3r31+14πε0q2q3r23=14πε0[q1q2r12+q1q3r31+q2q3r23]
This work done is stored in the form of potential energy
U=W= potential energy of three system of
=14πε0[q1q2r12+q1q3r31+q2q3r23]
or U=1214πε03i=13j=1,ijqiqjrij
(c) Electric potential energy due to four system of charges:
Suppose there are four charges in a system of charges, situated as shown in figure (3.18 (b)).
The total energy of this system can be obtained by adding the potential energies of each possible pair.
Hence U=U12+U23+U34+U41+U13+U24
=14πε0[q1q2r12+q2q3r23+q3q4r34+q4q1r41+q1q3r13+q2q4r24]
Similarly, the electrical potential energy of the system of n charges can be obtained. We can write theequivalent expression as under,
U=14πε0All pairsqiqjrij(4)
To obtain this sum, we have to use each pair of charge only once. Therefore the general formula forelectrical potential energy will be as under
U=12ni=1nj=1,ij14πε0qiqjrij
Here we have to multiply the expression by 12 because in this expression each pair comes two times.

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