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In the derivation of potential energy of two-point charge system, we define the potential energy of the system as the work done in giving both charges to their present configuration, and while doing the derivation, we assume the work done in bringing the first charge to its desired configuration to be zero, but I think it shouldn't be like that. Why is the force = qE acting on the first charge due to the second charge being ignored while calculating the work done in bringing the first charge to the desired configuration?

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    $\begingroup$ If both charges are initially at infinity, and far from each other, there is negligible E-field on charge 1 as it is brought to the desired location. $\endgroup$
    – Puk
    May 28, 2023 at 21:47
  • $\begingroup$ Sir I think I am wrong in the basic definition, the definition that I mentioned is correct? $\endgroup$
    – user363737
    May 28, 2023 at 21:59
  • $\begingroup$ Your definition is fine. The point is while you are bringing in charge 1, there is no force against which you must do work, because charge 2 is so far away as this is happening and its force on charge 1 is negligible. $\endgroup$
    – Puk
    May 28, 2023 at 22:04
  • $\begingroup$ Sir does that mean I have to place first charge at - ∞ and second charge at + ∞, and thank you so much for helping me with so much patience. $\endgroup$
    – user363737
    May 28, 2023 at 22:21
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    $\begingroup$ That's one way to do it. Both charges could also be at $+\infty$ and still be infinitely far apart. Note also that space has 3 dimensions so there is a lot of room "at infinity". $\endgroup$
    – Puk
    May 28, 2023 at 22:47

1 Answer 1

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An important idea is that it does not matter how the charges arrived at their final positions after travelling from being an infinite distance apart, the final electric potential energy of the system of two charges will be the same.

It is much easier to do the sums by assuming that one charge is put into position with no external work being done, as there is no force due to an electric field being present on the charge, and then evaluate the external work done in bringing the other charge in from infinity.
One could sum the work done by external forces when both charges move together from their initial positions at infinity to find that it is the same as starting with one charge already moved to its final position.

Put another way, suppose there was an isolated charge in a region where there was no electric field present then there would be no force acting on it and so no work could be done by/on the charge and so no work would be needed to move the charge to infinity.

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