Potential Energy Of Point Charges Smartphysics Homework

As Giorgio says, you are overcounting the terms when multiplying by 4. I would like to elaborate a little bit, just to show how this is more than just a math error. It is also a conceptual error:

You mention in your reasoning "The total potential energy = the sum of the PE of each of the 4 charges." But this is false. Not only that, but there is no such thing as "the PE of each of the 4 charges." The PE is not stored in the charges, but in the electric field. So one can speak of "the potential energy stored by adding a fourth charge to the other three charges" (this is the energy value which you multiplied by 4), but there is no physical reason that this would be 1/4 of the total potential energy of the electric field.

It is possible to solve this problem by considering one charge at a time, but to do so, you must "build" the charge configuration from scratch, starting with an isolated charge, then counting the added potential by adding a second charge, then the third, and finally the fourth. Each charge will add more potential than the one before it.

Or you can take the mathematically simpler approach shown in the second solution you photographed. It is just a bit harder to intuitively justify that method, but it is equivalent in the end.

Тогда почему они послали не профессионального агента, а университетского преподавателя. Выйдя из зоны видимости бармена, Беккер вылил остатки напитка в цветочный горшок. От водки у него появилось легкое головокружение. Сьюзан, подшучивая над ним, часто говорила, что напоить его не составляет никакого труда.

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