# Energy of a system of point charges

I'm a little confused about something that came up in a lecture a while back. There's nothing about it any books or online seemingly...

The energy for bringing four point charges from infinity to a their relative positions was calculated.

This was done by saying the work to bring in the first is zero, the work to bring in the second is the potential of the first times the charge of the second, the work to bring in the third is the sum of the potentials of the first and second times the charge of the third, and the work to bring in the fourth is the sum of the potentials of the first, second and third. Then the total work U=U1+U2+U3+U4.

Then the next calculation was such that for each charge, there was a sum over the product of the net potential due to the three other charges with that specific charge, and then the total energy of the system W was simply the sum of these products for each of the charges. With a bit of algebra, you could show W=2U.

Now, I'm very confused as to what is actually going on here. I'm a little confused about what the quantites U and W actually represent, and I'm a bit concerned about energy conservation because I feel it implies the work to form the system doesn't equal the potential energy. I realise this may be a bit vague without the actual calculations so if need be I can post some more. If anybody could explain that would be great, thanks.

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mfb
Mentor
Energy conservation is fine. In the second case, you count the potential energy twice - potential energy comes from pairs of charges, not from individual charges. If you add the potential energy for the pair (charge1, charge2) twice (once for charge1 and once for charge2), you get twice the real result.

Energy conservation is fine. In the second case, you count the potential energy twice - potential energy comes from pairs of charges, not from individual charges. If you add the potential energy for the pair (charge1, charge2) twice (once for charge1 and once for charge2), you get twice the real result.
Yeah that makes sense, but do you see any point in this exercise? It effectively proves 2=2x1...

mfb
Mentor
It shows that you have to be careful, otherwise you could calculate 2 and think you had 1.

Another possibly related issue - if I have a charge near an infinite grounded conducting plate, then I can find the force acting on the charge using an image charge and the find the work done to move this to infinity by integrating the force with respect to distance. However if I try to calculate this work finding the potential due to the image charge at the charge of interest, and then multiply by the charge on the charge of interest, I recieve a result which is too large by a factor of two.

I know the latter method is flawed but I don't understand why - surely if the image charge system is set up correctly I should be able to calculate all quantities above the sheet using it...