Energy of a system of point charges

• fayled
In summary, the lecturer was discussing the energy required to move charges from a specific location to another, and how this can be calculated. He said that there are two cases to consider - one where the charges are at a specific location and one where they are at infinity. In the first case, the work to bring the charges in is zero, and in the second case, the work to bring in the charges is the potential energy of the charges times the charge of the second charge. The total work is then the sum of these potential energies for each charge. He also mentioned that there is a flaw in the second case - if the image charge system is set up correctly, the calculations should be able to be done using it.
fayled
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 realize 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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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.

mfb said:
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...

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 receive 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...

1. What is the energy of a system of point charges?

The energy of a system of point charges is the amount of work that must be done to bring the charges together from an infinite distance apart to their current configuration. It is a measure of the potential energy stored in the system due to the interactions between the charges.

2. How is the energy of a system of point charges calculated?

The energy of a system of point charges can be calculated using the formula E = kq1q2/r, where k is the Coulomb constant, q1 and q2 are the magnitudes of the charges, and r is the distance between them. This formula is known as the Coulomb's law for potential energy.

3. Is the energy of a system of point charges always positive?

No, the energy of a system of point charges can be positive, negative, or zero. It depends on the configuration of the charges and their relative positions. A positive energy indicates that the charges repel each other, while a negative energy indicates that the charges attract each other.

4. Can the energy of a system of point charges change?

Yes, the energy of a system of point charges can change if the positions or magnitudes of the charges are altered. For example, if two like charges are brought closer together, the energy of the system will increase, while if two opposite charges are brought closer together, the energy will decrease.

5. How does the energy of a system of point charges affect its stability?

The energy of a system of point charges can be used to determine the stability of the system. If the energy is positive, the system is unstable and the charges will repel each other. If the energy is negative, the system is stable and the charges will attract each other. Zero energy indicates a neutral or balanced system.

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