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Electricity and Magnetism

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Potential Energy of Many Equally Spaced Charges

EANDM-1S2FTH

There are several equally spaced, equally charged particles — charges which are all the same distance $r$ from one another. Assuming that the potential energy is zero when all charges are infinitely far apart from one another, the potential energy of the configuration is equal to:

$$U = N \cfrac{kq^2}{r}$$

…where $k = 8.99 \times 10^9 \ \rm{\cfrac{{N \cdot m}^2}{C^2}}$ is the electric force constant and $N$ is the number of non-repeating pairs of charges.

So long as the number of charges to consider is reasonably small, it is easy enough to directly count the number of such non-repeating pairs. The table below, for example, shows the value of the number of non-repeating pairs of charges $N$ for several small systems of $n$ charges.

n N
2 1
3 3
4 6
5 10


However, one can no longer simply count the number of pairs directly when the number becomes substantial.