 #1
cwill53
 214
 40
 Homework Statement:

What is the potential energy of an arrangement of eight negative charges on the corners of a cube of side b, with a positive charge in the center of the cube? Suppose each negative charge
is an electron with charge −e, while the central particle carries a double positive
charge, 2e?
 Relevant Equations:
 $$U=\frac{1}{2}\sum_{j=1}^{N}\sum_{k=j}^{}\frac{1}{4\pi \epsilon _{0}}\frac{q_{j}q_{k}}{r_{jk}}$$
This question is an example in Durcell's Electricity and Magnetism.
The solution goes as follows:
[In this case] there are four different types of pairs. One type involves the center charge, while the other three involve the various edges and diagonals of the cube. Summing over all pairs yields
$$U=\frac{1}{4\pi \epsilon _0}(8\cdot \frac{(2e^2)}{(\frac{\sqrt{3}}{2})b}+12\cdot \frac{e^2}{b}+12\cdot \frac{e^2}{\sqrt{2}b}+4\cdot \frac{e^2}{\sqrt{3}b})$$
Then the book introduces the following equation to sum over pairs
$$U=\frac{1}{2}\sum_{j=1}^{N}\sum_{k=j}^{}\frac{1}{4\pi \epsilon _{0}}\frac{q_{j}q_{k}}{r_{jk}}$$
I'm just a bit confused at the meaning of finding the electrical potential energy. Can someone explain what these equations mean intuitively and then how to apply the equation to the above example? Thanks.
The solution goes as follows:
[In this case] there are four different types of pairs. One type involves the center charge, while the other three involve the various edges and diagonals of the cube. Summing over all pairs yields
$$U=\frac{1}{4\pi \epsilon _0}(8\cdot \frac{(2e^2)}{(\frac{\sqrt{3}}{2})b}+12\cdot \frac{e^2}{b}+12\cdot \frac{e^2}{\sqrt{2}b}+4\cdot \frac{e^2}{\sqrt{3}b})$$
Then the book introduces the following equation to sum over pairs
$$U=\frac{1}{2}\sum_{j=1}^{N}\sum_{k=j}^{}\frac{1}{4\pi \epsilon _{0}}\frac{q_{j}q_{k}}{r_{jk}}$$
I'm just a bit confused at the meaning of finding the electrical potential energy. Can someone explain what these equations mean intuitively and then how to apply the equation to the above example? Thanks.