# Total electric potential energy

1. Nov 24, 2015

### betelgeuse91

1. The problem statement, all variables and given/known data
My text book states that for point charges $q_1, q_2, ... ,$ where distance between $q_i$ and $q_j$ is $r_{ij},$ the total potential energy U is the sum $U = \dfrac{1}{4\pi\epsilon_0} \sum_{i<j} \dfrac{q_iq_j}{r_{ij}}$ and specifically mentions not to count them twice as (i,j) and (j,i), but I don't see why.

2. Relevant equations
Potential energy of a point charge $q_i$ is $U_i = \dfrac{q_i}{4\pi\epsilon_0}\sum_{i \neq j}\dfrac{q_j}{r_j}$

3. The attempt at a solution
Using the above equation, I think the total potential energy should be $U = \dfrac{1}{4\pi\epsilon_0} \sum \dfrac{q_iq_j}{r_{ij}}$, counting both cases (i,j) and (j,i). For instance, when there are two point charges $q_1$ and $q_2$, the potential energies of individuals is $U_1 = \dfrac{q_1}{4\pi\epsilon_0}\dfrac{q_2}{r_{12}}$, $U_2 = \dfrac{q_2}{4\pi\epsilon_0}\dfrac{q_1}{r_{21}}$ so that the total potential energy is the sum $U = U_1 + U_2 = \dfrac{1}{4\pi\epsilon_0} \sum_{1\leqslant i,j\leqslant2} \dfrac{q_iq_j}{r_{ij}}$ which counts both the cases (1,2) and (2,1).
Where am I wrong?

2. Nov 24, 2015

### betelgeuse91

This condition should add to the last equation: $i\neq j$

3. Nov 25, 2015

### J Hann

Suppose you have only 2 charges.
The work done in moving charge 1 into place is zero.
Then you can move charge 2 into place by doing work that depends on R12.
So you don't need R21, since you would be counting the work twice.