where does the term half in the formula of electrostatic potential energy comes from?(system of point charges)

Question

Electrostatic potential energy stored in a system of point charges (from wikipedia)

The electrostatic potential energy $U_E$ stored in a system of N charges q1, q2, ..., qN at positions r1, r2, ..., rN respectively, is:

$U_\mathrm{E} = \frac{1}{2}\sum_{i=1}^N q_i \Phi(\mathbf{r}_i) = \frac{1}{2} \sum_{i=1}^N q_i \sum_{j=1}^{N(j\ne i)} k_e \frac{q_j}{r_{ij}}$

where does the term $1\over2$ comes from?

Thank you

Answer 1

It is really a matter of combinations. Potential energy is a feature of a system, so between two particles there is one potential energy. The summation however, will cadd the potential energy between two particles twice (e.g., $q_1\phi(\mathbf{r}_2)$ and $q_2\phi(\mathbf{r}_1)$). Hence, the one half term has to be introduced so that the potential energy of each pairing is not added twice.