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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

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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.

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  • $\begingroup$ Your very welcome. If you found my answer useful you can upvote my answer and press the check, thereby making my answer the official answer, @yurihbss $\endgroup$
    – Cicero
    May 18, 2015 at 16:47
  • $\begingroup$ I needed to wait 4 minutes :P and I can't upvote yet $\endgroup$ May 18, 2015 at 16:48

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