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$$U=\frac{q_0}{4\pi\epsilon_0}\left(\frac{q_1}{r_1}+\frac{q_2}{r_2}+\frac{q_3}{r_3}+\dots\right)=\frac{q_0}{4\pi\epsilon_0}\sum_i\frac{q_i}{r_i}$$

$$U=\frac1{4\pi\epsilon_0}[C]\sum_{i<j}\frac{q_iq_j}{r_{ij}}$$

How I understand the first equation. Is its the potential energy of $q_0$ at its location due to the other charges around it with zero potential being at infinity, but I don't get the difference how this is different from the second equation.

Basically my question is what is the difference in meaning behind the two equations for electrical potential energy?

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    $\begingroup$ Hello! I have edited your question using MathJax (LaTeX) math typesetting. For future questions, you can refer to MathJax basic tutorial and quick reference. Thanks! $\endgroup$
    – jng224
    Nov 8, 2021 at 20:24
  • $\begingroup$ Is a minus sign missing? $\endgroup$
    – R. Romero
    Nov 8, 2021 at 22:21

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The second equation is the total electrostatic potential energy of a system of charges $q_1$, $q_2$, $\ldots$ at positions $\vec{r}_1$, $\vec{r}_2$, $\ldots$ so that the distance between charges $i$ and $j$ is $r_{ij} = |\vec{r}_i - \vec{r}_j|$. This is equivalent to the work done in assembling the charge configuration by moving the charges to their respective positions from infinitely far away.

The first equation is the change in potential energy caused by placing an additional charge $q_0$, given that we have already assembled a configuration of charges $q_1$, $q_2$, $\ldots$. This is equivalent to the work done on charge $q_0$ moving it into position given that the other charges are already in present.

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