PX157 - B7b - potential energy and potential of two point charges

F_{12} = \frac{q_{1} q_{2}}{4 π ϵ_{0}} \frac{\hat{r}}{r^{2}}

the path is described by position: $\vec{r} = r \hat{r} ⟹ d \vec{l} = d \vec{r} = d r \hat{r}$

\begin{aligned} W_{a \to b} & = \int_{r_{a}}^{r_{b}} \frac{q_{1} q_{2}}{4 π ϵ_{0}} \frac{\hat{r}}{r^{2}} \cdot \hat{r} d r \\ = \frac{q_{1} q_{2}}{4 π ϵ_{0}} \int_{r_{a}}^{r_{b}} \frac{1}{r^{2}} d r \\ = \frac{q_{1} q_{2}}{4 π ϵ_{0}} (\frac{1}{r_{a}} - \frac{1}{r_{b}}) \\ W_{a \to b} & = U_{a} - U_{b} \\ ⟹ U_{a} & = \frac{q_{1} q_{2}}{4 π ϵ_{0} r_{a}} \end{aligned}

defining that potential energy tends to zero as $r \to \infty : lim_{r_{b} \to \infty} U_{12} (b) = 0$ :

⟹ U_{12} = \frac{q_{1} q_{2}}{4 π ϵ_{0} r}

note: $U_{12} = U_{21}$ , because ${\vec{F}}_{12} = - {\vec{F}}_{21}$ , and $d {\vec{l}}_{12} = - d {\vec{l}}_{21}$
if there are $N$ charges interacting with a charge, $q_{0}$ , total potential energy:

U_{0} = \sum_{i = 1}^{N} U_{i 0} = \sum_{i = 0}^{N} \frac{q_{i} q_{0}}{4 π ϵ_{0} | {\vec{r}}_{0} - {\vec{r}}_{i} |}