Electrostatic Energy

#Physics

$\displaystyle W=k\sum_{i = 1}^{N}\sum_{j>1}^{N}\frac{q_{i}q_{j}}{\mathscr{r}{ij}}=\frac{k}{2}\sum{i = 1}^{N}\sum_{j\neq 1}^{N}\frac{q_{i}q_{j}}{\mathscr{r}_{ij}}$

• The work to assemble $\displaystyle N$ charges of magnitude $\displaystyle q_{i}$ from infinity to their distances from each other given by $\displaystyle \mathscr{r}_{ij}$
• Double counting requires you to divide by 2

$\displaystyle W=\frac{1}{2}\int _{V}\rho V , \mathrm{d}\tau$

• The work to bring all charges together

$\displaystyle W=\frac{{\varepsilon}{0}}{2}\int{\mathbb{R}^{n}} E^{2} , \mathrm{d}\tau$

• Integrate over all space to find the work needed to assemble charges