Electrostatic Energy

$W=k\sum _{i=1}^N\sum _{j>1}^N\frac{q_i{q}_j}{r_{ij}}=\frac{k}{2}\sum _{i=1}^N\sum _{j\ne 1}^N\frac{q_i{q}_j}{r_{ij}}$

• The work to assemble $N$ charges of magnitude $q_i$ from infinity to their distances from each other given by $r_{ij}$
• Double counting requires you to divide by 2

$W=\frac{1}{2}{\int }_V\rho V\mathrm{d}\tau$

• The work to bring all charges together

$W=\frac{{\epsilon }_0}{2}{\int }_{{\mathbb{R}}^n}{E}^2\mathrm{d}\tau$

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