Overview
(Following the logic present in Griffiths) Starting with empty space, it requires work to be able to assemble a set of charges into a given configuration. The first charge can be placed without requiring any work. The next charge, moved in from infinity, will then require {% q_2V_1(r_{1,2}) %} units of work. Here we have
- {% V_1 %} is the electric potential due to the first charge
- {% r_{1,2} %} is the distance vector between charge 1 and charge 2
- {% q_2 %} is the amount of charge 2
Then, the work to assemble the first two charges can be written as
{% W = \frac{1}{4 \pi \epsilon_0} q_2 \frac{q_1}{r_{1,2}} %}
The work to bring in the {% n^{th} %} charge is then
{% W = \frac{1}{4 \pi \epsilon_0} q_n \sum_{i=1}^{n-1}(\frac{q_i}{r_{i,n}}) %}