# Potential Energy of a System of Charges and in an External Field

Two point charge system contains charges q1 and qseparated by a distance r is given by

U = \frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1}q_{2}}{r}

Three point charge system U = \frac{1}{4 \pi \varepsilon_{0}} \cdot \left[\frac{q_{1}q_{2}}{r_{1}} + \frac{q_{2}q_{3}}{r_{2}} + \frac{q_{3}q_{1}}{r_{3}}\right]

### View the Topic in this video From 07:05 To 43:16

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1. Electric potential energy of a system of two charges is
U = \frac{1}{4\pi \varepsilon_{0}} \frac{q_{1}q_{2}}{r_{12}}

2. Electric field at the surface of a charged conductor
\overrightarrow{E} = \frac{\sigma}{\varepsilon_{0}}\hat{n}

3. Electric potential energy of a system of n point charges
U = \frac{1}{4 \pi \varepsilon_{0}} \sum_{all \ pairs} \frac{q_{j}q_{k}}{r_{jk}}

4. This work is stored as the potential energy of the system
U(\theta) = pE \cos\frac{\pi}{2} - \cos \theta = - pE \cos \theta = -p.E