UY1: Electric Potential Energy With Several Point Charges



Suppose the electric field $\vec{E}$ in which charge $q_{0}$ moves is caused by several point charges $q_{1}$, $q_{2}$, $q_{3}$, … at distances $r_{1}$, $r_{2}$, $r_{3}$, … from $q_{0}$.

The potential energy associated with the test charge $q_{0}$ at point a is the algebraic sum:

$$\begin{aligned} U_{a} &= \frac{q_{0}}{4 \pi \epsilon_{0}} \left( \frac{q_{1}}{r_{1}} + \frac{q_{2}}{r_{2}} + \frac{q_{3}}{r_{3}} + … \right) \\ &= \frac{q_{0}}{4 \pi \epsilon_{0}} \sum\limits_{i} \frac{q_{i}}{r_{i}} \end{aligned}$$

Note that the force experienced by the test charge is:

$$\begin{aligned} \vec{F} &= \sum\limits_{i} \vec{F}_{i} \\ &= \frac{q_{0}}{4 \pi \epsilon_{0}} \sum\limits_{i} \frac{q_{i}}{r_{i}^{2}} \hat{r}_{i} \end{aligned}$$

 

If we start with charges $q_{0}$, $q_{1}$, $q_{2}$, $q_{3}$, … all separated from each other by infinite distances and then bring them together so that the distancce between $q_{i}$ and $q_{j}$ is $r_{ij}$, the total potential energy is:

$$\begin{aligned} U &= \frac{q_{0} q_{1}}{4 \pi \epsilon_{0} r_{01}} + \frac{q_{0} q_{2}}{4 \pi \epsilon_{0} r_{02}} + \frac{q_{1} q_{2}}{4 \pi \epsilon_{0} r_{12}} + … \\ &= \frac{1}{4 \pi \epsilon_{0}} \sum\limits_{i < j} \frac{q_{i} q_{j}}{r_{ij}} \end{aligned}$$

 

Next: Electric Potential

Previous: Electric Potential Energy

Back To Electromagnetism

Back To University Year 1 Physics Notes



Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.



Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.