UY1: Electric Potential

The potential V at any point in an electric field is the potential energy U per unit charge associated with a test charge $q_{0}$ at that point:

$$V = \frac{U}{q_{0}}$$

$$\Delta U = q_{0} \Delta V$$

The SI unit of electric potential is volt.

$$1 V = 1 \, J C^{-1}$$

The work done per unit charge by the electric force when a charged body moves from a to b is equal to the potential at a minus the potential at b.

$$W_{a \rightarrow b} = \, – (U_{b} – U_{a})$$

$$\begin{aligned} \frac{W_{a \rightarrow b}}{q_{0}} &= \, – \left( \frac{U_{b}}{q_{0}} – \frac{U_{a}}{q_{0}} \right) \\ &= \, – (V_{b} – V_{a}) \\ &= V_{a} – V_{b} \\ &= V_{ab} \end{aligned}$$

where $V_{ab}$ is the potential of a with respect to b.




Example: Oppositely Charged Parallel Plates

Recall that the electric potential energy of a charge in uniform electric field is:

$$U =  q_{0}Ey$$

Since $V = \frac{U}{q_{0}}$,

$$V = Ey$$

$$\begin{aligned} V_{ab} &= V_{a}-V_{b} \\ &= E(y_{a}-y_{b}) \\ &= Ed \end{aligned}$$

Hence, you get the “potential gradient”:

$$E = \frac{V_{ab}}{d}$$

Recall that $E = \frac{\sigma}{\epsilon_{0}}$ for oppositely charged parallel plate of infinite size, where $\sigma$ is the surface charge density of the plate. You get an expression for the surface charge density:

$$\sigma = \epsilon_{0} \frac{V_{ab}}{d}$$

The above equation is useful for solving problems related to parallel plates.




Electric Potential Of A Point Charge

Recall that: $W_{a \rightarrow b} = \int\limits_{a}^{b} \vec{F}.d\vec{l} = \int\limits_{a}^{b} q_{0}\vec{E}.d\vec{l}$ and $\frac{W_{a \rightarrow b}}{q_{0}} = V_{a} – V_{b} = \int\limits_{a}^{b} \vec{E}.d\vec{l}$


Hence, the potential V due to a single point charge q:

$$\begin{aligned} V &= \frac{U}{q_{0}} \\ &= \frac{1}{4 \pi \epsilon_{0}} \frac{q}{r} \end{aligned}$$

where r is the distance from the point charge q to the point at which the potential is evaluated.

In general, moving with the direction of $\vec{E}$ means moving in the direction of decreasing V, and moving against the direction of \vec{E} means moving in the direction of increasing V.


Next: Potential Gradient

Previous: Electric Potential Energy With Several Point Charges

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.