# UY1: Equipotential Surfaces

An equipotential surface is a three-dimensional surface in which the electric potential V is the same at every point.

Using the example of a single positive charge q, the expression for V is:

$$V = \frac{1}{4 \pi \epsilon_{0}} \frac{q}{r}$$

If a test charge $q_{0}$ is moved from point to point on an equipotential surface, the electric potential energy $q_{0}V$ will remain constant. In equation form, this means that the work done is 0:

\begin{aligned} W &= \, – \Delta U \\ &= \, – q_{0} \, \Delta V \\ &= 0 \end{aligned}

It follows that $\vec{E}$ must be perpendicular to the equipotential surface at every point. How do you reach this conclusion? Recall that:

\begin{aligned} dV &= \frac{\partial V}{\partial x} \, dx + \frac{\partial V}{\partial y} \, dy + \frac{\partial V}{\partial z} \, dz \\ &= \vec{\nabla} V . d\vec{l} \\ &= \, – \vec{E}.d\vec{l} \end{aligned}

Since dV = 0, $\vec{E}.d\vec{l}$ is 0 $\rightarrow$ perpendicular.

At each point, the direction of $\vec{E}$ is the direction in which V decreases most rapidly.

\begin{aligned} dV &= \vec{\nabla}V . d\vec{l} \\ &= \, – \vec{E} . d\vec{l} \\ &= \, – E \, dl \, \text{cos} \, \phi \end{aligned}

where $\phi$ is the angle between electric field and displacement vector (direction a point charge move).

In a region where an electric field is present, we can construct an equipotential surface through any point.

Note: Equipontial surfaces for different potentials can never touch or interact.

When all the charges are at rest (equilibrium), the electric field just outside a conductor must be perpendicular to the surface at every point. If the electric field contains a non-zero parallel component, there will be a force on the charges at the surface which will cause the charges to move and distribute themselves.

When all charges are at rest, the surface of a conductor is always an equipotential surface.

Next: Gauss’s Law (Simple Version)

Previous: Electric Potential Of An Infinite Line Charge

Back To Electromagnetism (UY1)

##### 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.

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