Electric Potential - Yousef's Notes
Electric Potential

Electric Potential

The inverse squared dependency of the electric force is crucial, as it allows as to write the electric field as the gradient a scalar field: the electric potential $V$ (also frequently denoted by $\varphi$) $$ \vec{E}(\vec{r}) = -\nabla V(\vec{r}) $$ This equation thus implies $$ V_{a}-V_{b}=\int_{a}^{b} \vec E \cdot \vec{d l}=\int_{a}^{b} E \cos \phi d l $$

Properties

  • minus sign for convenience
  • Since the electric field satisfies the superposition principle, and being nabla a linear operator, the electric potential fulfills superposition.
$$ V=V_1+V_2+V_3+\dots $$
  • the SI unit is one volt (1V) in honor of Alessandro Volta.
  • Alternative SI unit for electric field: volt per meter $1 \mathrm{V} / \mathrm{m}=1 \mathrm{N} / \mathrm{C}$
  • In circuits, a difference in potential from one point to another is often called voltage.

#Calculating Electric Potentials

The potential of a point charge located at $\vec{r}’$ is simply

$$ V(\vec{r})=\frac{1}{4 \pi \epsilon_{0}} \frac{q_{1}}{|\vec{r}-\vec{r}'|} $$ Invoking the superposition principle, any potential can be calculated as sum (integral) of contributions. $$ V(\vec{r})= \frac{1}{4 \pi \epsilon_{0}} \sum_{i=1}^{N} \frac{q_{i}}{|\vec{r}-\vec{r}_{i}|}+\frac{1}{4 \pi \epsilon_{0}} \int_{V} \frac{\rho\left(\vec{r}^{\prime}\right)}{|\vec{r}-\vec{r}^{\prime}|} d V^{\prime}\\ +\frac{1}{4 \pi \epsilon_{0}} \int_{S} \frac{\sigma\left(\vec{r}^{\prime}\right)}{|\vec{r}-\vec{r}^{\prime}|} d S^{\prime} $$

#Simple Cases

Charged cylinder Charged ring
![[Pasted image 20240306143852.png]] ![[Pasted image 20240306143859.png]]
$V=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{R}{r}$ $V=\frac{1}{4 \pi \epsilon_{0}} \frac{Q}{\sqrt{x^{2}+a^{2}}}$

Read and understand the derivation by yourself: Sears & Zemansky examples 23.10, 23.11

#Equipotential Lines

A equipotential surface is an imaginary surface such that the electric potential $V$ is constant over it.

Examples ![[Pasted image 20240306143941.png]]

Properties:

  • Field lines and equipotential surfaces are always perpendicular to each other.
  • $E$ doesn’t have to be constant over an equipotential surface.
  • The larger the electric field, the closer the equipotential surfaces are.
  • The surface of a conductor in equilibrium is always an equipotential surface. Actually, all the volume is equipotential.
    • This follows from the fact that the electric field is always perpendicular to the surface of a conductor.