Voltage

Any vector with 0 curl can be written as the gradient of another function

Topics

• Electrostatic Energy
• Poisson’s Equation
• Normal Derivative
• Dipoles
• Multipole
• Retarded Potentials

$V(\vec{r})=-{\int }_{\mathcal{O}}^{\vec{r}}\vec{E}\cdot \mathrm{d}\vec{l}$

• Voltage at a point $\vec{r}$ due to an electric field $\vec{E}$
• $\mathrm{d}\vec{l}$ is any path element that leads from $\mathcal{O}$ to $\vec{r}$
• $\mathcal{O}$ is a reference point that is often set to infinite distance away by convention

$V(b)-V(a)=-{\int }_a^b\vec{E}\cdot \mathrm{d}\vec{l}$

• Voltage difference between points $a$ and $b$

$\vec{E}=-\nabla \vec{V}$

• Negative sign is from convention

$V(\vec{r})=k\int \frac{\rho ({\vec{r}}^{\prime })}{r}\mathrm{d}{\tau }^{\prime }$

• Voltage at a point $\vec{r}$ of a volume charge with charge density $\rho ({\vec{r}}^{\prime })$