Voltage

#Physics
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

$\displaystyle V(\vec{r})=-\int_{\mathcal{O}}^{\vec{r}} \vec{E}\cdot , \mathrm{d}\vec{l}$

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

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

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

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

• Negative sign is from convention

$\displaystyle V(\vec{r})=k\int \frac{\rho(\vec{r}')}{\mathscr{r}} , \mathrm{d}\tau'$

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