#### Fundamental Equations for the Electrostatic Potential
ems.ca.fe

- Gr 3.1

A generic configuration of static charges coupled via the Coulomb interaction defines an electrostatic problem, whose solution is in principle obtained from calculating either the field according to E_vcd

\begin{equation*} {\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} \end{equation*}

or (often simpler) by calculating the electrostatic potential, using either the explicit construction p_vcd

\[ \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}. \]

Alternately, we have also seen that the two fundamental equations for the electrostatic field, Gauss's law Gl_d and the vanishing curl condition curlE0 can be expressed as the single local differential condition (Poisson's equation) 🐟

\[ {\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}. \]

In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation Lap

\[ {\boldsymbol \nabla}^2 \phi = 0 \]

##### In this section:

- The Laplace Equation ems.ca.fe.L
- Green's Identities ems.ca.fe.g
- Uniqueness of Solution to Poisson's Equation ems.ca.fe.uP

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42