Pre-Quantum Electrodynamics

Fundamental Equations for the Electrostatic Potential ems.ca.fe

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 (\ref{eq:V_from_rho})

\[ V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}. \tag{\ref{eq:V_from_rho}} \]

Alternately, we have also seen that the two fundamental equations for the electrostatic field, Gauss's law (\ref{Gr(2.14)}) and the no-perpetual-machine (vanishing curl) condition (\ref{Gr(2.20)}) can be expressed as the single 'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})

\[ {\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}. \tag{\ref{eq:Poisson}} \]

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

\[ {\boldsymbol \nabla}^2 V = 0 \tag{\ref{eq:Laplace}} \]

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Author: Jean-Sébastien Caux

Created: 2022-02-10 Thu 08:32