Pre-Quantum Electrodynamics
Fundamental Equations for the Electrostatic Potentialems.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 Equationems.ca.fe.L
- Green's Identitiesems.ca.fe.g
- Uniqueness of Solution to Poisson's Equationems.ca.fe.uP
- Electrostatic Boundary Conditionsems.ca.fe.bc
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Created: 2024-02-27 Tue 10:31