# Pre-Quantum Electrodynamics
Author: Jean-Sébastien Caux

##### The Poisson Equation and the Laplace Equation
ems.es.ep.PL

Our two fundamental equations for the electrostatic field are \[ {\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0} \hspace{2cm} {\boldsymbol \nabla} \times {\bf E} = 0. \] For the electrostatic potential, Gauss' law becomes

The Poisson equation \[ {\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0} \label{eq:Poisson} \]

When the charge density vanishes, it becomes more simply

The Laplace Equation \[ {\boldsymbol \nabla}^2 V = 0 \label{eq:Laplace} \]

Since the curl of a gradient is always zero, we by construction have \({\boldsymbol \nabla} \times {\bf E} = - {\boldsymbol \nabla} \times ({\boldsymbol \nabla} V) = 0\).

Author: Jean-Sébastien Caux

Created: 2022-02-08 Tue 06:55

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