##### Poisson's and Laplace's Equations
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, by Emgp, Gauss's law becomes

- Gr (2.24)
- PM (2.72)

Poisson's equation \[ {\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0} \tag{🐟}\label{Poi} \]

When the charge density vanishes, it becomes more simply

- Gr (2.25)
- PM (2.75)

Laplace's Equation \[ {\boldsymbol \nabla}^2 \phi = 0 \tag{Lap}\label{Lap} \]

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

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42