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5.1 Poisson equation

Assuming a linear, isotropic and homogeneous medium, Gauss' law is written as follows:

$\displaystyle \v{\nabla} \cdot \v{E} = \frac{\rho}{\epsilon}$ (5.1)

In the absence of a changing magnetic field, $ \v{B} $, Faraday's law of induction gives:

$\displaystyle \v{\nabla} \times \v{E} = -\dfrac{\partial \v{B}} {\partial t} = 0$ (5.2)

Since the curl of the electric field is zero, it is defined by a scalar electric potential field $ \phi$:

$\displaystyle \v{E} = -\v{\nabla} \phi$ (5.3)

Substituting $ \v{E} $ provides us with a form of the Poisson equation:

$\displaystyle \v{\nabla} \cdot \v{\nabla} \phi = {\v{\nabla}}^2 \phi = \Delta \phi = -\frac{\rho}{\epsilon}.$ (5.4)

Where $ \Delta$ is the Laplace operator, which takes the following form in cartesian coordinates:

$\displaystyle \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}$ (5.5)

Equation 5.4 is a particular case of the inhomogeneous partial differential equation named Screened Poisson equation which with usual mathematical notation is the following:

$\displaystyle \left[ \Delta - \lambda^2 \right] u(\mathbf{r}) = - f(\mathbf{r})$ (5.6)

where $ \Delta$ is the Laplace operator, $ \lambda$ is a constant, $ f$ is an arbitrary function of position (known as the "source function") and $ u$ is the function to be determined. This equation is definned in unbounded space and is subject to the condition that $ u(r)$ vanishes sufficiy rapidly as $ r \rightarrow \infty $.

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