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Poissons and Laplace equations
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Obtaining Poisson's equation is exceedingly simple, for from the point form of Gauss's law, $$\nabla \cdot \mathbf{D}=\rho_{v}$$ ...(1)

the definition of D

$$\mathbf{D}=\epsilon \mathbf{E}$$ ...(2)

and the gradient relationship, $$\mathbf{E}=-\nabla V$$ ...(3)

by substitution we have

$$\nabla \cdot \mathbf{D}=\nabla \cdot(\epsilon \mathbf{E})=-\nabla \cdot(\epsilon \nabla V)=\rho_{v}$$

or

$\nabla \cdot \nabla V=-\frac{\rho_{v}}{\epsilon}$ ...(4)

for a homogeneous region in which $\epsilon$ is constant.

Equation 4 is Poisson's equation, but the "double $\nabla^{\prime \prime}$ operation must be interpreted and expanded, at least in cartesian coordinates, before the equation can be useful. In cartesian coordinates.

\begin{aligned} \nabla \cdot \mathbf{A} &=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z} \\ \nabla V &=\frac{\partial V}{\partial x} \mathbf{a}_{x}+\frac{\partial V}{\partial y} \mathbf{a}_{y}+\frac{\partial V}{\partial z} \mathbf{a}_{z} \end{aligned}

and therefore

\begin{aligned} \nabla \cdot \nabla V &=\frac{\partial}{\partial x}\left(\frac{\partial V}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial V}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial V}{\partial z}\right) \\ &=\frac{\partial^{2} V}{\partial x^{2}}+\frac{\partial^{2} V}{\partial y^{2}}+\frac{\partial^{2} V}{\partial z^{2}} \end{aligned} ...(5)

Usually the operation $\nabla \cdot \nabla$ is abbreviated $\nabla^{2}$ (and pronounced "del squared"), a good reminder of the second-order partial derivatives appearing in $(5),$ and we have

$\nabla^{2} V=\frac{\partial^{2} V}{\partial x^{2}}+\frac{\partial^{2} V}{\partial y^{2}}+\frac{\partial^{2} V}{\partial z^{2}}=-\frac{\rho_{v}}{\epsilon}$ ...(6)

in cartesian coordinates.

If $\rho_{v}=0,$ indicating zero volume charge density, but allowing point charges, line charge, and surface charge density to exist at singular locations as sources of the field, then

$\nabla^{2} V=0$ ...(7)

which is Laplace's equation. The $\nabla^{2}$ operation is called the Laplacian of V

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