Topic 0002: Electrostatic Divergence

In this topic we shall develop the vector equations for electrostatic divergence in Cartesian co-ordinates.

Divergence relates the nett electric flux crossing a surface to the charge enclosed within it.

In principle the divergence equation allows us to calculate the capacitance of complex electrode structures, such as microstrip and stripline transmission lines and PCB tracks in general. In reality it is only possible to solve the equations for a few simple conductor geometries. For more complex conductor systems and geometries we obtain numerical solutions using field plotting software. An understanding of the concepts of divergence and the meaning of the divergence equations allows us to use such software effectively.

Electric Flux Crossing a Small Area

Imagine a stationary point charge q as shown in Fig 1. Static electric flux surrounds the charge.

Figure 1 The Electric Flux Crossing a Small Area dS

$\begin{gathered} {\text{ The elecrtic flux density at any point is related to the electric field strength by: - }} \hfill \\ \hfill \\ {\mathbf{D}} = \varepsilon _0 {\mathbf{E}}{\text{ }}..........................................................................................................{\text{1}}{\text{.0}} \hfill \\ {\text{ }} \hfill \\ {\text{The amount of electrtic flux crossing the small surface is given by: - }} \hfill \\ d\psi = {\mathbf{D}}.d{\mathbf{S}}{\text{ where }}d\psi {\text{ is a scalar quantity}} \hfill \\ \hfill \\ {\text{The amount of flux crossing the area of the sphere of radius r is given by: - }} \hfill \\ \psi = \oint\limits_S {{\mathbf{D}}.d{\mathbf{S}}} {\text{ }}....................................................................................................{\text{2}}{\text{.0}} \hfill \\ {\text{This reduces to: - }} \hfill \\ \psi = \oint_S {D_n } dS{\text{ where D}}_{\text{n}} {\text{ is the component of D normal to the surface dS }}.....{\text{3}}{\text{.0}} \hfill \\ \psi = \oint\limits_S {\varepsilon _0 E_n } dS \hfill \\ \psi = \oint\limits_S {\varepsilon _0 \frac{q} {{4\pi \varepsilon _0 r^2 }}dS} \hfill \\ \psi = \frac{q} {{4\pi r^2 }}\oint\limits_S {dS{\text{ }}} {\text{because r is constant for the integration process}} \hfill \\ \psi {\text{ = }}\frac{q} {{4\pi r^2 }}4\pi r^2 \hfill \\ \psi = q{\text{ }}..............................................................................................................{\text{4}}{\text{.0}} \hfill \\ {\text{This is an important result: the amount of electric flux crossing any closed surface}} \hfill \\ {\text{surrounding a volume v = equal to the amount of charge enclosed within the volume}}{\text{.}} \hfill \\ \hfill \\ {\text{Electric flux has the dimensions of charge}}{\text{.}} \hfill \\ \end{gathered}$

$\begin{gathered} {\text{Now assume that the charge q is distributed in a volume dV with a }} \hfill \\ {\text{volume distribution of }}\rho \hfill \\ {\text{Then }}q = \mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot} {\rho dV{\text{ = }}} \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {{\mathbf{D}}.d{\mathbf{S}} = q} {\text{ }}...................................................................{\text{5}}{\text{.0}} \hfill \\ {\text{In general terms, the total electric flux crossing a closed surface }} \hfill \\ {\text{surrounding a charged region = the charge enclosed and electric flux has}} \hfill \\ {\text{the dimesnsions of charge, as already stated}}{\text{.}} \hfill \\ \end{gathered}$

Now consider an imaginary small cube embedded in an electric field, as shown in Fig 2.

Figure 2 A Small Imaginary Cube Embedded in an Static Electric Field

$\begin{gathered} {\text{Considering the field in the direction of the x - axis, the flux entering the left hand face}} \hfill \\ {\text{of the cube is }}D_x dydx \hfill \\ \hfill \\ {\text{The flux leaving the right hand face of the cube is: - }} \hfill \\ \left( {D_x + \frac{{\partial D_x }} {{\partial x}}dx} \right)dydz \hfill \\ {\text{The net flux leaving the cube in the }}x{\text{ direction is: - }} \hfill \\ {\text{d}}\psi _x {\text{ = }}\left( {D_x + \frac{{\partial D_x }} {{\partial x}}dx} \right)dydz - {\text{D}}_{\text{x}} dydz \hfill \\ d\psi _x = D_x dydz + \frac{{\partial D_x }} {{\partial x}}dxdydz - {\text{D}}_{\text{x}} dydz \hfill \\ d\psi _x = \frac{{\partial D_x }} {{\partial x}}dxdydz \hfill \\ {\text{We may write }}dxdydx{\text{ as }}dv,{\text{ a small element of volume}} \hfill \\ {\text{Then }}d\psi _x = \frac{{\partial D_x }} {{\partial x}}dv \hfill \\ {\text{We may repeat the above process for the other pairs of faces of the cube: - }} \hfill \\ d\psi _y = \frac{{\partial D_y }} {{\partial x}}dv \hfill \\ d\psi _z = \frac{{\partial D_z }} {{\partial x}}dv \hfill \\ {\text{The total flux leaving the cube is then given by: - }} \hfill \\ {\text{d}}\psi {\text{ = }}d\psi _x + d\psi _y + d\psi _z \hfill \\ d\psi = \frac{{\partial D_x }} {{\partial x}}dv + \frac{{\partial D_y }} {{\partial x}}dv + \frac{{\partial D_z }} {{\partial x}}dv \hfill \\ d\psi = \left( {\frac{{\partial D_x }} {{\partial x}} + \frac{{\partial D_y }} {{\partial x}} + \frac{{\partial D_z }} {{\partial x}}} \right)dv \hfill \\ \frac{{d\psi }} {{dv}} = \left( {\frac{{\partial D_x }} {{\partial x}} + \frac{{\partial D_y }} {{\partial x}} + \frac{{\partial D_z }} {{\partial x}}} \right) \hfill \\ {\text{The term }}\frac{{d\psi }} {{dv}} = {\text{ the charge density }}\rho {\text{ inside the cube}} \hfill \\ \left( {\frac{{\partial D_x }} {{\partial x}} + \frac{{\partial D_y }} {{\partial x}} + \frac{{\partial D_z }} {{\partial x}}} \right) = \rho {\text{ }}...................................................................................{\text{6}}{\text{.0}} \hfill \\ {\text{The term }}\left( {\frac{{\partial D_x }} {{\partial x}} + \frac{{\partial D_y }} {{\partial x}} + \frac{{\partial D_z }} {{\partial x}}} \right){\text{ is called the divergence of D or just }}div{\mathbf{D}} \hfill \\ \hfill \\ \end{gathered}$

$\begin{gathered} {\text{Note that }}D_x = \varepsilon E = \varepsilon \frac{{\partial V_x }} {{\partial x}} \hfill \\ {\text{So }}\frac{{\partial D_x }} {{\partial x}} = \frac{\partial } {{\partial x}}\left( {\varepsilon \frac{{\partial V_x }} {{\partial x}}} \right) = \varepsilon \frac{{\partial ^2 V_x }} {{\partial x^2 }} \hfill \\ {\text{The same process can be repeated for the other terms in the div equation: - }} \hfill \\ div{\mathbf{D}} = \varepsilon \frac{{\partial ^2 V_x }} {{\partial x^2 }} + \varepsilon \frac{{\partial ^2 V_y }} {{\partial y^2 }} + \varepsilon \frac{{\partial ^2 V_z }} {{\partial z^2 }} \hfill \\ {\text{Or}} \hfill \\ \rho {\text{ = }}\varepsilon \left( {\frac{{\partial ^2 V_x }} {{\partial x^2 }} + \frac{{\partial ^2 V_y }} {{\partial y^2 }} + \frac{{\partial ^2 V_z }} {{\partial z^2 }}} \right){\text{ }}..........................................................................{\text{7}}{\text{.0}} \hfill \\ {\text{If }}\rho \ne {\text{0, equation 7}}{\text{.0 is called Poissions equation}} \hfill \\ \hfill \\ {\text{If }}\rho = {\text{0, equation 7}}{\text{.0 is called Laplaces equation}} \hfill \\ \end{gathered}$

Field plotting software solves the above equations numerically, allowing us to calculate capacitance and other important electrical parameters.

Author: Ron Hood

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