Poisson's Equation in Electrostatics

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A detailed form of the Poisson's Equation[1] in Electrostatics is:


\frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial V(x,y,z)}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial V(x,y,z)}{\partial y} \right) 
+ \frac{\partial}{\partial z}\cdot \left(\varepsilon_{z} \cdot \frac{\partial V(x,y,z)}{\partial z} \right)  + \rho_v(x,y,z)=0


with:

D_x = -\varepsilon_{x} \frac{\partial V(x,y,z)}{\partial x} \quad D_y = -\varepsilon_{y} \frac{\partial V(x,y,z)}{\partial y} \quad D_z = -\varepsilon_{z} \frac{\partial V(x,y,z)}{\partial z}
E_x = -\frac{\partial V(x,y,z)}{\partial x} \qquad E_y = -\frac{\partial V(x,y,z)}{\partial y} \qquad E_z = -\frac{\partial V(x,y,z)}{\partial z}


References

  1. Poisson's Equation
  2. Electrostatics Summary
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