Formulations in Electrostatics
Electrostatic problems are described by the Poisson's equation for scalar electric potential U (E = -gradU, E - electric field intensity vector). The equation for planar case is
\[ \frac{\partial}{\partial x} \left( ε_x \frac{\partial U}{\partial x} \right) + \frac{\partial}{\partial y} \left( ε_y\frac{\partial U}{\partial y} \right) = -ρ, \]and for axisymmetric case is
\[ \frac{1}{r} \frac{\partial}{\partial r} \left( ε_r r \frac{\partial U}{\partial r} \right) + \frac{\partial}{\partial z} \left( ε_z \frac{\partial U}{\partial z} \right) = -ρ, \]where components of electric permittivity tensor εx, εy or εz, εr and electric charge density ρ are constants within each block of the model.