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== Static Modeling Methods ==
<math> \mathbf{E(r)} = - \nabla \Phi(\mathbf{r}) = \frac{1}{4\pi\epsilon} \int\int\int_V \frac{\mathbf{r - r^{\prime}} }{ | \mathbf{r - r^{\prime}} |^3 } \rho(\mathbf{r^{\prime}}) dv^{\prime} </math>
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== Static Fields Arising from Steady-State Conduction Currents ==
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In an Ohmic conductor, the current density is related to the electric field as follows:
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<math> \mathbf{J(r)} = \sigma \mathbf{E(r)} = -\sigma \nabla \Phi(\mathbf{r}) </math>
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where σ is the electric conductivity. On the other hand, the continuity equation for a stationary current requires no charge buildup or decay inside a closed region. This means that
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<math> \nabla . \mathbf{J(r)} = 0 </math>
These above two equations lead to the Laplace equation inside an Ohmic conductor medium:
<math>\nabla^2 \Phi(\mathbf{r}) = 0</math>
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In addition, the boundary condition at a conductor-dielectric interface requires a vanishing normal derivative of the electric potential:
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<math> \frac{\partial \Phi}{\partial n} = 0 </math>
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At the interface between two contiguous conductors, the normal component of the current density must be continuous.
<math> \hat{\mathbf{n}} . [ \mathbf{J_2(r)} - \mathbf{J_1(r)} ] = 0 </math>
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which can be written as:
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<math> \sigma_1 \hat{\mathbf{n}} . \mathbf{E_1(r)} = \sigma_2 \hat{\mathbf{n}} . \mathbf{E_2(r)} </math>
== Magnetostatics Analysis==
*Dirichlet boundary condition: ψ = k =const.
*Neumann boundary condition: ∂ψ /∂n = k = const. In the above, ∂ψ/∂n denotes the normal derivative of the potential at the surface of the domain boundary. [[EM.Ferma]]'s default domain boundary condition for both the electrostatic and magnetostatic solvers is Dirichlet. At the interface between different material media, additional boundary conditions must be applied. These boundary conditions involve electric or magnetic field components. The field components can be expressed as partial derivatives of the potential, i.e. in the form of ∂ψ/∂x, ∂ψ/∂y or ∂ψ/∂z. Using the respective finite difference approximations of these derivatives, one arrives at fairly complicated difference equations involving the constitutive parameters ε, μ and σ, which must be solved simultaneously with the primary potential difference equations.
[[EM.Ferma]]'s default domain boundary condition for both Note that the electrostatic Poisson and magnetostatic solvers is Dirichlet. At Laplace equations are of the interface between different material mediascalar type, additional boundary conditions must be appliedwhile the magnetostatic Poisson and Laplace equations are vectorial. These boundary conditions involve electric or magnetic field components. The field components can be expressed as partial derivatives As a result, the size of the potential, i.e. numerical problem in the form of ∂ψ/∂x, ∂ψ/∂y or ∂ψ/∂zlatter case is three times as large as the former case for the same mesh size.
== 2D Quasi-Static Solution of TEM Transmission Line Structures ==
and its characteristic impedance is given by:
<math> Z_0 = \frac{\eta_0}{\sqrt{ \epsilon_{eff}}} = \eta_0 \sqrt{ \frac{C_a}{C} } </math>
where η<sub>0</sub> = 120π Ω is the intrinsic impedance of the free space.