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== Static Modeling Methods Heat Diffusion Equation == The distribution of temperature can be modeled by the heat diffusion equation: <math>\Delta T(\mathbf{r}) = \nabla^2 T(\mathbf{r}) = -\frac{\rho(\mathbf{r})}{\epsilon}</math> where Φ(<b>r</b>) is the electric scalar potential expressed in Volts [v], ρ(<b>r</b>) is the volume charge density expressed in C/m<sup>3</sup>, and ε = ε<sub>r</sub> ε<sub>0</sub> is the permittivity of the medium having the units of F/m.
Under the static assumptions, Maxwell's equations reduce to elliptic partial differential equations known as the Poisson and Laplace equations. These equations can be solved analytically only for a few canonical geometries with very simple boundary conditions. For most practical and realistic problems, you need to utilize a numerical technique and seek a computer solution. The Poisson and Laplace equations can be solved numerically using the finite difference (FD) method.
