EM.Ferma solves the Poisson equation for the electric scalar potential subject to specified boundary conditions:
<math>\Delta\varPhiPhi(\mathbf{r}) = \nabla^2 \varPhiPhi(\mathbf{r}) = -\frac{\varrhorho(\mathbf{r})}{\varepsilonepsilon}</math>
In a source-free region, ρ(<b>r</b>) = 0, and Poisson's equation reduces to the familiar Laplace equation:
<math>\Delta\varphiPhi(\mathbf{r}) = \nabla^2 \varphiPhi(\mathbf{r}) = 0</math>
Once the electric scalar potential is computed, the electric fields field can easily be computed via the equation below:
<math> \mathbf{E(r)} = - \nabla \varPhiPhi(\mathbf{r})</math>
where <b>A(r)</b> is the magnetic vector potential, <b>J(r)</b> is the volume current density, and μ is the permeability of the medium. The magnetic Poisson equation is vectorial in nature and involves a system of three scalar differential equations corresponding to the three components of <b>A(r)</b>.
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Once the magnetic vector potential is computed, the magnetic field can easily be computed via the equation below:
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<math> \mathbf{H(r)} = - \frac{1}{\mu} \nabla \times \mathbf{A} (\mathbf{r})</math>