The electric field boundary conditions at the interface between two material media are:Â <math> \hat{\mathbf{n}} \dot [ \mathbf{D_2(r)} - \mathbf{D_1(r)} ] = \rho_s (\mathbf{r}) </math>
<math> \hat{\mathbf{n}} .[ \mathbf{D_2(r)} - \mathbf{D_1(r)} ] = \rho_s (\mathbf{r}) </math>
<math> \hat{\mathbf{n}} \times [ \mathbf{E_2(r)} - \mathbf{E_1(r)} ] = 0 </math>
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where <b>D(r)</b> = ε<b>E(r)</b> is the electric flux density, <b>E(r)</b> is the electric field vector, and ρ<sub>s</sub> is the surface charge density at the interface.
<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi(\mathbf{r}) = 0</math>
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It must be noted that in the absence of an electric charge source, you need to specify a non-zero potential somewhere in your structure. Otherwise, you will get a trivial zero solution of the Laplace equation.
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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The magnetic field boundary conditions at the interface between two material media are:
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<math> \hat{\mathbf{n}} \dot [ \mathbf{B_2(r)} - \mathbf{B_1(r)} ] = 0 </math>
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<math> \hat{\mathbf{n}} \times [ \mathbf{H_2(r)} - \mathbf{H_1(r)} ] = \mathbf{J_s(r)} </math>
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where <b>B(r)</b> = μ<b>H(r)</b> is the magnetic flux density, <b>H(r)</b> is the magnetic field vector, and <b>J<sub>s</sub></b> is the surface current density at the interface.