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Electrostatics and Magnetostatics

Applications involving electrostatics include high voltage apparatuses, electronic devices, and capacitors. In electrostatics, the time rate of change is slow, and the wavelengths are very large compared to the size of the domain of interest. The electrostatic scalar potential V is related to the electric field E by E = –∇V. Using the Maxwell's equation ∇ · D = ρ and the relationship D = εE, you can write the Poisson equation

–∇ · (εV) = ρ,

where ε is the dielectric permittivity and ρ is the space charge density.

For electrostatics problems, you can use Dirichlet boundary conditions specifying the electrostatic potential V on the boundary or Neumann boundary conditions specifying the surface charge n · (εV) on the boundary.

Applications involving magnetostatics include magnets, electric motors, and transformers. In magnetostatics, the time rate of change is slow.

Maxwell's equations for steady cases are ×H=J and B=0. Here, B=μH, where B is the magnetic flux density, H is the magnetic field intensity, J is the current density, and µ is the material's magnetic permeability.

Since B=0, there exists a magnetic vector potential A such that B=×A and ×(1μ×A)=J.

If the current flows are parallel to the z-axis, then A=(0,0,A) and J=(0,0,J). Using the common gauge assumption ·A=0, simplify the equation for A in terms of J to the scalar elliptic PDE:

·(1μA)=J, where J=J(x,y). For the 2-D case, B=(Ay,Ax,0).

For subdomain borders between regions of different material properties, H x n must be continuous. This implies the continuity of the derivative 1μAn. Also, in ferromagnetic materials, µ usually depends on the field strength |B| = |∇A|. The Dirichlet boundary condition specifies the value of the magnetostatic potential A on the boundary. The Neumann condition specifies the value of the normal component of n(1μA) on the boundary. This is equivalent to specifying the tangential value of the magnetic field H on the boundary.