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Electrostatics and Magnetostatics

Solve PDEs that model static electrical and magnetic fields

Electrostatics

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.

Magnetostatics

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 $\nabla ×H=J$ and $\nabla \cdot B=0$. Here, $B=\mu H$, 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 $\nabla \cdot B=0$, there exists a magnetic vector potential A such that .

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

$-\nabla \text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(\frac{1}{\mu }\nabla A\right)=J$, where $J=J\left(x,y\right)$. For the 2-D case, $B=\left(\frac{\partial A}{\partial y},-\frac{\partial A}{\partial x},0\right)$.

For subdomain borders between regions of different material properties, H x n must be continuous. This implies the continuity of the derivative $\frac{1}{\mu }\frac{\partial A}{\partial n}$. 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 $n\cdot \left(\frac{1}{\mu }\nabla A\right)$ on the boundary. This is equivalent to specifying the tangential value of the magnetic field H on the boundary.

Apps

 PDE Modeler Solve partial differential equations in 2-D regions

Topics

Programmatic Workflow

Poisson's Equation on Unit Disk

Use command-line functions to solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

PDE Modeler App Workflow

Poisson's Equation on Unit Disk: PDE Modeler App

Use the PDE Modeler app to solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

Electrostatic Potential in Air-Filled Frame

Find the electrostatic potential in an air-filled annular quadrilateral frame.

Magnetic Field in Two-Pole Electric Motor

Find the static magnetic field induced by the stator windings in a two-pole electric motor.

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