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

–∇ · (*ε*∇*V*)
= *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** · (*ε*∇*V*) 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 $$\nabla \times H=J$$ 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 $$B=\nabla \times A\text{and}\nabla \times \left(\frac{1}{\mu}\nabla \times A\right)=J$$.

If the current flows are parallel to the *z*-axis,
then $$A=\left(0,0,A\right)\text{and}J=\left(0,0,J\right)$$.
Using the common gauge assumption $$\nabla \xb7A=0$$, simplify the
equation for **A** in terms of **J** to the scalar elliptic PDE:

$$-\nabla \text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla A\right)=J$$, 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.

PDE Modeler | Solve partial differential equations in 2-D regions |

**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.

**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.