## Electrostatic Potential in Air-Filled Frame: PDE Modeler App

Find the electrostatic potential in an air-filled annular quadrilateral frame using the PDE Modeler app. For this example, use the following parameters:

Inner square side is 0.2 m

Outer square side is 0.5 m

Electrostatic potential at the inner boundary is 1000

*V*Electrostatic potential at the outer boundary is 0

*V*

The PDE governing this problem is the Poisson equation

–∇ · (*ε*∇*V*)
= *ρ*.

The PDE Modeler app uses the relative permittivity *ε*_{r} =
*ε*/*ε*_{0}, where *ε*_{0} is the absolute
dielectric permittivity of a vacuum (8.854 · 10^{-12 }
farad/meter). The relative permittivity for the air is 1.00059. Note that the coefficient of
permittivity does not affect the result in this example as long as the coefficient is
constant.

Assuming that there is no charge in the domain, you can simplify the Poisson equation to the Laplace equation,

Δ*V* = 0.

Here, the boundary conditions are the Dirichlet boundary conditions *V* =
1000 at the inner boundary and *V* = 0 at the outer
boundary.

To solve this problem in the PDE Modeler app, follow these steps:

Draw the following two squares.

pderect([-0.1 0.1 -0.1 0.1]) pderect([-0.25 0.25 -0.25 0.25])

Set both

*x*- and*y*-axis limits to`[-0.3 0.3]`

. To do this, select**Options**>**Axes Limits**and set the corresponding ranges. Then select**Options**>**Axes Equal**.Model the frame by entering

`SQ2-SQ1`

in the**Set formula**field.Set the application mode to

**Electrostatics**.Specify the boundary conditions. To do this, switch to the boundary mode by selecting

**Boundary**>**Boundary Mode**. Use**Shift**+click to select several boundaries. Then select**Boundary**>**Specify Boundary Conditions**.For the inner boundaries, use the Dirichlet boundary condition with

`h = 1`

and`r = 1000`

.For the outer boundaries, use the Dirichlet boundary condition with

`h = 1`

and`r = 0`

.

Specify the coefficients by selecting

**PDE**>**PDE Specification**or clicking the**PDE**button on the toolbar. Specify`epsilon = 1`

and`rho = 0`

.Initialize the mesh by selecting

**Mesh**>**Initialize Mesh**.Solve the PDE by selecting

**Solve**>**Solve PDE**or clicking the**=**button on the toolbar.Plot the equipotential lines using a contour plot. To do this, select

**Plot**>**Parameters**and choose the contour plot in the resulting dialog box.Improve the accuracy of the solution by refining the mesh close to the reentrant corners where the gradients are steep. To do this, select

**Solve**>**Parameters**. Select**Adaptive mode**, use the**Worst triangles**selection method, and set the maximum number of triangles to 500. Select**Mesh**>**Refine Mesh**.Solve the PDE using the refined mesh. To display equipotential lines at every 100th volt, select

**Plot**>**Parameters**and enter`0:100:1000`

in the**Contour plot levels**field.