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# Electrostatics in 3D

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# Electrostatics in 3D   We can convert this example quite simply to a three dimensional calculation.  The modifications that must be made are:

 • Specify cartesian3 coordinates.
 • Add an extrusion section listing the dividing surfaces.
 • Provide boundary conditions for the end faces.
 • Qualify plot commands with the cut plane in which the plot is to be computed.

In the following descriptor, we have divided the extrusion into three layers.  The dielectric constant in the first and third layer are left at the default of k=1, while layer 2 is given a dielectric constant of 50 in the dielectric region only.

A contour plot of the potential in the plane x=0 has been added, to show the resulting vertical cross section.  The plots in the z=0.15 plane reproduce the plots shown above for the 2D case.

Modifications to the 2D descriptor are shown in red.

#### Descriptor 1.2: 3D Dielectric.pde

title

'Electrostatic Potential'

coordinates

cartesian3

variables

V

definitions

eps = 1

equations

extrusion

surface "bottom"  z=0

surface "dielectric_bottom" z=0.1

layer "dielectric"

surface "dielectric_top"  z=0.2

surface "top"  z=0.3

boundaries

surface "bottom" natural(V)=0

surface "top" natural(V)=0

region 1

start (0,0)

value(V) = 0       line to (1,0)

natural(V) = 0     line to (1,1)

value(V) = 100     line to (0,1)

natural(V) = 0     line to close

region 2

layer "dielectric"  eps = 50

start (0.4,0.4)

line to (0.8,0.4) to (0.8,0.8)

to (0.6,0.8) to (0.6,0.6)

to (0.4,0.6) to close

monitors

contour(V) on z=0.15 as 'Potential'

plots

contour(V) on z=0.15 as 'Potential'

vector(-dx(V),-dy(V)) on z=0.15 as 'Electric Field'

contour(V) on x=0.5 as 'Potential'

end

The following potential plot on x=0 shows the vertical cross section of the extruded domain.  Notice that the potential pattern is not symmetric, due to the influence of the extended leg of the dielectric in the y direction. 