FEATool Multiphysics
v1.16.1
Finite Element Analysis Toolbox

Model of an electrostatic spherical capacitor. The example makes use of a 2D axisymmetric approximation with three subdomains, where an electric charge is applied between the middle insulating layer. For this configuration the analytic capacitance can be calculated as 4*πε_{r}ε_{0}/dr, where dr is the thickness of the insulating layer.
This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Electromagnetics > Spherical Capacitor from the File menu. Or alternatively, follow the stepbystep instructions below.
Select the Electrostatics physics mode from the Select Physics dropdown menu.
Start by creating three circles all centered at (0, 0) with radius 0.003, 0.01, and 0.012, respectively.
0.003
into the radius edit field.0.01
into the radius edit field.0.012
into the radius edit field.Press OK to finish and close the dialog box.
Create three rectangles on the left side of the symmetry axis (r<0) so that they cover the left side of the circles (extending a somewhat above, below and to the left).
0.013
into the x_{min} edit field.0
into the x_{max} edit field.0.013
into the y_{min} edit field.0.013
into the y_{max} edit field.Press OK to finish and close the dialog box.
One by one subtract the rectangles from the corresponding circles so that only half circles on right side half plane remain.
C1  R1
into the Geometry Formula edit field.Press the  / Subtract geometry objects Toolbar button.
Switch to Grid mode by clicking on the corresponding Mode Toolbar button.
Press the Equation mode button to switch from grid mode to physics and equation/subdomain specification mode. In the Equation Settings dialog box that automatically opens, select all Subdomains (13) and enter sigma+epsr*eps0/tscale
for the Permittivity, ε, and rho/tscale
for the Space charge density, ρ.
Name  Expression 

r1  0.003 
r2  0.01 
r3  0.012 
sigma  0 6e7 6e7 
eps0  8.85e12 
epsr  3.9 1 1 
tscale  1e17 
rho  0 q0*3/4/pi/(r1^3) q0*3/4/pi/(r3^3r2^3) 
q0  6e11 
Select Insulation/symmetry from the Electrostatics dropdown menu.
Select Ground/antisymmetry from the Electrostatics dropdown menu.
Switch to Solve mode by clicking on the corresponding Mode Toolbar button, and press the = Toolbar button to call the solver. After the problem has been solved, FEATool automatically changes to Postprocessing mode and plots the electric potential V.
Open the Postprocessing Settings dialog box and plot the current density which can be expressed as sigma*sqrt(Vr^2+Vz^2)
To verify the solution, one can calculate the capacitance as q_{0}/( V_{max}  V_{min} ) where V_{max} can be found by using the Min/Max Evaluation... option from the Post menu (V_{min} will be 0).
q0/V
into the edit field.Press the Apply button.
Alternatively, one can integrate the expression for the energy E = ε_{0}ε_{r}(V_{r}^{2}+V_{z}^{2})πr over all subdomains, where the capacitance then can be calculated as q_{0}^{2}/(2·E).
Compare the computed capacitances with the analytical expression 4*πε_{r}ε_{0}/( 1/0.003  1/0.01 ) which in this example should be equal to 1.8588e12.
The spherical capacitor electromagnetics model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB mscript text file (available as the example ex_electrostatics2 script file), or GUI script (.fes) file.
To visualize the full 3D solution from the axisymmetic model, the data can be exported and processed on the MATLAB command line interface (CLI) console with the Export Model Data Struct to MATLAB option from the File menu. The postrevolve and postplot functions can then be applied to revolve and visualize the data, for example
fea_revolved = postrevolve( fea, 24, 0.75 ); postplot( fea_revolved, 'surfexpr', 'V', ... 'parent', figure, 'axis', 'off', 'colorbar', 'off' ) view(70, 25)