FEATool Multiphysics
v1.10 Finite Element Analysis Toolbox |

Electrostatic Spherical Capacitor

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

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 step-by-step instructions below.

- To start a new model click the
**New Model**toolbar button, or select*New Model...*from the*File*menu. - Select the
**Axi 2D**radio button. Select the

**Electrostatics**physics mode from the*Select Physics*drop-down menu.- Press
**OK**to finish the physics mode selection.

Start by creating three circles all centered at (0, 0) with radius *0.003*, *0.01*, and *0.012*, respectively.

- Select
**Circle**from the*Geometry*menu. - Enter
`0.003`

into the*radius*edit field. - Press
**OK**to finish and close the dialog box. - Select
**Circle**from the*Geometry*menu. - Enter
`0.01`

into the*radius*edit field. - Press
**OK**to finish and close the dialog box. - Select
**Circle**from the*Geometry*menu. - Enter
`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).

- Select
**Rectangle**from the*Geometry*menu. - Enter
`-0.013`

into the*x*edit field._{min} - Enter
`0`

into the*x*edit field._{max} - Enter
`-0.013`

into the*y*edit field._{min} - Enter
`0.013`

into the*y*edit field._{max} - Press
**OK**to finish and close the dialog box. - Select
**R1**in the geometry object*Selection*list box. - Press the
**Copy and/or transform selected geometry object***Toolbar*button. - Select the
**Make copy of geometry object**check box. - Press
**OK**to finish and close the dialog box. - Select
**R2**in the geometry object*Selection*list box. - Press the
**Copy and/or transform selected geometry object***Toolbar*button. - Select the
**Make copy of geometry object**check box. 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.

- Select
**Combine Objects...**from the*Geometry*menu. - Enter
`C1 - R1`

into the*Geometry Formula*edit field. - Press
**OK**to finish and close the dialog box. - Select
**C2**and**R2**in the geometry object*Selection*list box. - Press the
**- / Subtract geometry objects***Toolbar*button. - Select
**C3**and**R3**in the geometry object*Selection*list box. 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 (**1-3**) and enter`sigma+epsr*eps0/tscale`

for the*Permittivity*, ε, and`rho/tscale`

for the*Space charge density*, ρ.- Press the
**Constants***Toolbar*button, or select the corresponding entry from the*Equation*menu, and enter the following variables for the capacitor parameters in the*Model Constants and Expressions*dialog box. Press*Enter*after the last expression or use the**Add Row**button to expand the expression list. Also note that values for different subdomains are entered a a space separated list.

Name | Expression |
---|---|

r1 | 0.003 |

r2 | 0.01 |

r3 | 0.012 |

sigma | 0 6e7 6e7 |

eps0 | 8.85e-12 |

epsr | 3.9 1 1 |

tscale | 1e-17 |

rho | 0 q0*3/4/pi/(r1^3) -q0*3/4/pi/(r3^3-r2^3) |

q0 | 6e-11 |

- Switch to
**Boundary**mode by clicking on the corresponding*Mode Toolbar*button. - Select
**1**,**2**,**3**,**6**, and**7**in the*Boundaries*list box. Select

**Insulation/symmetry**from the*Electrostatics*drop-down menu.- Select
**4**and**5**in the*Boundaries*list box. Select

**Ground/antisymmery**from the*Electrostatics*drop-down menu.- Press
**OK**to finish the boundary condition specification. 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

- Select
**Min/Max Evaluation...**from the*Post*menu. - Enter
`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

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

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 m-script text file, or GUI script (.fes) file.