# FEM Example Capacitance Two Balls

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Topic Tutorial Finite Element Analysis Beginner Sudhanshu Dubey 0.19 or above None

## Introduction

This example is meant to show how to simulate the 6th example of Elmer GUI Tutorials, Electrostatic equation – Capacitance of two balls, using the new FEM Examples. It illustrates how to setup the example, study it's various parts, solve it using the Elmer Solver and visualize the results using Clip Filter.

## Requirements

• A compatible version of FreeCAD designated in the tutorial overview.
• No external software is needed for loading the example, viewing the mesh and geometry as well as for visualizing the results.
• For solving the finite element analysis (FEA), the solver software Elmer must be installed on your computer. See this page for how to install Elmer.

## Understanding the Simulation Case

This case presents the solution of the capacitance of perfectly conducting balls in free space. A voltage difference between the balls results to electric charge being introduced to the system. The balls have also self-capacitance that comes from the voltage difference with the far field. Therefore a symmetric capacitance matrix with of size 2 × 2 needs to be solved. The capacitances may be computed from two different voltage configurations.

## Understanding the Model

• The model contains three spheres.
1. The two smaller ones are the perfectly conducting balls.
2. The bigger one is to simulate the surrounding air.
• The two smaller spheres are fused together and then that fusion is is cut from the bigger sphere.

## Analysis container and its objects

1. SolverElmer
2. Electrostatic, the electrostatics equation
3. FemMaterial, a fluid material to represent the surrounding air
4. ElectrostaticPotential, constraints (3 of them)
5. ConstantVaccumPermittivity, optional
6. Mesh, a Gmsh mesh
7. MeshRegion, a mesh region for the smaller spheres

## Running the FEA

If you get an error message on solver binary or similar when triggering the analysis, check the installation of Elmer.

## Post Processing the Result

Now we can clearly see that potential distribution in and around the balls.

Note that when Apply Changes is on, you would have been able to select the "Field" in the clip dialog directly and not to reopen it after the plane was created.

## Finding the Capacitance

```StatElecSolve: Capacitance matrix computation performed (i,j,C_ij)
StatElecSolve:   1  1    5.07016E+00
StatElecSolve:   1  2    1.69328E+00
StatElecSolve:   2  2    5.07201E+00
```
• Here, our desired result is `C12 = 1.69328`. This value is close to the `1.691` given in the Elmer GUI Tutorials. We can get an even closer value by making a finer Mesh Region but this activity is left for the user. Also, the user is advised to play with the Clip Filter to get a visual result similar to the first picture of this tutorial.