Simulation of the Electrostatic Field Between Two Charged Spheres

Self-guided learning

This example of the electrostatic field between two charged spheres contains a short description of the theory, detailed information on how to construct the model, a video showing how to construct the model, and the fully constructed model ready for you to download.

When working through the example, you may notice some small differences in your model compared to ours – this is usually simply due to the fact that you are using a different software version.

: Figure 1: Potential distribution of two charged conducting spheres.

The Physics

The investigated setup consists of two conducting, charged spheres with radius R₀=1m in free space. In classic textbooks this setup is often used to present an enhanced example for the method of imaging, see for example [1].

The electrostatic problem (no time dependency) consists of the two PEC spheres; the midpoints are separated by C=R₀/0.2 and the potential difference between both spheres is defined as U=1V. Following [1], the highest electric field strength E_max can be found at the points on the spheres which are closest to each other (P₁ and P₂ in Fig. 2). The expected, analytical result is E_max ≈ 0.74U. Please note that the exact value depends on the truncation of the infinite series.

: Figure 2: Diagram of the problem with all relevant variables labeled, according to [1].

The Model

The model can be easily constructed and simulated with CST Studio Suite®.

The CAD modeler has a predefined object ‘Sphere’ and so the setup is quickly done. The option of a potential definition can be found in the simulation ribbon. By choosing open boundary conditions, appropriate symmetry planes and a tetrahedral mesh with curved elements, the simulation can be fast and accurately performed. In order to improve the accuracy of the simulation, a vacuum sphere is defined around each PEC sphere, with a radius 0.1m greater than that of the PEC sphere. This produces a denser mesh around the sphere, where the gradient of the field is expected to be greatest.

: Figure 3: Problem setup in CST Studio Suite. All relevant distances are shown.

Parameter	Value	Description
R0	1 m	PEC sphere radius
Distance	5 m	Separation of sphere midpoints
U	1 V	Potential between spheres

In order to extract the field value at a certain point in the 3 dimensional calculation domain, the 3D post processing template "Evaluate field in arbitrary coordinates" is used.

Discussion of Results

: Figure 4: Field plot, showing the absolute value (left) and vector representation (right) of the E-field.

The provided images show the results in a 2D cut-plane. From the absolute field pattern, it is clear that the highest electric field is at the inner faces of the two spheres. The vector plot helps to understand the field distribution. The arrows are always perpendicular to the PEC surfaces, indicating that the field is normal to the surface of the spheres, and the open boundaries of the simulation setup have no visual influence on the direction and strength of the fields close to the boundaries of the calculation domain.

By varying the distance between spheres, the maximum field strength can plotted vs the distance as shown in Figure 5. It can be seen that the analytical value Emax=0.74V/m is met even with a quite coarse mesh. Separating the spheres further decreases the field strength - bringing them closer together leads to higher field values.

: Figure 5: Calculated electric field strength (absolute value) at the point P2

Questions

Will the electric field introduce a force on the spheres? If so, in which direction does it point? Use the "Forces" tool under the post processing ribbon to calculate the force acting on the spheres relative to the origin.