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Non-concentric spheres capacitance

spherical capacitor simulation, spherical electrodes, electrostatic field between spheres

Problem Type:
Axisymmetric problem of electrostatics.

Geometry:
Non-concentric spheres capacitance Finding the mutual capacitance between two spheres and comparison with analytical solution a Air d +q -q

a = 100 mm, d = 500 mm.

Given:
Relative permittivity of air ε = 1,
The charge q = 10-9 C

Task:
Find the mutual capacitance between two spheres and compare its value with analytical solution:
C = 2π·ε·ε0 · a non-concentric spheres capacitance analytical solution [F] *,
where D = d/ (2a).

Solution:
Sphere's surfaces are marked as 'floating conductor', i.e. isolated conductors with unknown potential. At some point on each of sphere's surface the charge is applied. The charge is then redistributed along the conductor surface automatically.

Results:


Potential distribution around spheres.
The capacitance can be calculated as C = q / (U2 - U1). The measured potential difference is U2 - U1 = 143.4 V.
The capacitance is C = 10-9 / 143.4 = 6.97·10-12 F.

sphere in front of wall capacitance

Electric field stress distribution around spheres.
The capacitance can be calculated as C = q / (U2 - U1) = 6.4e-12 / 1 = 6.4·10-12 F.

sphere to sphere capacitance

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Theoretical result

2D 3D

C, pF

6.97

6.40

6.99

*Wikipedia, Capacitance.

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