Conducting sphere inside capacitor - QuickField simulation example
Electric field stress in the dielectric around an uncharged conducting sphere placed between charged parallel capacitor plates at fixed potentials.
How to calculate the electric field stress in the dielectric around a conducting sphere inside a parallel-plate capacitor?
Answer Typical applications Geometry
Given
Task
Solution
To model the conducting body with unknown potential we set the electric permittivity to a very high value (10000). This effectively makes the body's surface equipotential (as the real conductor' surface would be).
Results
Electric field distribution inside the dielectric:
Electric potential distribution inside the dielectric:
*Reference: Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207.
Engineering question
Model the uncharged sphere as an equipotential conductor, solve the axisymmetric electrostatic field, and compare the local electric stress with the analytical solution.
capacitor dielectric inclusions, insulation defect sites, dielectric particle inclusions
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Simulation problem
Problem Type
Axisymmetric problem of electrostatics
Sphere radius: R = 1 mm;
Distance between the plates: d = 10 mm;
Dielectric permittivity: ε = 4;
Plate size: A = 100x100 mm²;
Voltage applied: V+ = 24 V.
Find the electric field stress in the dielectric.
To overcome 3D import limitations (only single body topology is allowed) we subtract the sphere from the dielectric body and import the resulting single body.
This problem could be simulated in 2D axisymmetric formulation if we replace square plates with discs of the same area (πD²/4 = A).
This modification preserves the inner field strength and the geometric capacitance values (~A/d).
Analytical solution:
External uniform electric field E0x = U / d = 24/10 = 2.4 [V/mm] = 2400 [V/m].
Electric potential outside the sphere U(x) = E0x * (R³/x² - x) [V] *
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