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Dielectric sphere in a uniform electric field

QuickField simulation example

The dielectric sphere is submerged into the uniform electric field.

Problem Type
3D-import electrostatics.

Geometry
Dielectric sphere in a uniform electric field The dielectric sphere is submerged into the uniform electric field. Dielectric ε Air 10 kV E0

Given
Sphere diameter d = 50 mm
Relative permittivity of dielectric ε = 4,
External field strength E0 = 10 kV/m.

Task
Find the electric field distribution inside the sphere.

Solution
Analytical solution [1, page 115] gives the average electric field stress inside the sphere: E = E0 * 3/(2 + ε)

In QuickField to simulate uniform external electric field we apply 0 and +10 kV on two faces separated by 1 meter distance. Ther problem can be simulated both in 2D and in 3D.

Results
Analytical solution: E = 10 * 3/(2 + 4) = 5 kV/m = 5000 V/m (average in a volume)
QuickField 2D: E = 5002 V/m (center of the sphere)
QuickField 3D: E = 5026 V/m (center of the sphere)

Electric field distribution inside and outside the dielectric sphere (2D and 3D):
Dielectric sphere in a uniform electric field
Dielectric sphere in a uniform electric field. 3D

Reference:
[1] J.Jackson. Classical Electrodynamics. 3-d edition. Hoboken, NJ, USA: John Wiley & Sons, 1998.