Dielectric ellipsoid in a uniform electric field

The dielectric ellipsoid is submerged into the uniform electric field. The external field vector is aligned with the ellipsoid major axis.

Problem Type: 3D electrostatics.

Geometry: Axisymmetric / 3D import.

Given:
Length along Z axis: dz = 0.05 m.
Length along X,Y axes: dx = dy = 0.02 m.
Relative permittivity of air ε₀ = 1,
Relative permittivity of dielectric ε_r = 4,
External field strength E_ext = 1 kV/m.

Problem:
Find the electric field distribution inside the ellipsoid.

Solution:
2D analytical solution (external field vector E_ext is aligned with the ellipse Z axis):
electric field stress inside the ellipse E_z = E_ext / (1+(ε_r - 1)·n_z), where
     depolarization coefficient n_z = (1 - e²)/e³ * (Artanh(e) - e),
     ellipse eccentricity e = sqrt(1 - dx²/dz²).

In case the external field E_ext vector is aligned with one of the ellipsoid's axes, the geometry model could be constructed as 2D axisymmetric, with the model geometry presented in QuickField as an upper half of the ellipse cross-section (in general case the problem requires 3D analysis).

Results:
Ellipse eccentricity e = sqrt(1 - 0.02²/0.05²) = 0.917,
depolarization coefficient n_z = (1 - 0.917²)/0.917³ * (Artanh(0.917) - 0.917) = 0.135.
Uniform electric field strength inside the ellipsoid (analytical solution):
E_z = 1000 / (1 + (4 - 1)·0.135) = 712 V/m.

QuickField 2D: E_z = 712 V/m.

QuickField 3D: E_z = 712 V/m.

Electric field distribution inside and outside the dielectric ellipsoid (2D and 3D):

See the dielectric_ellipsoid_2d.pbm, dielectric_ellipsoid_3d.pbm problems in the Examples folder.