
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 ε_{0} = 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  e^{2})/e^{3} * (Artanh(e)  e),
ellipse eccentricity e = sqrt(1  dx^{2}/dz^{2}).
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 crosssection (in general case the problem requires 3D analysis).
Results:
Ellipse eccentricity e = sqrt(1  0.02^{2}/0.05^{2}) = 0.917,
depolarization coefficient n_{z} = (1  0.917^{2})/0.917^{3} * (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.