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Dielectric ellipsoid in the uniform electric field

# 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:** 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 cross-section (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) = 711 V/m.

QuickField 2D: *E*_{z} = 712 V/m.

QuickField 3D: *E*_{z} = 711 V/m.

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

Video:

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