]> Dielectric sphere in an electric field

MEMS World - Electrostatics

Part II: Dielectric sphere in a surrounding electrical field

Hypothesises

Here we study a dielectric material made sphere. It is polarized by a polarization vector P and we can decompose its internal electric field Eint in a sum of contributions from the external surrounding electric field and the induced electric field.

Dielectric polarized sphere
Schematic of a dielectric polarized sphere

Depolarizing electric field

We must calculate the contributions from the different phenomena. The induced field Ed is calculated by decomposing the sphere in two virtual spheres separated by an infinitesimal space we will note a along Oz so that the sphere 1 has a -ρ volume density charge and the sphere 2 has a +ρ volume density charge. You can easily prove that these spheres together are equivalent to a single polarized one.

Schematic of the two virtual spheres
Schematics of the two virtual charged sphere equivalent to the polarized sphere

In the space commmon to both spheres, charges compensate so that the electrical field is not affected. In the infinitely small space at the end of each sphere that is not shared with the other one, we have:

dq = ρ with = a.cosθdS

so:

dq = ρacosθdS

and then we have:

σ = dq dS = ρa cosθ

Applying the Gauss theorem to each sphere and then the supeposition theorem give, for the common part:

Ed = - P 3ϵ0

The polarization induced electric field is colinear but in the opposite direction to the external field. It is called depolarizing field.

Knowing the relation between the polarization vector and the electric field, we can deduce a global relation:

Eint = E0 + Ed
Eint = E0 - χr Eint 3

So, for a linear homogeneous isotropic material, we get:

Eint = 3ϵ0 2ϵ0+ϵ E0

Complex permittivity sphere in a complex permittivity environment

The electrical permittivity can be a complex number. This represents a lossy material, losses being due to electrical conduction inside the material. In that case, the electrostatic energy is converted in a conductive energy. The complex permittivity is noted ϵ*. We will see later what the complex component is exactly. Just remember it is dependant on the electrical field variation frequency. So we consider from now to the end that the electric field amplitude follows an alternative mode function.

So, the formula giving the electric field inside the material is different. The polarization is still linked to the induced electric fieldEd, but the surrounding environment is now also polarized. The surface charges density is a result of both the polarization of the sphere and an equilibrium with the environment polarization. The permittivity is now:

ϵint* = ϵ0 ( ϵext,r* + χr )

where ϵext* is the complex electric permittivity of the surrounding environment of the sphere, and ϵint* the sphere material one.
So, redoing the same calculations, we get:

Eint = 3ϵ0 2ϵext* + ϵint* . E0

Dipolar moment of the sphere

The dipolar moment, that we note mint to be coherent with the rest of the notation, can be calculated by integrating the polarisation vector P on the sphere:

mint = 43 πR3 P

This allows us to establish the polarizability of the sphere in function of the electric field E0, that we define as:

αint = 4πϵ0R3 ϵint* - ϵext* ϵint* + 2 ϵext*

In this equation, we find a essential scalar factor for all kind of fields studies. It is called Clausius-Mossotti factor here, but you can find its equivalent in thermal fields, conductivity (Maxwell), optical refraction (Lorentz), etc.
Here, we will note it K(ω):

K(ω) = ϵint* - ϵext* ϵint* + 2 ϵext*

Three cases depending on Clausius-Mossotti factor
Schematics of the three different cases of polarization depending on the Clausius-Mossotti factor under an electric field. From left to right: the particle is not much polarizable before the surrounding environment; the particle is very polarizable before the surrounding environment; the particle is as polarizable as the surrounding environment.

Depending on the differences between ϵint* and ϵext*, we find different configuration for the electric fields. The external charges compensate more or less the ones inside the sphere.

The basic mechanism of the dipole generation in a dielectric sphere can be generalize to any kind of objects. The geometry and properties of the object make calculation more or less complicated. But we can imagine a way to use this dipole generation to get an applied mechanical force on the object just with electric fields. The most common method used in this way involves non-uniform electric fields. It is called dielectrophoresis.