# 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 $\overrightarrow{P}$ and we can decompose its internal electric field $\overrightarrow{{E}_{\mathrm{int}}}$ in a sum of contributions from the external surrounding electric field and the induced electric field.

Schematic of a dielectric polarized sphere

### Depolarizing electric field

We must calculate the contributions from the different phenomena. The induced field $\overrightarrow{{E}_{\mathrm{d}}}$ is calculated by decomposing the sphere in two virtual spheres separated by an infinitesimal space we will note $a$ along ${O}_{z}$ so that the sphere 1 has a $-\rho $ volume density charge and the sphere 2 has a $+\rho $ volume density charge. You can easily prove that these spheres together are equivalent to a single polarized one.

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:

$$\mathrm{dq}=\rho \mathrm{d\tau}$$ with $\mathrm{d\tau}=a.\mathrm{cos}\theta \mathrm{dS}$

so:

$$\mathrm{dq}=\rho a\mathrm{cos}\theta \mathrm{dS}$$

and then we have:

$$\sigma =\frac{\mathrm{dq}}{\mathrm{dS}}=\rho a\mathrm{cos}\theta $$

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

$$\overrightarrow{{E}_{\mathrm{d}}}=-\frac{\overrightarrow{P}}{3{\u03f5}_{0}}$$

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:

$$\overrightarrow{{E}_{\mathrm{int}}}=\overrightarrow{{E}_{0}}+\overrightarrow{{E}_{\mathrm{d}}}$$

$$\overrightarrow{{E}_{\mathrm{int}}}=\overrightarrow{{E}_{0}}-\frac{{\chi}_{r}\overrightarrow{{E}_{\mathrm{int}}}}{3}$$

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

$$\overrightarrow{{E}_{\mathrm{int}}}=\frac{3{\u03f5}_{0}}{2{\u03f5}_{0}+\u03f5}\overrightarrow{{E}_{0}}$$

### 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 ${\u03f5}^{*}$. 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 field$\overrightarrow{{E}_{\mathrm{d}}}$, 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:

$${\u03f5}_{\mathrm{int}}^{*}={\u03f5}_{0}({\u03f5}_{\mathrm{ext,r}}^{*}+{\chi}_{r})$$

where ${\u03f5}_{\mathrm{ext}}^{*}$ is the complex electric permittivity of the surrounding environment of the sphere, and ${\u03f5}_{\mathrm{int}}^{*}$ the sphere material one.

So, redoing the same calculations, we get:

$$\overrightarrow{{E}_{\mathrm{int}}}=\frac{3{\u03f5}_{0}}{2{\u03f5}_{\mathrm{ext}}^{*}+{\u03f5}_{\mathrm{int}}^{*}}.\overrightarrow{{E}_{0}}$$

### Dipolar moment of the sphere

The dipolar moment, that we note $\overrightarrow{{m}_{\mathrm{int}}}$ to be coherent with the rest of the notation, can be calculated by integrating the polarisation vector $\overrightarrow{P}$ on the sphere:

$$\overrightarrow{{m}_{\mathrm{int}}}=\frac{4}{3}\pi {R}^{3}\overrightarrow{P}$$

This allows us to establish the *polarizability* of the sphere in function of the electric field $\overrightarrow{{E}_{0}}$, that we define as:

$${\alpha}_{\mathrm{int}}=4\pi {\u03f5}_{0}{R}^{3}\frac{{\u03f5}_{\mathrm{int}}^{*}-{\u03f5}_{\mathrm{ext}}^{*}}{{\u03f5}_{\mathrm{int}}^{*}+2{\u03f5}_{\mathrm{ext}}^{*}}$$

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 $K\left(\omega \right)$:

$$K\left(\omega \right)=\frac{{\u03f5}_{\mathrm{int}}^{*}-{\u03f5}_{\mathrm{ext}}^{*}}{{\u03f5}_{\mathrm{int}}^{*}+2{\u03f5}_{\mathrm{ext}}^{*}}$$

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 ${\u03f5}_{\mathrm{int}}^{*}$ and ${\u03f5}_{\mathrm{ext}}^{*}$, we find different configuration for the electric fields. The external charges compensate more or less the ones inside the sphere.

- The sphere is weakly polarizable in front of the surrounding environment, ${\u03f5}_{\mathrm{int}}$. The sphere acts like a capacity. You can find an excess of charges in the surrounding environment. In this case, the electrical field path around the sphere have a tendancy to converge towards the center of the sphere, making a right angle with the sphere's surface. The electric field
has a weak amplitude. The resulting dipole is colinear but with an opposite direction toE int → E 0 → - The sphere is polarizable in a weakly polarizable environment, ${\u03f5}_{\mathrm{int}}$. Here we can consider that the environment is rather a good conductor before the dielectric sphere. So the charges collect to the internal surface of the sphere and generate a strong electrical field $\overrightarrow{{E}_{\mathrm{int}}}$. Electric field path around the sphere avoid it. The resulting electrostatic dipole has the same direction as $\overrightarrow{{E}_{0}}$.
- The sphere and the surrounding environment are approximately equally polarizable, ${\u03f5}_{\mathrm{int}}\approx {\u03f5}_{\mathrm{ext}}$. The absence of polarizability makes external and internal charges compensating. The resulting dipole will be weak if even existing in the case of a perfect equality.

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*.