Dielectric Polarization: How Charges are Shifted in a Dielectric

A material that is non-conductive but polarizable is called a dielectric. When a dielectric with a relative permittivity \( \varepsilon_{\text r} \) is placed inside a parallel plate capacitor, filling the entire space between the plates, the voltage \( U \) across the capacitor decreases by a factor \( \varepsilon_{\text r} \) when the capacitor is isolated from the voltage source. Since the homogeneous electric field \( \class{purple}{E} \) of the parallel plate capacitor is proportional to the voltage, the electric field also decreases by the same factor \( \varepsilon_{\text r} \). This effect is caused by the polarization of the dielectric.

A dielectric consists of neutral atoms. An atom generally consists of several positive and negative charges. The positive charges within an atom are described by the positive center of charge, while the negative charges are described by the negative center of charge. Since the atom is overall neutral, the positive and negative charge centers coincide.

When a dielectric material (such as quartz glass, ceramics, or water) is placed in an external electric field \( \class{purple}{\boldsymbol{E}} \) of a parallel plate capacitor, the two charge centers within an atom of the dielectric material shift relative to each other. The negative center of charge is attracted towards the positive capacitor plate, while the positive center of charge is attracted towards the negative capacitor plate. As a result, the two centers of charge are no longer located at the same position but have a certain distance \(r\) to each other. This phenomenon is known as the induction of an electric dipole by the external electric field, and it is referred to as dielectric polarization.

Note that electrostatic induction in conductors, such as metals, also involves a displacement of charges due to the external electric field. However, in conductors, the negative charges can leave the atoms and move throughout the entire conductor. This is the difference between electrostatic induction and polarization.

Two opposite charges (the charge centers) with magnitude \( q \) separated by a distance \( \boldsymbol{r} \) form an electric dipole described by the electric dipole moment \( \boldsymbol{d} \) , given by the following equation:

In this case, the displacement vector \( \boldsymbol{r} \) points from the negative pole to the positive pole. The displacement is incredibly small. While the typical scale of an atom is around \(10^{-10} \, \mathrm{m}\), the displacement of the charge center is on the order of \(10^{-15} \, \mathrm{m}\). So how can we even notice the effect of polarization in our everyday lives? It is due to the sheer number of dipoles formed in the material!

The dielectric is composed of many atoms, and each of these atoms can form an electric dipole when the dielectric is placed in an external electric field. Under real conditions, such as room temperature, not all atoms of the dielectric will necessarily form a dipole. The physical quantity that takes into account the formation and alignment of these dipoles is called polarization \( \class{red}{\boldsymbol{P}} \). Polarization \( \class{red}{\boldsymbol{P}} \) is defined as the sum of all \(N\) induced dipole moments \( \boldsymbol{d}_i \) in the dielectric per unit volume \(V\) of the dielectric material:

Polarization \(\class{red}{\boldsymbol{P}}\) is therefore the vector sum of all dipoles \(\boldsymbol{d}_i\) per unit volume \(V\) of the dielectric. The direction of polarization is determined by the vector sum of these dipole moments!

The polarization in equation 2 has the unit of surface charge density: \( [\class{red}{\boldsymbol{P}}] = [\text{C}/\text{m}^2] \) (Coulomb per square meter). You will soon understand the intuitive reason behind this.

The above example illustrates that, in order to achieve a large (strong) polarization \(\class{red}{\boldsymbol{P}}\) of the dielectric material, it is necessary to have a parallel alignment of all dipoles. However, in reality, various factors can hinder the alignment of the dipoles, such as the temperature of the dielectric material.

However, in the following, it is assumed that all disruptive influences affecting dipole alignment are absent, and the dielectric is ideal. To continue with the example of the dielectric material between the electrodes of a parallel plate capacitor: All dipoles in the dielectric align themselves parallel to the homogeneous electric field \( \class{purple}{\boldsymbol{E}} \) of the capacitor. When the voltage across the capacitor is reversed, the direction of the electric field changes, and the dipoles in the dielectric also immediately rotate along with it. This would be an ideal system without any disturbances!

The reason why the dipoles align in the same direction as the electric field is due to the definition 1 of the dipole moment \(\boldsymbol{d}\). The displacement vector \(\boldsymbol{r}\) is defined as pointing from the negative pole to the positive pole of the dipole, which is exactly opposite to the definition of the direction of the electric field, which goes from the positive pole to the negative pole. The capacitor plate with the positive pole attracts the negative pole of the dipole, and the capacitor plate with the negative pole attracts the positive pole of the dipole. As a result, the dipole aligns itself in such a way that the dipole moment \(\boldsymbol{d}\) points in the same direction as the external electric field. That is why the definition of the dipole moment is chosen from negative to positive and not the other way around.

In an electric field where all \(N\) dipoles are aligned parallel to the electric field and have the same magnitude \( d \) (meaning they are atoms/molecules of the same kind), the vector sum in equation 2 can be simplified:

Here we refer to the ratio \(N/V\) as dipole density. The greater the dipole density of the induced dipoles and the greater the dipole moment \( d \) of a dipole, the more strongly the material is polarized in the external electric field.

Let us return to the question of what it means that the polarization \( \class{red}{\boldsymbol{P}} \) has the unit of the surface charge density \( \mathrm{C}/\mathrm{m}^2 \).

As a result of the displacement of the charge centers in the dielectric, polarization charges are formed at the boundary surfaces of the dielectric, which are orthogonal to the external electric field (as shown in Illustration 1). At the positive pole of the parallel plate capacitor, the negative charge centers of the dipoles located at the boundary surfaces are present, while at the negative pole of the capacitor, the positive charge centers of the dipoles located at the boundary surfaces are found.

Let \(Q_{\text p}\) be the total charge of all polarization charges located at the boundary surfaces. Only these charges are important when studying polarization. The reason why only the charge of the polarization charges at the boundary surfaces is relevant is because only these boundary charges do not find an oppositely charged partner for charge compensation (as shown in Illustration 1). Inside the dielectric, all dipoles neutralize each other, resulting in a total charge of zero in the interior. What remains are only the boundary charges, which are the polarization charges. The polarization charge \(Q_{\text p}\) located on a boundary surface \(A\) of the dielectric can be expressed using the surface charge density \(\sigma_{\text p}\). Surface charge density is defined as charge per unit area: \(\sigma_{\text p} = \frac{Q_{\text p}}{A} \).

Between the oppositely charged boundary surfaces of the dielectric, an electric field \( \class{red}{\boldsymbol{E}_{\text p}} \) forms, pointing from the positive to the negative boundary surface. Therefore, \( \class{red}{\boldsymbol{E}_{\text p}} \) is opposite in direction to the external field \( \class{purple}{\boldsymbol{E}} \). Using the first Maxwell's equation, we can establish the relationship between the magnitude \( \class{red}{E_{\text p}} \) of the polarization field and the surface charge density \( \sigma_{\text p} \):

Now you understand why the polarization \(\class{red}{\boldsymbol{P}}\) has the unit of area charge density and IS an area charge density.

It can be easily shown that the magnitude of polarization \(\class{red}{ P }\) is equal to the surface charge density of the polarization charges: \(P = \sigma_{\text p} \). However, since polarization \(\class{red}{\boldsymbol{P}}\) is opposite in direction to the field \( \class{red}{ \boldsymbol{E}_{\text p} } \) of the polarization charges, the magnitude equation 5 can be expressed in vector form using \(\class{red}{\boldsymbol{P}}\) as follows:

Unlike in a metal, where the external electric field is fully compensated inside (due to electrical induction), in a dielectric, there remains a residual field \( \class{purple}{\boldsymbol{E}_{\text{m}}} \) that points in the same direction as the external electric field, as we always have \( \class{purple}{\boldsymbol{E}} \geq \class{red}{\boldsymbol{E}_{\text p}} \). The residual field \( \class{purple}{\boldsymbol{E}_{\text{m}}} \) (\( \class{purple}{\text{m}}\) stands for material) is the attenuated external field inside the material and corresponds to the vector sum of the external field and the polarization field:

Thus, the electric field in the dielectric, as given by equation 7, can be expressed in terms of the polarization \(\class{red}{\boldsymbol{P}}\) from equation 6:

The average electric field \( \class{purple}{\boldsymbol{E}_{\text{m}}} \) in the dielectric exerts a force \( \boldsymbol{F}_{\text{m}} = q\, \class{purple}{\boldsymbol{E}_{\text{m}}} \) on the positive and negative charge centers of a dipole in the dielectric. The attractive force \(-\boldsymbol{F}_{\text{m}}\) between the oppositely charged charge centers opposes the force \(\boldsymbol{F}_{\text{m}}\). In equilibrium, when the dipoles have reached a fixed \(\boldsymbol{r}\), the two forces must be equal in magnitude but opposite in direction.

Since the displacement \(\boldsymbol{r}\) of the charge centers that occurs in equilibrium is very small, the restoring force \(-\boldsymbol{F}_{\text{m}}\) (always!) can be approximated using the Hooke's law.

Hooke's law states that the restoring force \(-\boldsymbol{F}_{\text{m}}\), which wants to return the dipole to its undeflected state, is proportional to the deflection \(\boldsymbol{r}\): \( \boldsymbol{F}_{\text{m}} = -\alpha \, \boldsymbol{r} \).

Since \(\boldsymbol{r}\) is proportional to \(\class{purple}{\boldsymbol{E}_{\text{m}}}\), it follows from Hooke's law that \(\boldsymbol{r}\) is also proportional to \(\class{purple}{\boldsymbol{E}_{\text{m}}}\). According to equation 1, \(\boldsymbol{r}\) is proportional to the dipole moment \(\boldsymbol{d}\) is also proportional to the field \(\class{purple}{\boldsymbol{E}_{\text{m}}}\):

Here the proportionality constant \( \class{blue}{\alpha} \) is called polarizability. This material quantity tells how easy or difficult it is to polarize a dielectric.

To establish the relationship between polarization \( \class{red}{\boldsymbol{P}} \) and the electric field \( \class{purple}{\boldsymbol{E}_{\text{m}}} \) in the dielectric, equation 9 can be substituted into equation 4:

Since polarization \(\class{red}{\boldsymbol{P}}\) and electric field \( \class{purple}{\boldsymbol{E}_{\text m}} \) in the dielectric are experimentally challenging to access directly, polarization can be expressed in terms of the applied electric field \( \class{purple}{\boldsymbol{E}} \), which is more easily accessible experimentally. To do this, equation 10 is substituted into equation 8:

Here, the factor \( \class{green}{\chi_{\text e}} := \frac{N \, \alpha}{V \, \varepsilon_0} \) is called electric susceptibility. Thus we can write Eq. 12 as follows:

The electric susceptibility \( \class{green}{\chi_{\text e}} \) depends on the dipole density \(N/V\) and the polarizability \( \class{blue}{\alpha} \) of the dielectric. According to equation 12, the susceptibility \( \class{green}{\chi_{\text e}} \) determines how strong the external field \( \class{purple}{\boldsymbol{E}} \) is weakened in the dielectric. The greater the electric susceptibility, the weaker the electric field in the dielectric.

• In the limiting case \(\class{green}{\chi_{\text e}} \rightarrow \infty\), the dielectric is a metal and its interior is field-free.
• In the limiting case \(\class{green}{\chi_{\text e}} \rightarrow 0\) the dielectric becomes a vacuum.

The following table shows values for the electric susceptibility of some materials: