KAHIBARO
Table of Contents
Polarization in Dielectrics
When a dielectric material is placed in an electric field, the charges inside its atoms and molecules do not flow freely as they do in a conductor. Instead, positive and negative charges shift slightly in opposite directions. This small separation of charge inside the material is called polarization.
Polarization is the microscopic response of a dielectric to an external electric field. It helps explain why inserting a dielectric into a capacitor changes the capacitor's behavior.
Microscopic Picture
In an atom or molecule, the center of positive charge and the center of negative charge may move apart slightly under the action of an electric field. This creates an electric dipole moment. Some molecules already have permanent dipole moments, while others become polarized only when a field is applied.
There are two common ways polarization happens. In nonpolar molecules, the external field distorts the electron cloud and induces a dipole moment. In polar molecules, the molecules already have dipole moments, but these are normally oriented randomly. The field tends to align them.
The figure shows that many tiny dipoles tend to point in the direction of the applied field. Because of this alignment, the dielectric develops bound charges on its surfaces.
Polarization Vector
To describe polarization in a material, physics uses the polarization vector $\vec P$. It is defined as the electric dipole moment per unit volume.
$$
\vec P = \frac{\text{dipole moment in a small volume}}{\text{that volume}}
$$
So if a small volume $\Delta V$ contains total dipole moment $\Delta \vec p$, then
$$
\vec P = \lim_{\Delta V \to 0} \frac{\Delta \vec p}{\Delta V}
$$
The direction of $\vec P$ is the direction from negative bound charge toward positive bound charge inside the material.
Important definition:
$$
\vec P = \frac{d\vec p}{dV}
$$
Polarization is dipole moment per unit volume.
Bound Charges
Polarization creates charges that are attached to the material, called bound charges. These are different from free charges, which can move through conductors or be placed on capacitor plates.
If the polarization is uniform, bound charges appear mainly on the surfaces of the dielectric. The surface bound charge density is
$$
\sigma_b = \vec P \cdot \hat n
$$
where $\hat n$ is the outward normal unit vector to the surface.
If the polarization changes from point to point inside the material, there can also be a bound volume charge density:
$$
\rho_b = - \nabla \cdot \vec P
$$
These relations connect the microscopic dipole picture to measurable charge distributions.
Key bound-charge formulas:
$$
\sigma_b = \vec P \cdot \hat n
$$
$$
\rho_b = - \nabla \cdot \vec P
$$
Surface bound charge comes from polarization meeting a boundary. Volume bound charge appears when polarization is nonuniform.
Polar and Nonpolar Dielectrics
Different materials polarize in different ways.
| Type of material | Behavior without field | Behavior in applied field |
|---|---|---|
| Nonpolar dielectric | No permanent dipole moment | Dipole moments are induced |
| Polar dielectric | Permanent dipole moments exist | Dipoles tend to align with the field |
In a nonpolar dielectric, polarization is mainly due to distortion of charge clouds. In a polar dielectric, polarization comes mainly from partial alignment of permanent dipoles, though thermal motion prevents perfect alignment.
Linear Dielectrics
For many materials, if the electric field is not too strong, polarization is approximately proportional to the electric field:
$$
\vec P = \varepsilon_0 \chi_e \vec E
$$
Here, $\chi_e$ is the electric susceptibility of the material, and $\varepsilon_0$ is the permittivity of free space.
This is called linear polarization. It is a good approximation for many common insulating materials under ordinary conditions.
For a linear dielectric,
$$
\vec P = \varepsilon_0 \chi_e \vec E
$$
The stronger the electric field, the larger the polarization, as long as the material remains in the linear regime.
Relation to Dielectric Constant
Polarization is closely related to the dielectric constant of a material. In a linear, isotropic dielectric, the permittivity is
$$
\varepsilon = \varepsilon_0 (1 + \chi_e)
$$
and the relative permittivity is
$$
\kappa = \frac{\varepsilon}{\varepsilon_0} = 1 + \chi_e
$$
So a material with greater susceptibility polarizes more strongly and usually produces a larger increase in capacitance.
| Quantity | Meaning | Relation |
|---|---|---|
| $\chi_e$ | Electric susceptibility | Measures how easily a material polarizes |
| $\varepsilon$ | Permittivity of material | $\varepsilon = \varepsilon_0(1+\chi_e)$ |
| $\kappa$ | Relative permittivity, dielectric constant | $\kappa = 1+\chi_e$ |
Physical Effect of Polarization in a Capacitor
Inside a capacitor, polarization of the dielectric produces bound charges on the dielectric surfaces. These bound charges create an electric field that opposes the original field due to the free charges on the capacitor plates. As a result, the net electric field inside the dielectric is reduced.
Because the electric field is reduced for a given free charge, the potential difference becomes smaller. This is why the capacitance increases when a dielectric is inserted.
The details of capacitance and dielectric effects belong to nearby chapters, but polarization is the microscopic reason behind those effects.
Limits of Polarization
Polarization does not increase without limit. In very strong electric fields, the simple linear relation may fail. Some materials reach saturation, meaning that permanent dipoles are almost fully aligned and cannot contribute much more. If the field becomes too strong, dielectric breakdown may occur, and the material may begin to conduct.
So the equation $\vec P = \varepsilon_0 \chi_e \vec E$ is useful, but only within the appropriate range.
Summary
Polarization is the separation or alignment of charges inside a dielectric under an electric field. It is described by the polarization vector $\vec P$, which is dipole moment per unit volume. Polarization leads to bound charges, given by $\sigma_b = \vec P \cdot \hat n$ on surfaces and $\rho_b = -\nabla \cdot \vec P$ in the volume. In many materials, polarization is proportional to the electric field, $\vec P = \varepsilon_0 \chi_e \vec E$. This microscopic behavior is what allows dielectrics to modify electric fields and increase capacitance.
Essential ideas of polarization:
$$
\vec P = \frac{d\vec p}{dV}
$$
$$
\sigma_b = \vec P \cdot \hat n
$$
$$
\rho_b = - \nabla \cdot \vec P
$$
For linear dielectrics,
$$
\vec P = \varepsilon_0 \chi_e \vec E
$$
Polarization is the dielectric response that produces bound charge and weakens the internal electric field.