Polarization (dielectric)

Polarization $\mathbf{P}$ of a Dielectric medium is the dipole moment per unit volume induced by an applied Electric field:

$\mathbf{P}={ϵ}_0{\chi }_e\mathbf{E}\text{(linear medium)},$

where ${\chi }_e$ is the electric susceptibility of the material (dimensionless).

Units of $\mathbf{P}$: C/m² — same as a surface charge density, because bound dipole charges produce a kind of effective surface charge at the boundaries of the polarized region.

The microscopic picture

In a dielectric with no applied field, atoms are charge-neutral with the electron cloud centered on the nucleus — no dipole moment.

Apply an external $\mathbf{E}$. Electrons (negative) shift slightly against the field; nuclei (positive) shift slightly along it. Each atom develops a small dipole moment in the direction of $\mathbf{E}$.

The medium is now polarized — its volume-averaged dipole moment per unit volume is $\mathbf{P}\ne 0$.

The induced dipoles set up their own field, opposing $\mathbf{E}$ (negative end of each dipole faces the $+\mathbf{E}$ source). This is why the field inside a dielectric is smaller than the field that would exist if the same free charges were in free space.

How it connects to $\mathbf{D}$ and $ϵ$

The fundamental relation:

$\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}.$

In free space $\mathbf{P}=0$ and $\mathbf{D}={ϵ}_0\mathbf{E}$. In a linear dielectric, $\mathbf{P}={ϵ}_0{\chi }_e\mathbf{E}$, so

$\mathbf{D}={ϵ}_0(1+{\chi }_e)\mathbf{E}=ϵ\mathbf{E},$

with Permittivity $ϵ={ϵ}_0{ϵ}_r$ and ${ϵ}_r=1+{\chi }_e$. So $\mathbf{D}$ is the field “seen” by free charges only; the polarization $\mathbf{P}$ is folded into the permittivity once you trust the medium to respond linearly.

Linear, isotropic, homogeneous

Electromagnetics (like most introductory EM courses) assumes:

• Linear: $\mathbf{P}\propto \mathbf{E}$. Nonlinear corrections appear only at very high fields (nonlinear optics).
• Isotropic: $\mathbf{P}$ is parallel to $\mathbf{E}$. Anisotropic crystals would have ${\mathbf{P}}_i={ϵ}_0{\chi }_{ij}{E}_j$ with a tensor susceptibility.
• Homogeneous: ${\chi }_e$ doesn’t depend on position.

Under these conditions $ϵ$ is a single scalar constant.

Bound vs free charge

Two distinct kinds of charge live in a dielectric:

• Free charge ${\rho }_v$: charges that can move (e.g., conduction electrons in a doped semiconductor, or charges deliberately placed on a capacitor plate).
• Bound charge ${\rho }_b$: the displaced charges within the dielectric atoms, immobile but contributing to the local field.

Bound charge density is derivable from $\mathbf{P}$: ${\rho }_b=-\nabla \cdot \mathbf{P}$. Where $\mathbf{P}$ has nonzero divergence, bound charges accumulate.

$\mathbf{D}$ is engineered to be sourced only by free charge: $\nabla \cdot \mathbf{D}={\rho }_v$. The bound charges are quietly inside the constitutive relation. This is the practical reason to work with $\mathbf{D}$ rather than $\mathbf{E}$ in dielectric problems.

Dielectric breakdown

A dielectric polarizes linearly up to a critical field $E_{\text{ds}}$ — the dielectric strength. Beyond that, the field is strong enough to ionize bound electrons, freeing them; an ionization cascade flows, and the material conducts.

Typical strengths:

• Air: ~3 MV/m
• Glass: 25–40 MV/m
• Mica: ~200 MV/m

Breakdown is irreversible damage in solid dielectrics; in gases (air, SF6) it self-heals when the field is removed.

The breakdown voltage of a capacitor is $V_{\text{br}}=E_{\text{ds}}d$ where $d$ is the dielectric thickness. This sets a hard ceiling on operating voltage for a given physical design.