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Hidden Electric Dipole: Do you know how polarization density is defined?
In classical electromagnetism, polarization density is the vector field that describes the permanent or induced electric dipole moment inside a dielectric material. When a dielectric material is placed in an external electric field, its molecules acquire an electric dipole moment and are called polarized. For a specific dielectric material sample, the electric polarization can be defined as the ratio of the electric dipole moment to the volume, i.e. the polarization density.
Polarization density is mathematically denoted P and is expressed in SI units of coulombs per square meter (C/m²). Not only does it describe how a material responds to an applied electric field, it can also be used to calculate the forces resulting from this interaction.
When an external electric field is applied to a dielectric material, the charged elements within the material are displaced. It is worth noting that these displaced charged elements do not move freely, but are bound to the atoms or molecules inside the material. Positively charged elements will move in the direction of the electric field, while negatively charged elements will move in the opposite direction, so that an electric dipole moment will be formed even if the molecule remains neutral.
When considering a small volume element ΔV inside a dielectric material, if the volume element carries an electric dipole moment Δp, we can define the polarization Density P:
P =
Δp/ΔV
In general, the electric dipole moment Δp varies point by point within a dielectric material. Therefore, for an infinitesimal volume dV of dielectric material, its polarization density P can also be expressed as:
P =
dp/dV
The net charge that appears due to the polarization process is called the bound charge, usually denoted Qb. This definition of the electric dipole moment per unit volume is widely adopted, although it can lead to ambiguities and paradoxes in some cases.
Other expressions of polarization
Consider a volume dV inside the dielectric material, where the positive bound charges dqb⁺ are sparse relative to the negative bound charges dqb⁻ due to polarization. code> displacement, forming an electric dipole moment:
dp = dqb * d
Substituting this expression into the definition of polarization density, we obtain:
P =
dqb/dV
Since dqb is the charge bounded in a volume dV, it can be expressed as ρb * dV. Therefore, the polarization density is directly related to the charge density inside the material.
Gauss's law and polarization density
For the bound charge Qb in the closed volume V, it is related to the flux of polarization P, that is,
-Qb = Φ(P)
This means that, under certain circumstances, the relationship between polarization and the electric field generated by the material can be expressed by Gauss's law.
Relationship between polarization density and electric field
In a homogeneous, linear, non-dispersive, isotropic dielectric material, there is a proportional relationship between polarization and the electric field E:
P =
χ * ε₀ * E
Where ε₀ is the electrical constant and χ is the potential energy of the medium. Such a relationship shows that the polarization density can be closely related to the change of the external electric field in most cases.
Polarization of nonlinear dielectric materials
When the polarization is no longer linear with the electric field, the material is called a nonlinear dielectric. At this point, the polarization density P can be expressed as the Taylor expansion of the electric field E, further refining the relationship between the quadratic and cubic responses:
P =
Σχ(1) * E + Σχ(2) * E² + Σχ(3) * E³ + …
Therefore, materials may exhibit more complex polarization behaviors when facing different electric fields.
As the electric field strength and time change, we can't help but wonder how far-reaching the impact of polarization density is in the study of materials science and electromagnetism?
