Polarization density
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In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the volumetric density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized.
Electric polarization of a given dielectric material sample is defined as the quotient of electric dipole moment (a vector quantity, expressed as
Polarization density is denoted mathematically by P;[2] in SI units, it is expressed in coulombs per square meter (C/m2).Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism.
Similar to
Definition
An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements.
A bound charge is a charge that is associated with an atom or molecule within a material. It is called "bound" because it is not free to move within the material like free charges. Positive charged elements are displaced in the direction of the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms.[3][4]
For a certain volume element in the material, which carries a dipole moment , we define the polarization density P:
In general, the dipole moment changes from point to point within the dielectric. Hence, the polarization density P of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole moment dp is:
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(1)
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The net charge appearing as a result of polarization is called bound charge and denoted .
This definition of polarization density as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes.[5]
Other expressions
Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge will be displaced a distance relative to the negative bound charge , giving rise to a dipole moment . Substitution of this expression in (1) yields
Since the charge bounded in the volume dV is equal to the equation for P becomes:[3]
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(2)
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where is the density of the bound charge in the volume under consideration. It is clear from the definition above that the dipoles are overall neutral and thus is balanced by an equal density of opposite charges within the volume. Charges that are not balanced are part of the free charge discussed below.
Gauss's law for the field of P
For a given volume V enclosed by a surface S, the bound charge inside it is equal to the flux of P through S taken with the negative sign, or
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(3)
Let a surface area S envelope part of a dielectric. Upon polarization negative and positive bound charges will be displaced. Let d1 and d2 be the distances of the bound charges and , respectively, from the plane formed by the element of area dA after the polarization. And let dV1 and dV2 be the volumes enclosed below and above the area dA.
It follows that the negative bound charge moved from the outer part of the surface dA inwards, while the positive bound charge moved from the inner part of the surface outwards.
By the law of conservation of charge the total bound charge left inside the volume after polarization is:
Since
The above equation becomes
By (2) it follows that , so we get:
And by integrating this equation over the entire closed surface S we find that
which completes the proof.
Differential form
By the divergence theorem, Gauss's law for the field P can be stated in differential form as:
By the divergence theorem we have that
Relationship between the fields of P and E
Homogeneous, isotropic dielectrics
In a
where ε0 is the
Taking into account this relation between P and E, equation (3) becomes:[3]
The expression in the integral is Gauss's law for the field E which yields the total charge, both free and bound , in the volume V enclosed by S.[3] Therefore,
which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume):
Since within a homogeneous dielectric there can be no free charges , by the last equation it follows that there is no bulk bound charge in the material . And since free charges can get as close to the dielectric as to its topmost surface, it follows that polarization only gives rise to surface bound charge density (denoted to avoid ambiguity with the volume bound charge density ).[3]
may be related to P by the following equation:[8]
Anisotropic dielectrics
The class of dielectrics where the polarization density and the electric field are not in the same direction are known as
In such materials, the i-th component of the polarization is related to the j-th component of the electric field according to:[7]
This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field of crystal optics.
As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. The polarizability of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius–Mossotti relation.
In general, the susceptibility is a function of the
If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:
where is the linear susceptibility, is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and is the third-order susceptibility (describing third-order effects such as the Kerr effect and electric field-induced optical rectification).
In
Polarization density in Maxwell's equations
The behavior of electric fields (E, D), magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations in matter.
Relations between E, D and P
In terms of volume charge densities, the free charge density is given by
where is the total charge density. By considering the relationship of each of the terms of the above equation to the divergence of their corresponding fields (of the electric displacement field D, E and P in that order), this can be written as:[9]
This is known as the
In general, P varies as a function of E depending on the medium, as described later in the article. In many problems, it is more convenient to work with D and the free charges than with E and the total charge.[1]
Therefore, a polarized medium, by way of
- The bound volumetric charge density:
- The bound surface charge density:
- The free volumetric charge density:
- The free surface charge density:
Time-varying polarization density
When the polarization density changes with time, the time-dependent bound-charge density creates a polarization current density of
so that the total current density that enters Maxwell's equations is given by
where Jf is the free-charge current density, and the second term is the magnetization current density (also called the bound current density), a contribution from atomic-scale magnetic dipoles (when they are present).
Polarization ambiguity[dubious ]
Crystalline materials
The polarization inside a solid is not, in general, uniquely defined. Because a bulk solid is periodic, one must choose a unit cell in which to compute the polarization (see figure).[11][12] In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of P, and neither of them will be wrong. For example, if Alice chooses a unit cell with positive ions at the top and Bob chooses the unit cell with negative ions at the top, their computed P vectors will have opposite directions. Alice and Bob will agree on the microscopic electric field E in the solid, but disagree on the value of the displacement field .
On the other hand, even though the value of P is not uniquely defined in a bulk solid, variations in P are uniquely defined.
The non-uniqueness of P is not problematic, because every measurable consequence of P is in fact a consequence of a continuous change in P.[11] For example, when a material is put in an electric field E, which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift. This changes P, and the result is electric susceptibility (and hence permittivity). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing P. The result is pyroelectricity. In all cases, the properties of interest are associated with a change in P.
Even though the polarization is in principle non-unique, in practice it is often (not always) defined by convention in a specific, unique way. For example, in a perfectly
This can be seen in a pyroelectric material. Above the Curie temperature the material is not polarized and it has a centrosymmetric configuration. Lowering the temperature below the Curie temperature induces a structural phase transition that breaks the centrosymmetricity. The P of the material grows proportionally to the distortion, thus allowing to define it unambiguously.
Amorphous materials
Another problem in the definition of P is related to the arbitrary choice of the "unit volume", or more precisely to the system's scale.[5] For example, at microscopic scale a plasma can be regarded as a gas of free charges, thus P should be zero. On the contrary, at a macroscopic scale the same plasma can be described as a continuous medium, exhibiting a permittivity and thus a net polarization P ≠ 0.
See also
References and notes
- ^ ISBN 81-7758-293-3
- ^ ISBN 0-07-051400-3
- ^ ISBN 81-239-0306-5
- ^ Matveev. A. N. (1986). Electricity and Magnetism. Mir Publishers.
- ^ .
- ^ Based upon equations from Gray, Andrew (1888). The theory and practice of absolute measurements in electricity and magnetism. Macmillan & Co. pp. 126–127., which refers to papers by Sir W. Thomson.
- ^ ISBN 0-201-02117-X
- ISBN 978-0-471-92712-9
- ISBN 978-0-471-35832-9.
- .
- ^ , an introductory-level powerpoint.
- S2CID 55374298.
External links
- Media related to Electric polarization at Wikimedia Commons