# Polarization density
**Polarization density** (symbol **P**, also called electric polarization or dielectric polarization) is the vector field that expresses the change in electric dipole moment per unit volume of a dielectric material in response to an applied electric field.[1](https://goldbook.iupac.org/terms/view/08856) Its SI unit is coulombs per square meter (C m⁻²), the same dimensions as charge per area, because a dipole moment (charge times distance) spread over a volume reduces to that unit.[1](https://goldbook.iupac.org/terms/view/08856) • [2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html) When a dielectric is placed in an external field, its molecules acquire dipole moments and the material is said to be polarized; P is the macroscopic vector sum of those dipole moments per unit volume.[1](https://goldbook.iupac.org/terms/view/08856)
| Key fact | Detail |
|---|---|
| Definition | Change in electric dipole moment per unit volume of dielectric in response to an electric field; symbol P, unit C m⁻²[1](https://goldbook.iupac.org/terms/view/08856) |
| Dipole model | P = Np, with N the number of dipoles per unit volume and p = qd the dipole moment of each[2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html) |
| Bound charges | Volume density ρ_b = −∇·P; surface density σ_b = P·n̂[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) |
| Displacement field | D = ε₀E + P, with ∇·D = ρ_f, so Gauss's law for D sees only free charge[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) |
| Linear response | P = ε₀χ_e E in a linear isotropic medium; relative permittivity ε_r = 1 + χ_e[4](https://www.egr.msu.edu/emrg/sites/default/files/content/module2_fundamental_behavior.pdf) |
| Typical susceptibilities | Air 0.00059 (0 °C), benzene 1.29, acetone 25.6, distilled water 79.1[5](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html) |
| Ferroelectrics | Spontaneous polarization persists without a field, typically 1–100 μC/cm²[6](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2018/EECS-2018-131.pdf) |
## Dipole model and macroscopic averaging
The standard microscopic picture treats a polarized medium as a collection of electric dipoles, each an opposite pair of charges ±q separated by a vector distance d directed from the negative to the positive charge, with moment p = qd.[2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html) For N such dipoles per unit volume, the polarization density is P = Np.[2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html) Because the separation d is smaller than the size of the atom or molecule, it is small compared with any macroscopic dimension of interest, so P is a smooth vector function of position on the atomic scale and is a macroscopic variable.[2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html) • [7](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_40_Macroscopic_Equations.html) The macroscopic fields used in the equations are averages over volumes containing very large numbers of atoms, since the microscopic fields vary wildly inside individual molecules.[7](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_40_Macroscopic_Equations.html)
Equivalently, P equals the product of the number density n of dipoles and the average dipole moment of each molecule, P = nq**d**, where **d** is the offset of the positive charge relative to the negative charge.[8](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_%28Staelin%29/02%3A_Introduction_to_Electrodynamics/2.05%3A_Electric_and_magnetic_fields_in_media) P points from negative to positive polarization charge; in an ordinary linear isotropic dielectric with positive susceptibility, P aligns with the applied electric field, while the field opposite to P is the depolarizing field produced by the bound charges.[8](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_%28Staelin%29/02%3A_Introduction_to_Electrodynamics/2.05%3A_Electric_and_magnetic_fields_in_media) • [15](https://openlearninglibrary.mit.edu/assets/courseware/v1/91d812aee7f8e5bb80a7ebaee9201997/asset-v1%3AMITx%2B8.02.1x%2B1T2019%2Btype%40asset%2Bblock/textbook_Chapter_5_small.pdf)
## Bound charges
Polarization produces real charge, but of a special kind. Any divergence of the polarization field gives rise to an effective bound charge density ρ_b = −∇·P in the medium, attributable to charges that arise from the polarization of neutral atoms rather than to free charges, which represent a net surplus or deficit of electrons.[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) At a surface where the normal component of P jumps, for example where a polarized material meets free space and P falls from a finite value to zero, there is a bound surface charge density σ_b = P·n̂, the outward normal component of P.[2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html) • [9](https://www.pas.rochester.edu/~stte/phy415F24/units/unit_3-1.pdf)
**Uniform versus non-uniform polarization.** A uniform polarization has zero divergence, so a uniformly polarized body carries no bound volume charge at all; everything happens on the surface.[9](https://www.pas.rochester.edu/~stte/phy415F24/units/unit_3-1.pdf) A net bound volume charge appears only where P is non-uniform.[9](https://www.pas.rochester.edu/~stte/phy415F24/units/unit_3-1.pdf) The total bound charge of an isolated polarized body is zero: integrating ρ_b over the whole body and adding the surface contribution gives zero, because P vanishes on a surface enclosing the body.[2](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html)
The field and potential of a polarized object are the same as those produced by the volume charge ρ_b together with the surface charge σ_b, a result obtained by integrating the dipole field over the body and applying the divergence theorem.[10](https://gcm.upc.edu/en/members/roberto-macovez/NEW_EMAG_TOPICS_1_2_3_A4.pdf)
## The displacement field
Gauss's law for E involves all charge, free and bound alike: ∇·E = (ρ_f + ρ_b)/ε₀.[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) Substituting ρ_b = −∇·P and rearranging defines the electric displacement D = ε₀E + P, for which ∇·D = ρ_f.[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) The flux of D out of a closed surface equals the total free charge enclosed, so D lets one calculate electric fields in the presence of dielectrics without first knowing the distribution of bound charges, provided a constitutive relation connects D and E.[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) Unlike E (force per unit charge) or P (dipole moment per unit volume), D itself has no direct physical interpretation; it is a bookkeeping field.[3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html) A time-varying polarization also contributes a polarization current density j_p = ∂P/∂t, which enters Ampère's law through the displacement current term ∂D/∂t.[11](https://labs.phys.utk.edu/mbreinig/phys421/modules/m1/lih.html)
## Linear, isotropic, and anisotropic response
In a linear, isotropic medium, polarization is directly proportional to the electric field intensity: P = ε₀χ_e E, where χ_e is the electric susceptibility.[4](https://www.egr.msu.edu/emrg/sites/default/files/content/module2_fundamental_behavior.pdf) Substitution into the definition of D gives D = ε₀(1 + χ_e)E = ε₀ε_r E = εE, where ε_r = 1 + χ_e is the relative permittivity, or dielectric constant.[4](https://www.egr.msu.edu/emrg/sites/default/files/content/module2_fundamental_behavior.pdf) All isotropic media behave as linear media when the applied field is sufficiently small, because any continuous function P(E) can be expanded in a [Taylor series](https://www.edgepedia.org/taylor-series) in E and truncated after the first term.[5](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html) For strong fields the response ceases to be linear and P saturates; at sufficiently strong fields neutral molecules are disrupted and the dielectric concept itself fails.[5](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html) • [3](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html)
Susceptibility varies enormously between materials. Measured values include 0.00059 for air at 0 °C, 1.29 for benzene, 4.6–5.0 for mica, 25.6 for acetone, 55.2 for glycerine at 15 °C, and 79.1 for distilled water.[5](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html) Because nuclei are bound tightly to their electron clouds, χ is generally less than 3 for most atoms, although some molecules and crystals, particularly in fluid form, show much higher values.[8](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_%28Staelin%29/02%3A_Introduction_to_Electrodynamics/2.05%3A_Electric_and_magnetic_fields_in_media)
**Anisotropy and frequency dependence.** In anisotropic media such as crystals with asymmetric unit cells, the susceptibility is a second-rank tensor χ_ij relating the three components of E to the three components of P; by a suitable choice of coordinates it can be diagonalized, and the diagonal entries are the principal susceptibilities.[12](https://ocw.mit.edu/courses/3-063-polymer-physics-spring-2007/168531914042ab1dc7b9365fc335003d_lec22_07.pdf) The microscopic mechanisms also differ in speed. Electronic polarization, the displacement of electrons, takes a femtosecond and stays in phase with an alternating field such as that of light; ionic polarization in ionic crystals is also essentially instantaneous; orientation polarization, the reorientation of permanent dipoles, is significantly slower and is often out of phase with the light field.[1](https://goldbook.iupac.org/terms/view/08856)
## Permanent and spontaneous polarization
Most of the relations above assume the polarization is induced by the field, but some materials are polarized permanently, with P specified independently of E.[5](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html) In ferroelectric materials a spontaneous polarization exists in the absence of an external field below a characteristic [Curie temperature](https://www.edgepedia.org/curie-temperature), and the polarization versus field curve shows a hysteresis loop analogous to magnetic hysteresis in ferromagnets.[13](https://122.physics.ucdavis.edu/?q=node%2F34) Spontaneous polarizations are typically in the range of 1–100 μC/cm².[6](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2018/EECS-2018-131.pdf) In a ferroelectric capacitor the surface charge density equals the displacement field at the surface, D = ε₀E + P; in practice P greatly exceeds ε₀E, so the polarization and the capacitor charge can be treated as interchangeable in most contexts.[6](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2018/EECS-2018-131.pdf) Pyroelectric materials, whose polarization depends on temperature, and piezoelectric materials, whose polarization depends on strain, exploit these other variables in transducer applications.[5](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html)
## Historical note
The modern separation of D and P is not Maxwell's own. In his Treatise, Maxwell defined his displacement vector D as the electric polarization and did not introduce the concept of a separate polarization P or the associated bound volume and surface charge densities −div P and Pn.[14](https://arxiv.org/abs/1911.06387) The P field and its bound-charge formalism were developed later as the macroscopic theory was refined from the microscopic equations, in which every charge is accounted for individually and no polarization term appears.[7](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_40_Macroscopic_Equations.html)
## References
1. [IUPAC Compendium of Chemical Terminology, 'polarization density' (08856)](https://goldbook.iupac.org/terms/view/08856)
2. [Electromagnetic Fields and Energy, Chapter 6.1: Polarization Density, MIT](https://web.mit.edu/course/6/6.013_book/www/chapter6/6.1.html)
3. [R. Fitzpatrick, Polarization, Electromagnetism lecture notes, University of Texas at Austin](https://farside.ph.utexas.edu/teaching/em/lectures/node69.html)
4. [Fundamental Behavior of Electric and Magnetic Fields in Materials, Michigan State University EMRG](https://www.egr.msu.edu/emrg/sites/default/files/content/module2_fundamental_behavior.pdf)
5. [Electromagnetic Fields and Energy, Chapter 6.4: Constitutive Laws of Polarization, MIT](https://www.mit.edu/course/6/6.013_book/www/chapter6/6.4.html)
6. [Ferroelectric Negative Capacitance, EECS Technical Report EECS-2018-131, UC Berkeley](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2018/EECS-2018-131.pdf)
7. [Derivation of the Equations of Macroscopic Electromagnetism, University of Virginia](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_40_Macroscopic_Equations.html)
8. [Electric and Magnetic Fields in Media, Physics LibreTexts (Staelin, Electromagnetics and Applications)](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_%28Staelin%29/02%3A_Introduction_to_Electrodynamics/2.05%3A_Electric_and_magnetic_fields_in_media)
9. [Macroscopic Maxwell's Equations in Matter, PHY 415 Unit 3, University of Rochester](https://www.pas.rochester.edu/~stte/phy415F24/units/unit_3-1.pdf)
10. [Microscopic Constituents of Matter, R. Macovez, Universitat Politècnica de Catalunya](https://gcm.upc.edu/en/members/roberto-macovez/NEW_EMAG_TOPICS_1_2_3_A4.pdf)
11. [Linear, Isotropic, Homogeneous Materials, Physics 421, University of Tennessee](https://labs.phys.utk.edu/mbreinig/phys421/modules/m1/lih.html)
12. [Optical Properties of Materials, 3.063 Polymer Physics, MIT OpenCourseWare](https://ocw.mit.edu/courses/3-063-polymer-physics-spring-2007/168531914042ab1dc7b9365fc335003d_lec22_07.pdf)
13. [Ferroelectricity Experiment Guide, Physics 122, UC Davis](https://122.physics.ucdavis.edu/?q=node%2F34)
14. [Maxwell's Definition of Electric Polarization as Displacement, arXiv:1911.06387](https://arxiv.org/abs/1911.06387)
15. [Textbook Chapter 5 Small.pdf, openlearninglibrary.mit.edu](https://openlearninglibrary.mit.edu/assets/courseware/v1/91d812aee7f8e5bb80a7ebaee9201997/asset-v1%3AMITx%2B8.02.1x%2B1T2019%2Btype%40asset%2Bblock/textbook_Chapter_5_small.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Electrostatics › Dielectrics and polarization*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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