# Electric susceptibility (χe)

**Electric susceptibility** (\( \chi_{\mathrm{e}} \)) is a dimensionless constant that measures how strongly a dielectric material polarizes in response to an applied electric field. In SI units it is defined by the relation \( \mathbf{P} = \chi_{\mathrm{e}} \, \varepsilon_{0} \mathbf{E} \), where \( \mathbf{P} \) is the polarization (dipole moment per unit volume), \( \mathbf{E} \) is the electric field, and \( \varepsilon_{0} \) is the vacuum permittivity<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup>. It is one of three related parameters, together with the relative permittivity \( \kappa = 1 + \chi_{\mathrm{e}} \) and the permittivity \( \varepsilon = \kappa \varepsilon_{0} \), that describe the same linear dielectric property<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup>.

| Key fact | Detail |
|---|---|
| Definition (SI) | \( \mathbf{P} = \chi_{\mathrm{e}} \, \varepsilon_{0} \mathbf{E} \); \( \chi_{\mathrm{e}} \) is dimensionless<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup> |
| Relation to permittivity | \( \varepsilon_{r} = 1 + \chi_{\mathrm{e}} \) and \( \varepsilon = \varepsilon_{r} \varepsilon_{0} \)<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup> |
| Displacement field | \( \mathbf{D} = \varepsilon_{0} \mathbf{E} + \mathbf{P} = \varepsilon \mathbf{E} \)<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup> |
| Vacuum | \( \chi_{\mathrm{e}} = 0 \), so \( \varepsilon_{r} = 1 \) and \( \varepsilon = \varepsilon_{0} \)<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup> |
| Typical values | Air at 0 °C: 0.00059; diamond: 15.5; glycerine at 15 °C: 55.2; distilled water: 79.1<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup> |
| Unit systems | In Gaussian units \( (\chi_{\mathrm{e}})_{\text{Gaussian}} = (\chi_{\mathrm{e}})_{\text{SI}} / 4\pi \)<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup> |
| Frequency behavior | In liquid water \( \chi_{\mathrm{e}} \) is constant below \( 10^{10} \) Hz and becomes negligible at higher frequencies; in ice just below 0 °C the transition is around \( 10^{3} \) Hz<sup>[4](https://www2.physics.ox.ac.uk/sites/default/files/profiles/terquem/em-lecturenotes-45580.pdf)</sup> |

## Definition and role in electrostatics

The susceptibility enters electrostatics through the displacement field, introduced to keep bound charge out of [Gauss's law](https://www.edgepedia.org/gausss-law). For a linear, isotropic dielectric,

\[ \mathbf{D} = \varepsilon_{0} \mathbf{E} + \mathbf{P} = \varepsilon_{0} (1 + \chi_{\mathrm{e}}) \mathbf{E} = \varepsilon_{0} \varepsilon_{r} \mathbf{E} = \varepsilon \mathbf{E}, \]

where \( \varepsilon_{r} = 1 + \chi_{\mathrm{e}} \) is the relative dielectric constant and \( \varepsilon = \varepsilon_{r} \varepsilon_{0} \) is the permittivity of the dielectric<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup>. In free space the susceptibility is zero, so \( \varepsilon_{r} = 1 \) and the permittivity reduces to that of free space<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup>. The relation between \( \mathbf{D} \) and \( \mathbf{E} \) is a constitutive relation and cannot be derived from first principles<sup>[4](https://www2.physics.ox.ac.uk/sites/default/files/profiles/terquem/em-lecturenotes-45580.pdf)</sup>.

**Why linear?** In an isotropic medium with no preferred direction, the polarization must be parallel to the field, and for sufficiently small fields any continuous function \( \mathbf{P}(\mathbf{E}) \) can be expanded in a [Taylor series](https://www.edgepedia.org/taylor-series) truncated after the first term in \( \mathbf{E} \); an isotropic medium cannot have a term independent of \( \mathbf{E} \)<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup>. All isotropic media therefore behave as linear media when the applied field is small enough<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup>. For large fields the polarization saturates, and the linear description fails<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup>.

## Microscopic origin

The susceptibility connects the macroscopic field to the polarizability \( \alpha \) of the individual atoms or molecules. A dipole moment \( \mathbf{p} \) responds to the local field, which in the Lorentz model for an isotropic dielectric is the macroscopic field with the depolarization contribution \( -\mathbf{P}/3\varepsilon_{0} \) reversed in sign, so the local field is raised rather than lowered<sup>[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC4553869/)</sup>. Solving \( \mathbf{P} = N \alpha (\mathbf{E} + \mathbf{P}/3\varepsilon_{0}) \) for a dipole density \( N \) gives<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup>

\[ \chi_{\mathrm{e}} = \frac{N \alpha / \varepsilon_{0}}{1 - N \alpha / 3 \varepsilon_{0}}. \]

Two mechanisms dominate ordinary dielectrics: orientation of molecules that carry permanent dipole moments, such as water, and induction of dipole moments in otherwise nonpolar atoms and molecules<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup>. The determination of the susceptibility itself, the intrinsic restoring force of the medium, belongs to quantum mechanics rather than classical electrodynamics; polarization is a macroscopic variable summarizing atomic-scale quantum effects<sup>[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC2775418/)</sup>.

## Typical values

Susceptibility spans several orders of magnitude across common materials<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup>:

| Material (condition) | \( \chi_{\mathrm{e}} \) |
|---|---|
| Air, 0 °C | 0.00059 |
| Air, 80 atm | 0.0439 |
| Carbon dioxide, 0 °C | 0.000985 |
| Benzene | 1.29 |
| Castor oil | 3.67 |
| Diamond | 15.5 |
| Acetone, 0 °C | 25.6 |
| Glycerine, 15 °C | 55.2 |
| Water, distilled | 79.1 |

The pattern follows molecular polarity. Gases, whose molecules are sparse and weakly interacting, have susceptibilities near zero; compressed air at 80 atmospheres reaches 0.0439, about 74 times its value at 0 °C and one atmosphere, because susceptibility scales with dipole density<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup>. Polar liquids such as water and glycerine have large values because their permanent dipoles align with the field; the polarity of water is what gives it a dielectric constant of about 80<sup>[6](https://openstax.org/books/college-physics-2e/pages/19-5-capacitors-and-dielectrics)</sup>. High-dielectric-constant materials are usually composed of highly polar molecules<sup>[2](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)</sup>.

## Frequency dependence and complex susceptibility

The simple proportionality \( \mathbf{P} = \chi_{\mathrm{e}} \varepsilon_{0} \mathbf{E} \) assumes that polarization follows the field instantaneously. Real polarization mechanisms have finite response times, so the susceptibility depends on frequency. In a dielectric made of polar molecules, the dipoles must reorient as the field varies, and the timescales are relatively long<sup>[4](https://www2.physics.ox.ac.uk/sites/default/files/profiles/terquem/em-lecturenotes-45580.pdf)</sup>. In liquid water the electric susceptibility is constant below \( 10^{10} \) Hz (microwaves) and becomes negligible at higher frequencies, when the dipoles cannot keep up; in ice just below 0 °C the transition occurs around \( 10^{3} \) Hz<sup>[4](https://www2.physics.ox.ac.uk/sites/default/files/profiles/terquem/em-lecturenotes-45580.pdf)</sup>.

**Loss.** When the polarization lags the field by a time \( \delta t \), a harmonic field \( \mathbf{E}(t) = \hat{\mathbf{E}} e^{-i\omega t} \) produces a polarization with a phase shift, \( \hat{\mathbf{P}} = \chi \varepsilon_{0} e^{+i\omega \delta t} \hat{\mathbf{E}} \), so the susceptibility becomes complex and the dielectric constant acquires an imaginary part even in a perfectly non-conducting dielectric<sup>[7](https://web2.ph.utexas.edu/~vadim/Classes/2026s/skin.pdf)</sup>. The imaginary part represents energy dissipated as heat; the real part measures energy storage<sup>[8](https://digital.lib.washington.edu/bitstreams/b8409a2d-6b58-4615-b986-64e8802316e3/download)</sup>.

## Beyond the linear scalar case

Anisotropic crystals, such as silicon, require a susceptibility tensor rather than a scalar to relate \( \mathbf{P} \) and \( \mathbf{E} \) exactly, although most important crystals are only weakly anisotropic and are reasonably characterized by an angle-averaged scalar<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup>. Quartz illustrates the effect even in modest form: its susceptibility is 3.69 along one axis and 4.06 perpendicular to it<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup>.

Some materials fall outside the linear framework entirely. In pyroelectric materials the polarization is a function of temperature, and in piezoelectric materials it is a function of strain; ferroelectrics, whose spontaneous polarization can be reoriented by an applied field, show hysteretic polarization-versus-field loops characterized by a saturation polarization \( P_{s} \), a remanent polarization \( P_{r} \), and a coercive field \( E_{c} \)<sup>[3](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)</sup><sup> • </sup><sup>[9](https://www.mri.psu.edu/sites/default/files/542/Gross%2C%20Steven%20J.%20thesis%202004.pdf)</sup>.

## Unit systems

The numerical value of \( \chi_{\mathrm{e}} \) depends on the unit system. In [Gaussian units](https://www.edgepedia.org/gaussian-units) the susceptibility is defined by \( \mathbf{P} = \chi_{\mathrm{e}} \mathbf{E} \) without the \( \varepsilon_{0} \) factor, so \( (\chi_{\mathrm{e}})_{\text{Gaussian}} = (\chi_{\mathrm{e}})_{\text{SI}} / 4\pi \), a frequent source of confusion since the parameter is dimensionless in both systems<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup>. Notation also varies by community: older physics literature writes the dielectric constant as \( \varepsilon_{r} \), while electrical engineering publications often use \( K \)<sup>[1](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)</sup>.

## References

1. [Polarization of Dielectrics, Essential Graduate Physics – Classical Electrodynamics (Likharev), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.03%3A_Polarization_of_Dielectrics)
2. [Polarization, Electromagnetic Field Theory: A Problem Solving Approach (Zahn), Engineering LibreTexts](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_%28Zahn%29/03%3A_Polarization_and_Conduction/3.01%3A_Polarization)
3. [Constitutive Laws of Polarization, MIT 6.013 Electromagnetics and Applications](https://web.mit.edu/6.013_book/www/chapter6/6.4.html)
4. [Electromagnetism lecture notes, University of Oxford](https://www2.physics.ox.ac.uk/sites/default/files/profiles/terquem/em-lecturenotes-45580.pdf)
5. [Rigorous treatment of electrostatics for spatially varying dielectrics based on energy minimization, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC2775418/)
6. [Capacitors and Dielectrics, College Physics 2e, OpenStax](https://openstax.org/books/college-physics-2e/pages/19-5-capacitors-and-dielectrics)
7. [Dielectric and Magnetic Losses (skin depth notes), University of Texas at Austin](https://web2.ph.utexas.edu/~vadim/Classes/2026s/skin.pdf)
8. [Dielectric Loss Mechanisms, University of Washington thesis](https://digital.lib.washington.edu/bitstreams/b8409a2d-6b58-4615-b986-64e8802316e3/download)
9. [Gross, Steven J. (2004), thesis on ferroelectricity, Pennsylvania State University](https://www.mri.psu.edu/sites/default/files/542/Gross%2C%20Steven%20J.%20thesis%202004.pdf)
10. [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC4553869/)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Electrostatics › Dielectrics and polarization*

*Initially written Sep 17, 2026 · Reviewed: Sep 18, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 18, 2026; Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license