# Electric displacement field

The electric displacement field, usually written **D**, is a macroscopic vector field used in electromagnetism that combines the electric field **E** with the polarization **P** of a dielectric medium, defined in SI units as \( \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} \), where \( \varepsilon_0 \) is the [vacuum permittivity](https://www.edgepedia.org/vacuum-permittivity).<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup> Its purpose is practical: [Gauss's law](https://www.edgepedia.org/gausss-law) written for **D** involves only the free charges, the charges that can be placed and controlled on electrodes, and not the bound charges that polarization induces inside and on the surface of a dielectric.<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup><sup> • </sup><sup>[2](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/2cccce322d387cccbc78c3ed65843f49_MITRES_6_002S08_chp03_text.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition (SI) | \( \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} \)<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup> |
| Gauss's law for D | \( \nabla \cdot \mathbf{D} = \rho_{\mathrm{free}} \); in integral form \( \oint_S \mathbf{D} \cdot d\mathbf{A} = Q_{\mathrm{free,\,encl}} \)<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup><sup> • </sup><sup>[2](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/2cccce322d387cccbc78c3ed65843f49_MITRES_6_002S08_chp03_text.pdf)</sup> |
| SI units | Coulombs per square meter (C/m²), the same units as **P** and as surface charge density<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> |
| Linear dielectric | \( \mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_0 (1 + \chi_e) \mathbf{E} \), with relative permittivity \( \varepsilon_r = 1 + \chi_e \)<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> |
| Boundary conditions | The normal component of **D** is continuous across an interface with no free surface charge; the tangential component of **E** is continuous<sup>[4](https://farside.ph.utexas.edu/teaching/em/lectures/node70.html)</sup> |
| Gaussian units | \( \mathbf{D} = \mathbf{E} + 4\pi \mathbf{P} \) and \( \nabla \cdot \mathbf{D} = 4\pi \rho \), so **D** and **E** share units<sup>[5](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.02%3A_Dipole_Media)</sup> |
| Time dependence | A time-varying **D** enters Ampère's law as the displacement current \( \partial \mathbf{D}/\partial t \) [A/m²]<sup>[6](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/5d9f5c06b6e82844f59979eeb093388c_MIT6_013S09_chap02.pdf)</sup> |

## Why the field is defined

Polarizing a dielectric produces bound charge: a volume density \( \rho_{\mathrm{bound}} = -\nabla \cdot \mathbf{P} \) and a surface density \( \sigma_{\mathrm{bound}} = \mathbf{P} \cdot \hat{n} \).<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> The ordinary electric field responds to the total charge, free plus bound, so \( \varepsilon_0 \nabla \cdot \mathbf{E} = \rho_{\mathrm{free}} + \rho_{\mathrm{bound}} \). Substituting \( \rho_{\mathrm{bound}} = -\nabla \cdot \mathbf{P} \) and moving the polarization term to the left side gives \( \nabla \cdot (\varepsilon_0 \mathbf{E} + \mathbf{P}) = \rho_{\mathrm{free}} \), which motivates the definition of **D**.<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup>

**What each field tracks.** Gauss's law for **D** says the displacement field is sensitive to the free charge in a situation; Gauss's law for **E** says the electric field is sensitive to the total charge; and the analogous relation for **P** says the polarization is sensitive to the bound charge.<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> Because the bound surface charge cancels between the \( \varepsilon_0 \mathbf{E} \) and **P** terms, the discontinuity in the normal component of **D** at a boundary equals the free surface charge there, \( \mathrm{disc}(D_\perp) = \sigma_{\mathrm{free}} \).<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup>

The field also has a physical interpretation: \( \mathbf{D}/\varepsilon_0 \) may be read as the "would-be" electric field that stand-alone charges would create in the absence of the medium's polarization, while **E** is the genuine macroscopic field.<sup>[5](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.02%3A_Dipole_Media)</sup> The name comes from the fact that **D** differs from \( \varepsilon_0 \mathbf{E} \) because of the slight charge displacements in electric dipoles.<sup>[2](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/2cccce322d387cccbc78c3ed65843f49_MITRES_6_002S08_chp03_text.pdf)</sup>

## Gauss's law and its uses

In integral form, \( \oint_S \mathbf{D} \cdot d\mathbf{A} = Q_{\mathrm{free}} \) for any closed surface S, whether S lies inside the dielectric, outside it, or crosses its boundary, as long as the enclosed free charge is counted correctly.<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup> In highly symmetric problems this determines **D** directly. For a long straight charged wire surrounded by rubber insulation, cylindrical symmetry gives \( \mathbf{D}(r) = (\lambda / 2\pi r) \hat{r} \), a formula that holds both inside the dielectric and outside it; outside, where **P** vanishes, \( \mathbf{E} = \mathbf{D}/\varepsilon_0 \).<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> Similarly, for a dielectric ball with a free point charge \( q \) at its center, spherical symmetry gives \( \mathbf{D}(r) = q\hat{r}/(4\pi r^2) \) everywhere, and the exterior field \( \mathbf{E} = \mathbf{D}/\varepsilon_0 \) follows regardless of the unknown radial profile of the polarization.<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup>

**Limits of the shortcut.** Outside these symmetric cases, **D** cannot be computed from free charges alone. Unlike the electrostatic field, which is conservative and derivable from a scalar potential, \( \nabla \times \mathbf{D} = \nabla \times \mathbf{P} \), which need not vanish, so in general \( \mathbf{D} \) is not the gradient of anything; Coulomb-like integrals over the free charges do not work for general geometries.<sup>[1](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)</sup> A constitutive relation connecting **P** (or **D**) to **E** is then needed to close the problem.<sup>[2](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/2cccce322d387cccbc78c3ed65843f49_MITRES_6_002S08_chp03_text.pdf)</sup>

## Constitutive relations

For a linear, homogeneous, isotropic dielectric, the polarization is proportional to the field, \( \mathbf{P} = \varepsilon_0 \chi_e \mathbf{E} \), so \( \mathbf{D} = \varepsilon_0 (1 + \chi_e) \mathbf{E} = \varepsilon \mathbf{E} \), with \( \varepsilon_r = 1 + \chi_e \).<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> In free space, where **P** is zero, \( \mathbf{D} = \varepsilon_0 \mathbf{E} \) and the displacement-field form of Gauss's law reduces to the vacuum law.<sup>[2](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/2cccce322d387cccbc78c3ed65843f49_MITRES_6_002S08_chp03_text.pdf)</sup>

In anisotropic dielectrics the relation becomes tensorial, \( \overline{\mathbf{D}} = \overline{\varepsilon}\, \overline{\mathbf{E}} \), with each component of **D** a linear combination of all components of **E**. When the permittivity tensor is diagonal in principal axes with \( \varepsilon_x \neq \varepsilon_y \neq \varepsilon_z \), **D** is parallel to **E** only when **E** points along one of those axes.<sup>[7](https://www.ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/fb6fb170f238e1c9a7af7fcb41a8ba8d_MIT6_013S09_lec08.pdf)</sup> Constitutive laws can also be nonlinear or anisotropic in more general electromechanical materials.<sup>[8](https://ocw.mit.edu/courses/res-6-001-electromagnetic-fields-and-energy-spring-2008/0f814aee1c72517df7d7e7e96b7365e1_11.pdf)</sup>

## Boundary conditions

At a sharp interface between two dielectrics, applying Gauss's law to a pillbox straddling the boundary shows that the normal component of **D** is continuous when the interface carries no free surface charge: \( D_{\perp 2} - D_{\perp 1} = 0 \).<sup>[4](https://farside.ph.utexas.edu/teaching/em/lectures/node70.html)</sup><sup> • </sup><sup>[5](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.02%3A_Dipole_Media)</sup> The corresponding statement for **E** is generally false, because **P** typically jumps at the interface, producing a sheet of bound (effective) surface charge.<sup>[5](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.02%3A_Dipole_Media)</sup> In the static case, Faraday's law gives \( \nabla \times \mathbf{E} = 0 \), so the tangential component of **E** is continuous across the interface.<sup>[4](https://farside.ph.utexas.edu/teaching/em/lectures/node70.html)</sup>

## Displacement current and loss

When **D** varies with time it becomes the displacement current \( \partial \mathbf{D}/\partial t \) [A/m²], which enters Ampère's law alongside the conduction current: \( \nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t \).<sup>[6](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/5d9f5c06b6e82844f59979eeb093388c_MIT6_013S09_chap02.pdf)</sup> In oscillating fields, real dielectrics are lossy because the field dithers the induced dipoles and part of that motion heats the material; this is represented by a complex permittivity, in which **D** lags **E** by a loss angle \( \delta \), quantified by the loss tangent \( \tan \delta \).<sup>[6](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/5d9f5c06b6e82844f59979eeb093388c_MIT6_013S09_chap02.pdf)</sup><sup> • </sup><sup>[8](https://ocw.mit.edu/courses/res-6-001-electromagnetic-fields-and-energy-spring-2008/0f814aee1c72517df7d7e7e96b7365e1_11.pdf)</sup>

## Units and notation across systems

In SI units **D** has dimensions of C/m², different from **E** (V/m).<sup>[3](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)</sup> In Gaussian units the displacement is defined as \( \mathbf{D} = \mathbf{E} + 4\pi \mathbf{P} \), so \( \nabla \cdot \mathbf{D} = 4\pi \rho \) and **D** and **E** share the same dimensions; Likharev, a physicist and author of the Essential Graduate Physics lecture series, notes that this dimensional coincidence is a perceptual handicap, since it is often convenient to treat components of **E** as generalized forces and components of **D** as [generalized coordinates](https://www.edgepedia.org/generalized-coordinates).<sup>[5](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.02%3A_Dipole_Media)</sup>

## References

1. [Vadim, Bound charges and the displacement field, University of Texas lecture notes](https://web2.ph.utexas.edu/~vadim/Classes/2018f/bound.pdf)
2. [Haus & Melcher, Electromagnetic Field Theory: A Problem-Solving Approach, Ch. 3, MIT OpenCourseWare](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/2cccce322d387cccbc78c3ed65843f49_MITRES_6_002S08_chp03_text.pdf)
3. [Steven Errede, UIUC Physics 435, Lecture Notes 10: The Macroscopic Electric Displacement Field](https://hep.physics.illinois.edu/home/serrede/P435/Lecture_Notes/P435_Lect_10.pdf)
4. [Richard Fitzpatrick, Boundary conditions for E and D, University of Texas](https://farside.ph.utexas.edu/teaching/em/lectures/node70.html)
5. [Konstantin K. Likharev, Essential Graduate Physics: Classical Electrodynamics, Sec. 3.2 Dipole Media, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/03%3A_Dipoles_and_Dielectrics/3.02%3A_Dipole_Media)
6. [Staelin, Morgenthaler & Kong, Electromagnetics and Applications, Ch. 2, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/5d9f5c06b6e82844f59979eeb093388c_MIT6_013S09_chap02.pdf)
7. [6.013 Electromagnetics and Applications, Lecture 8: Anisotropic Dielectrics, MIT OpenCourseWare](https://www.ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/fb6fb170f238e1c9a7af7fcb41a8ba8d_MIT6_013S09_lec08.pdf)
8. [Haus & Melcher, Electromagnetic Fields and Energy, Ch. 11, MIT OpenCourseWare](https://ocw.mit.edu/courses/res-6-001-electromagnetic-fields-and-energy-spring-2008/0f814aee1c72517df7d7e7e96b7365e1_11.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Field constants and interface conditions › Constitutive relations as field definitions*

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

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