# Electric displacement field

In physics, the electric displacement field, denoted by D, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In SI, it is expressed in units of coulomb per metre squared (C⋅m−2).

## Definition

In a dielectric material, the presence of an electric field E causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment. The electric displacement field D is defined as

${\displaystyle \mathbf {D} \equiv \varepsilon _{0}\mathbf {E} +\mathbf {P} ,}$

where ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity (also called permittivity of free space), and P is the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the polarization density.

The displacement field satisfies Gauss's law in a dielectric:

${\displaystyle \nabla \cdot \mathbf {D} =\rho -\rho _{\text{b}}=\rho _{\text{f}}}$

Electrostatic forces on ions or electrons in the material are governed by the electric field E in the material via the Lorentz Force. Also, D is not determined exclusively by the free charge. As E has a curl of zero in electrostatic situations, it follows that

${\displaystyle \nabla \times \mathbf {D} =\nabla \times \mathbf {P} }$

The effect of this equation can be seen in the case of an object with a "frozen in" polarization like a bar electret, the electric analogue to a bar magnet. There is no free charge in such a material, but the inherent polarization gives rise to an electric field, demonstrating that the D field is not determined entirely by the free charge. The electric field is determined by using the above relation along with other boundary conditions on the polarization density to yield the bound charges, which will, in turn, yield the electric field.

In a linear, homogeneous, isotropic dielectric with instantaneous response to changes in the electric field, P depends linearly on the electric field,

${\displaystyle \mathbf {P} =\varepsilon _{0}\chi \mathbf {E} ,}$

where the constant of proportionality ${\displaystyle \chi }$ is called the electric susceptibility of the material. Thus

${\displaystyle \mathbf {D} =\varepsilon _{0}(1+\chi )\mathbf {E} =\varepsilon \mathbf {E} }$

where ε = ε0 εr is the permittivity, and εr = 1 + χ the relative permittivity of the material.

In linear, homogeneous, isotropic media, ε is a constant. However, in linear anisotropic media it is a tensor, and in nonhomogeneous media it is a function of position inside the medium. It may also depend upon the electric field (nonlinear materials) and have a time dependent response. Explicit time dependence can arise if the materials are physically moving or changing in time (e.g. reflections off a moving interface give rise to Doppler shifts). A different form of time dependence can arise in a time-invariant medium, as there can be a time delay between the imposition of the electric field and the resulting polarization of the material. In this case, P is a convolution of the impulse response susceptibility χ and the electric field E. Such a convolution takes on a simpler form in the frequency domain: by Fourier transforming the relationship and applying the convolution theorem, one obtains the following relation for a linear time-invariant medium:

${\displaystyle \mathbf {D(\omega )} =\varepsilon (\omega )\mathbf {E} (\omega ),}$

where ${\displaystyle \omega }$ is the frequency of the applied field. The constraint of causality leads to the Kramers–Kronig relations, which place limitations upon the form of the frequency dependence. The phenomenon of a frequency-dependent permittivity is an example of material dispersion. In fact, all physical materials have some material dispersion because they cannot respond instantaneously to applied fields, but for many problems (those concerned with a narrow enough bandwidth) the frequency-dependence of ε can be neglected.

At a boundary, ${\displaystyle (\mathbf {D_{1}} -\mathbf {D_{2}} )\cdot {\hat {\mathbf {n} }}=D_{1,\perp }-D_{2,\perp }=\sigma _{\text{f}}}$, where σf is the free charge density and the unit normal ${\displaystyle \mathbf {\hat {n}} }$ points in the direction from medium 2 to medium 1.[1]

## History

Gauss's law was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867, meaning that the formulation and use of D were not earlier than 1835, and probably not earlier than the 1860s.

The earliest known use of the term is from the year 1864, in James Clerk Maxwell's paper A Dynamical Theory of the Electromagnetic Field. Maxwell used calculus to exhibit Michael Faraday's theory, that light is an electromagnetic phenomenon. Maxwell introduced the term D, specific capacity of electric induction, in a form different from the modern and familiar notations.[2]

It was Oliver Heaviside who reformulated the complicated Maxwell's equations to the modern form. It wasn't until 1884 that Heaviside, concurrently with Willard Gibbs and Heinrich Hertz, grouped the equations together into a distinct set. This group of four equations was known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations, and is sometimes still known as the Maxwell–Heaviside equations; hence, it was probably Heaviside who lent D the present significance it now has.

## Example: Displacement field in a capacitor

A parallel plate capacitor. Using an imaginary box, it is possible to use Gauss's law to explain the relationship between electric displacement and free charge.

Consider an infinite parallel plate capacitor placed in space (or in a medium) with no free charges present except on the capacitor. In SI units, the charge density on the plates is equal to the value of the D field between the plates. This follows directly from Gauss's law, by integrating over a small rectangular box straddling one plate of the capacitor:

${\displaystyle \oint _{A}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{\text{free}}}$

On the sides of the box, dA is perpendicular to the field, so the integral over this section is zero. For the space inside the capacitor where the fields of the two plates add,

${\displaystyle |\mathbf {D} |={\frac {Q_{\text{free}}}{A}}}$,

where A is surface area of the top face of the box and Qfree / A is the free surface charge density on the positive plate. Outside the capacitor, the fields of the two plates cancel each other and |E| = |D| = 0. If the space between the capacitor plates is filled with a linear homogeneous isotropic dielectric with permittivity ε, and the charge Qfree is kept constant, the total electric field E between the plates will be smaller than D by a factor of ε: |E| = Qfree / (εA). Alternatively, if the voltage and E are kept constant, the stored charges will increase by a factor of ε.

If the distance d between the plates of a finite parallel plate capacitor is much smaller than its lateral dimensions we can approximate it using the infinite case and obtain its capacitance as

${\displaystyle C={\frac {Q_{\text{free}}}{V}}\approx {\frac {Q_{\text{free}}}{|\mathbf {E} |d}}={\frac {A}{d}}\varepsilon ,}$

where V is the potential difference sustained between the two plates. The partial cancellation of fields in the dielectric allows a larger amount of free charge to dwell on the two plates of the capacitor per unit potential drop than would be possible if the plates were separated by vacuum.

## References

1. ^ David Griffiths. Introduction to Electrodynamics (3rd 1999 ed.).
2. ^ A Dynamical Theory of the Electromagnetic Field PART V. — THEORY OF CONDENSERS, page 494
Ampère's circuital law

In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law that André-Marie Ampère discovered in 1823) relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell (not Ampère) derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

Brush (electric)

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Cycloconverter

For the rotating electrical machine, see Rotary converter

A cycloconverter (CCV) or a cycloinverter converts a constant voltage, constant frequency AC waveform to another AC waveform of a lower frequency by synthesizing the output waveform from segments of the AC supply without an intermediate DC link (Dorf 1993, pp. 2241–2243 and Lander 1993, p. 181). There are two main types of CCVs, circulating current type or blocking mode type, most commercial high power products being of the blocking mode type.

Differential capacitance

Differential capacitance in physics, electronics, and electrochemistry is a measure of the voltage-dependent capacitance of a nonlinear capacitor, such as an electrical double layer or a semiconductor diode. It is defined as the derivative of charge with respect to potential.

Displacement current

In electromagnetism, displacement current density is the quantity ∂D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials (as opposed to vacuum), there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

The idea was conceived by James Clerk Maxwell in his 1861 paper On Physical Lines of Force, Part III in connection with the displacement of electric particles in a dielectric medium. Maxwell added displacement current to the electric current term in Ampère's Circuital Law. In his 1865 paper A Dynamical Theory of the Electromagnetic Field Maxwell used this amended version of Ampère's Circuital Law to derive the electromagnetic wave equation. This derivation is now generally accepted as a historical landmark in physics by virtue of uniting electricity, magnetism and optics into one single unified theory. The displacement current term is now seen as a crucial addition that completed Maxwell's equations and is necessary to explain many phenomena, most particularly the existence of electromagnetic waves.

Displacement field

Displacement field may refer to:

Displacement field (mechanics)

Electric displacement field

Duality (electricity and magnetism)

In physics, the electromagnetic dual concept is based on the idea that, in the static case, electromagnetism has two separate facets: electric fields and magnetic fields. Expressions in one of these will have a directly analogous, or dual, expression in the other. The reason for this can ultimately be traced to special relativity where applying the Lorentz transformation to the electric field will transform it into a magnetic field.

The electric field (E) is the dual of the magnetic field (H).

The electric displacement field (D) is the dual of the magnetic flux density (B).

Faraday's law of induction is the dual of Ampère's circuital law.

Gauss's law for electric field is the dual of Gauss's law for magnetism.

The electric potential is the dual of the magnetic potential.

Permittivity is the dual of permeability.

Electrostriction is the dual of magnetostriction.

Piezoelectricity is the dual of piezomagnetism.

Ferroelectricity is the dual of ferromagnetism.

An electrostatic motor is the dual of a magnetic motor;

Electrets are the dual of permanent magnets;

The Faraday effect is the dual of the Kerr effect;

The Aharonov–Casher effect is the dual to the Aharonov–Bohm effect;

The hypothetical magnetic monopole is the dual of electric charge.

Gauss's law

In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field.

The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1813, both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which form the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law, and vice versa.

Gaussian units

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The most common alternative to Gaussian units are SI units. SI units are predominant in most fields, and continue to increase in popularity at the expense of Gaussian units. (Other alternative unit systems also exist, as discussed below.) Conversions between Gaussian units and SI units are not as simple as normal unit conversions. For example, the formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on what system of units one uses. As another example, quantities that are dimensionless (loosely "unitless") in one system may have dimension in another.

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Interface conditions for electromagnetic fields

Interface conditions describe the behaviour of electromagnetic fields; electric field, electric displacement field, and the magnetic field at the interface of two materials. The differential forms of these equations require that there is always an open neighbourhood around the point to which they are applied, otherwise the vector fields and H are not differentiable. In other words, the medium must be continuous. On the interface of two different media with different values for electrical permittivity and magnetic permeability, that condition does not apply.

However, the interface conditions for the electromagnetic field vectors can be derived from the integral forms of Maxwell's equations.

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Poynting vector

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Statcoulomb

The statcoulomb (statC) or franklin (Fr) or electrostatic unit of charge (esu) is the physical unit for electrical charge used in the esu-cgs (centimetre–gram–second system of units) and Gaussian units. It is a derived unit given by

1 statC = dyn1/2 cm = cm3/2 g1/2 s−1.It can be converted using

1 newton = 105 dyne

1 cm = 10−2 mThe SI system of units uses the coulomb (C) instead. The conversion between C and statC is different in different contexts. The most common contexts are:

For electric charge:

1 C ↔ 2997924580 statC ≈ 3.00×109 statC

⇒ 1 statC ↔ ~3.33564×10−10 C.

For electric flux (ΦD):

1 C ↔ 4π × 2997924580 statC ≈ 3.77×1010 statC

⇒ 1 statC ↔ ~2.65×10−11 C.The symbol "↔" is used instead of "=" because the two sides are not necessarily interchangeable, as discussed below. The number 2997924580 is 10 times the value of the speed of light expressed in meters/second, and the conversions are exact except where indicated. The second context implies that the SI and cgs units for an electric displacement field (D) are related by:

1 C/m2 ↔ 4π × 2997924580×10−4 statC/cm2 ≈ 3.77×106 statC/cm2

⇒ 1 statC/cm2 ↔ ~2.65×10−7 C/m2due to the relation between the metre and the centimetre. The coulomb is an extremely large charge rarely encountered in electrostatics, while the statcoulomb is closer to everyday charges.

Stator

The stator is the stationary part of a rotary system, found in electric generators, electric motors, sirens, mud motors or biological rotors. Energy flows through a stator to or from the rotating component of the system. In an electric motor, the stator provides a rotating magnetic field that drives the rotating armature; in a generator, the stator converts the rotating magnetic field to electric current. In fluid powered devices, the stator guides the flow of fluid to or from the rotating part of the system.

Thyristor drive

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Wound rotor motor

A wound-rotor motor is a type of induction motor where the rotor windings are connected through slip rings to external resistance. Adjusting the resistance allows control of the speed/torque characteristic of the motor. Wound-rotor motors can be started with low inrush current, by inserting high resistance into the rotor circuit; as the motor accelerates, the resistance can be decreased.Compared to a squirrel-cage rotor, the rotor of the slip ring motor has more winding turns; the induced voltage is then higher, and the current lower, than for a squirrel-cage rotor. During the start-up a typical rotor has 3 poles connected to the slip ring. Each pole is wired in series with a variable power resistor. When the motor reaches full speed the rotor poles are switched to short circuit. During start-up the resistors reduce the field strength at the stator. As a result the inrush current is reduced. Another important advantage over squirrel-cage motors is higher starting torque.

The construction of slip ring induction motor is quite different compared to other induction motor. Slip rings Induction motor provides some advantages like provides high starting torque, low starting current and it improves the power factor. We can add external variable resistance to the rotor of this type of motor. So, we can able to control the speed of this type of motor easily.

A wound-rotor motor can be used in several forms of adjustable-speed drive. Certain types of variable-speed drives recover slip-frequency power from the rotor circuit and feed it back to the supply, allowing wide speed range with high energy efficiency. Doubly fed electric machines use the slip rings to supply external power to the rotor circuit, allowing wide-range speed control. Today speed control by use of slip ring motor is mostly superseded by induction motors with variable-frequency drives.

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