Electric displacement: Difference between revisions

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In [[physics]], the '''electric displacement''', usually denoted by '''D''', is a vector field in a non-conducting medium, a [[dielectric]]. The displacement '''D''' is proportional to the electric field '''E''' in which the dielectric is placed. In [[SI]] units,
In [[physics]], the '''electric displacement''', also known as '''dielectric displacement''' and usually denoted by '''D''', is a vector field in a non-conducting medium, a [[dielectric]]. The displacement '''D''' is proportional to an external electric field '''E''' in which the dielectric is placed.  
 
In [[SI]] units the proportionality is,
:<math>
:<math>
\mathbf{D}(\mathbf{r}) = \epsilon_0\epsilon_r \mathbf{E}(\mathbf{r}),
\mathbf{D}(\mathbf{r}) = \epsilon_0\epsilon_r \mathbf{E}(\mathbf{r}),
</math>
</math>
where &epsilon;<sub>0</sub> is the [[electric constant]] and &epsilon;<sub>r</sub> is the [[relative permittivity]].  In [[Gaussian units]] &epsilon;<sub>0</sub> does not occur and may put equal to unity.  In vacuum the dimensionless quantity &epsilon;<sub>r</sub> = 1 (both for SI and Gaussian units) and '''D''' is simply related (SI), or equal (Gaussian), to '''E'''. Often '''D''' is termed an auxiliary field with '''E'''  the principal field. An other auxiliary field is the [[electric polarization]] '''P''' of the dielectric,
where &epsilon;<sub>0</sub> is the [[electric constant]] and &epsilon;<sub>r</sub> is the [[relative permittivity]] of the dielectric.  In [[Gaussian units]] &epsilon;<sub>0</sub> does not occur and can be put equal to unity in this equation.  In vacuum the dimensionless quantity &epsilon;<sub>r</sub> = 1 (both for SI and Gaussian units) and '''D''' is simply related (SI), or equal (Gaussian), to '''E'''. Often '''D''' is termed an auxiliary field with '''E'''  the principal field.  
 
An other auxiliary field is the [[electric polarization]] '''P''' of the dielectric,
:<math>
:<math>
\begin{align}
\begin{align}
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\end{align}
\end{align}
</math>
</math>
The vector field '''P'''  describes the polarization (displacement of charges) occurring in a chunk of dielectric when it is brought into an electric field. The fact that for any insulator &epsilon;<sub>r</sub> > 1 (i.e., that '''D''' is not simply equal to &epsilon;<sub>0</sub>'''E''') has as its cause the polarization of the dielectric.
The vector field '''P'''  describes the polarization (displacement of charges) occurring in a chunk of dielectric when it is brought into an electric field. The fact that for any insulator &epsilon;<sub>r</sub> > 1 is caused by the [[polarization]] of the dielectric.


The electric displacement appears in one of the macroscopic [[Maxwell equations]],
The electric displacement appears in one of the macroscopic [[Maxwell equations]] (in SI),
:<math>
:<math>
\boldsymbol{\nabla} \cdot \mathbf{D}(\mathbf{r}) = \rho(\mathbf{r}),
\boldsymbol{\nabla} \cdot \mathbf{D}(\mathbf{r}) = \rho(\mathbf{r}),
</math>
</math>
where the symbol '''&nabla;'''&sdot; gives the [[divergence]] of '''D'''('''r''') and &rho;('''r''') is the charge density at the point '''r'''.
where the symbol '''&nabla;'''&sdot; gives the [[divergence]] of '''D'''('''r''') and &rho;('''r''') is the volume charge density at the point '''r'''.
 
In [[Gaussian units]] '''D''' and '''E''' have the same dimension: statV/cm or dyne/statC. In [[SI]] units '''D''' has the dimension charge per area,  C/m<sup>2</sup>.


==Relation of D to surface charge density &sigma;==
==Relation of D to surface charge density &sigma;==
Line 24: Line 30:


In the special case of a [[parallel-plate capacitor]], often used to study and exemplify problems in electrostatics,  the electric displacement ''D'' has an interesting interpretation.  In this case ''D'' (the magnitude of vector '''D''') is equal to the ''true surface charge density'': &nbsp;''&sigma;''<sub>true</sub> &nbsp; (the surface density on the plates of the  right-hand capacitor in the figure).<ref> The nomenclature of the several surface charge distributions is not standardized. Here we will follow by and large  R. Kronig, ''Textbook of physics'',  Pergamon Press London, New York (1959). (English translation from the Dutch ''Leerboek der Natuurkunde'')</ref>  
In the special case of a [[parallel-plate capacitor]], often used to study and exemplify problems in electrostatics,  the electric displacement ''D'' has an interesting interpretation.  In this case ''D'' (the magnitude of vector '''D''') is equal to the ''true surface charge density'': &nbsp;''&sigma;''<sub>true</sub> &nbsp; (the surface density on the plates of the  right-hand capacitor in the figure).<ref> The nomenclature of the several surface charge distributions is not standardized. Here we will follow by and large  R. Kronig, ''Textbook of physics'',  Pergamon Press London, New York (1959). (English translation from the Dutch ''Leerboek der Natuurkunde'')</ref>  
In this figure two  parallel-plate capacitors are shown that are identical, except for the matter between the plates: on the left no matter (vacuum), on the right a dielectric. Note in  particular that the plates have the same voltage difference ''V'' and the same area ''A''. When the capacitor on the right discharges, it will deliver  ''Q'' = ''A'' &times; ''&sigma;''<sub>true</sub> [[coulomb (unit)|coulomb]].
In this figure two  parallel-plate capacitors are shown that are identical, except for the matter between the plates: on the left no matter (vacuum), on the right a dielectric. Note in  particular that the plates have the same voltage difference ''V'' and the same area ''A''. When the capacitor on the right discharges, it will deliver  ''Q''<sub>true</sub> = ''A'' &times; ''&sigma;''<sub>true</sub> [[coulomb (unit)|coulomb]].


To explain that  ''D'' = ''&sigma;''<sub>true</sub>,  we recall that the [[relative permittivity]] may be defined as the ratio of two [[capacitances]] of parallel-plate capacitors,  (capacitance is total charge on the plates divided by voltage difference). Namely, the relative permittivity is the ratio of the capacitance ''C'' of the capacitor filled with dielectric to the capacitance ''C''<sub>vac</sub> of a capacitor in vacuum,  
To explain that  ''D'' = ''&sigma;''<sub>true</sub>,  we recall that the [[relative permittivity]] may be defined as the ratio of the [[capacitances]] of two parallel-plate capacitors,  (capacitance is total charge on the plates divided by voltage difference). Namely,  the ratio of the capacitance ''C'' of a capacitor filled with dielectric to the capacitance ''C''<sub>vac</sub> of an identical capacitor in vacuum,  
:<math>
:<math>
\epsilon_\mathrm{r} \equiv \frac{C}{C_\mathrm{vac}} = \frac{Q_\mathrm{true}}{V} \left[ \frac{Q_\mathrm{free}}{V}\right]^{-1} =
\epsilon_\mathrm{r} \equiv \frac{C}{C_\mathrm{vac}} = \frac{Q_\mathrm{true}}{V} \left[ \frac{Q_\mathrm{free}}{V}\right]^{-1} =
Line 32: Line 38:
\quad\Longrightarrow\quad \sigma_\mathrm{true} = \epsilon_\mathrm{r} \sigma_\mathrm{free},
\quad\Longrightarrow\quad \sigma_\mathrm{true} = \epsilon_\mathrm{r} \sigma_\mathrm{free},
</math>
</math>
where we used that ''Q'' is &sigma; &times; ''A'', with ''A'' the area of the plates.
where we used again that ''Q'' is &sigma; &times; ''A''. Clearly, the charge density on the plates increases by a factor &epsilon;<sub>r</sub> when the dielectric is inserted in between the plates. This means that the external source (of voltage ''V'') must deliver a current during this insertion (it must move negative charge&mdash;[[electron]]s&mdash;from the positive plate to the negative plate).  
Clearly, the charge density on the plates increases by a factor &epsilon;<sub>r</sub> when the dielectric is moved in between the plates.  


The extra charge on the plates is compensated by the ''polarization'' of the dielectric, that is, the build-up of a positive polarization surface charge density &sigma;<sub>p</sub> on the side of the negative plate and a negative surface charge density on the positive side. Note, parenthetically, that only the absolute values of the charge densities are indicated and that the vectors '''E'''<sub>vac</sub> and '''D''' are parallel.  The total charge is conserved, for instance on the side of the positively charged plate:
The extra charge on the plates is compensated by the ''polarization'' of the dielectric, that is, the build-up of a positive polarization surface charge density &sigma;<sub>p</sub> on the side of the negative plate and a negative surface charge density on the positive side. Note, parenthetically, that only the absolute values of the charge densities are indicated and that the vectors '''E'''<sub>vac</sub> and '''D''' are parallel.  The total charge is conserved, for instance on the side of the positively charged plate:
Line 61: Line 66:
D_i(\mathbf{r})  = \epsilon_0 \sum_{j=1}^3 (\epsilon_r)_{ij} E_j(\mathbf{r}),\quad i,j=1,2,3= x,y,z,
D_i(\mathbf{r})  = \epsilon_0 \sum_{j=1}^3 (\epsilon_r)_{ij} E_j(\mathbf{r}),\quad i,j=1,2,3= x,y,z,
</math>
</math>
so that '''D''' and '''E''' are not necessarily parallel.
so that '''D''' and '''E''' are not necessarily parallel in non-isotropic dielectrics.
 
==Note==
<references />

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In physics, the electric displacement, also known as dielectric displacement and usually denoted by D, is a vector field in a non-conducting medium, a dielectric. The displacement D is proportional to an external electric field E in which the dielectric is placed.

In SI units the proportionality is,

where ε0 is the electric constant and εr is the relative permittivity of the dielectric. In Gaussian units ε0 does not occur and can be put equal to unity in this equation. In vacuum the dimensionless quantity εr = 1 (both for SI and Gaussian units) and D is simply related (SI), or equal (Gaussian), to E. Often D is termed an auxiliary field with E the principal field.

An other auxiliary field is the electric polarization P of the dielectric,

The vector field P describes the polarization (displacement of charges) occurring in a chunk of dielectric when it is brought into an electric field. The fact that for any insulator εr > 1 is caused by the polarization of the dielectric.

The electric displacement appears in one of the macroscopic Maxwell equations (in SI),

where the symbol ⋅ gives the divergence of D(r) and ρ(r) is the volume charge density at the point r.

In Gaussian units D and E have the same dimension: statV/cm or dyne/statC. In SI units D has the dimension charge per area, C/m2.

Relation of D to surface charge density σ

Two capacitors at same static voltage difference V. On the left vacuum between the plates and on the right a dielectric with relative permittivity εr. Absolute values of surface charge densities are indicated by σ.

In the special case of a parallel-plate capacitor, often used to study and exemplify problems in electrostatics, the electric displacement D has an interesting interpretation. In this case D (the magnitude of vector D) is equal to the true surface charge density:  σtrue   (the surface density on the plates of the right-hand capacitor in the figure).[1] In this figure two parallel-plate capacitors are shown that are identical, except for the matter between the plates: on the left no matter (vacuum), on the right a dielectric. Note in particular that the plates have the same voltage difference V and the same area A. When the capacitor on the right discharges, it will deliver Qtrue = A × σtrue coulomb.

To explain that D = σtrue, we recall that the relative permittivity may be defined as the ratio of the capacitances of two parallel-plate capacitors, (capacitance is total charge on the plates divided by voltage difference). Namely, the ratio of the capacitance C of a capacitor filled with dielectric to the capacitance Cvac of an identical capacitor in vacuum,

where we used again that Q is σ × A. Clearly, the charge density on the plates increases by a factor εr when the dielectric is inserted in between the plates. This means that the external source (of voltage V) must deliver a current during this insertion (it must move negative charge—electrons—from the positive plate to the negative plate).

The extra charge on the plates is compensated by the polarization of the dielectric, that is, the build-up of a positive polarization surface charge density σp on the side of the negative plate and a negative surface charge density on the positive side. Note, parenthetically, that only the absolute values of the charge densities are indicated and that the vectors Evac and D are parallel. The total charge is conserved, for instance on the side of the positively charged plate:

(Here the minus sign appears because the polarization charge density σp is negative on the positive side of the capacitor).

Assuming that the plates are very much larger than the distance between the plates, we may apply the following formula for Evac (the magnitude of the vector Evac),

(This electric field strength does not depend on the distance of a field point to the plates: the electric field between the plates is homogeneous.) Now

It is of some interest to note that the polarization vector P (pointing from minus to plus polarization charges, i.e., parallel to Evac) has magnitude P equal to the polarization surface charge density σp. Indeed, the magnitudes of the three parallel vectors are related by,

Tensor character of relative permittivity

As defined above, D and E are parallel, i.e., εr is a number (a scalar). For a non-isotropic dielectric εr may be a second rank tensor,

so that D and E are not necessarily parallel in non-isotropic dielectrics.

Note

  1. The nomenclature of the several surface charge distributions is not standardized. Here we will follow by and large R. Kronig, Textbook of physics, Pergamon Press London, New York (1959). (English translation from the Dutch Leerboek der Natuurkunde)