Electric displacement: Difference between revisions

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imported>Paul Wormer
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[[Image: Dielectric.png|thumb|right|350px|Two capacitors at same voltage ''V''. On the left vacuum between the plates and on the right  a dielectric with relative permittivity &epsilon;<sub>r</sub>. Absolute values of surface charge densities are indicated by &sigma;.]]
[[Image: Dielectric.png|thumb|right|350px|Two capacitors at same voltage ''V''. On the left vacuum between the plates and on the right  a dielectric with relative permittivity &epsilon;<sub>r</sub>. Absolute values of surface charge densities are indicated by &sigma;.]]


The electric displacement ''D'' (is the magnitude of vector '''D''', which points from &minus; to + charge) is equal to the true surface charge density &sigma;<sub>true</sub> indicated in the figure,
In the special case of a [[parallel-plate capacitor]], often used to study and exemplify problems in electrostatics,  the electric displacement ''D'' is easy to visualize.  In this example ''D'' (the magnitude of vector '''D''', which points from &minus; to + charge) 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). 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''.
where two  [[parallel-plate capacitor]]s are shown that are completely identical except for the matter between the plates. In particular the plates have the same voltage difference ''V''.


To explain this interpretation of ''D'',  we recall that the [[relative permittivity]] may be defined as the ratio of two [[capacitances]],  ''C'' &equiv; ''Q'' / ''V'', namely 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 this result for ''D'',  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,  
:<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} =
\frac{Q_\mathrm{true}}{Q_\mathrm{free}} = \frac{\sigma_\mathrm{true}}{\sigma_\mathrm{free}},
\frac{Q_\mathrm{true}}{Q_\mathrm{free}} = \frac{\sigma_\mathrm{true}}{\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 that ''Q'' is &sigma; &times; ''A'', with ''A'' the area of the plates.
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 charge density &sigma;<sub>p</sub> on the side of the negative plate and a negative charge density on the other side. Note, parenthetically, that only the absolute values of the charge densities are indicated and that all vectors are parallel (pointing from &minus; to +).  The total charge is conserved, for instance on the side of the positively charged plate:
:<math>
\sigma_\mathrm{free} =  \sigma_\mathrm{true} - \sigma_\mathrm{p}= \epsilon_\mathrm{r}\sigma_\mathrm{free} - \sigma_\mathrm{p}.\,
</math>
(Here the minus sign appears  because the polarization charge density &sigma;<sub>p</sub> 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 ''E''<sub>vac</sub> (the magnitude of the vector '''E'''<sub>vac</sub>),
:<math>
E_\mathrm{vac} = \frac{\sigma_\mathrm{free}}{\epsilon_0}.
</math>
(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
:<math>
D \equiv \epsilon_0 \epsilon_\mathrm{r}  E_\mathrm{vac} = \epsilon_\mathrm{r} \sigma_\mathrm{free} =
\sigma_\mathrm{true}\,.
</math>
It is of some interest to note that the polarization vector '''P''' (pointing from minus to plus) has magnitude ''P'' equal to the polarization charge density &sigma;<sub>p</sub>. Indeed,
:<math>
P \equiv D - \epsilon_0 E_\mathrm{vac} =  \sigma_\mathrm{true} - \sigma_\mathrm{free} = \sigma_\mathrm{p}.
</math>


<!--
The value of the electric displacement D may be thought of as equal to the amount of free charge on one plate divided by the area of the plate. From this point of view D is frequently called the electric flux density, or free charge surface density, because of the close relationship between electric flux and electric charge. The dimensions of electric displacement, or electric flux density, in the metre-kilogram-second system are charge per unit area, and the units are coulombs per square metre. In the centimetre-gram-second system the dimensions of D are the same as those of the primary electric field E, the units of which are dynes per electrostatic unit, or statvolts per centimetre.
-->
==Tensor character of D==
==Tensor character of D==
As defined here, '''D''' and '''E''' are proportional, i.e., &epsilon;<sub>r</sub> is a number (a scalar). For a ''non-isotropic'' dielectric &epsilon;<sub>r</sub> may be a second rank [[tensor]],
As defined here, '''D''' and '''E''' are proportional, i.e., &epsilon;<sub>r</sub> is a number (a scalar). For a ''non-isotropic'' dielectric &epsilon;<sub>r</sub> may be a second rank [[tensor]],

Revision as of 09:40, 5 December 2008

In physics, electric displacement, usually denoted by D, is a vector field in a non-conducting medium, a dielectric, that is proportional to the electric field E. In SI units,

where ε0 is the electric constant and εr is the relative permittivity. In Gaussian units ε0 is not defined and may put equal to unity. In vacuum the dimensionless quantity εr = 1 (both for SI and Gaussian units) and D is simply related, or equal, to E. Often D is termed an auxiliary field with the principal field being E. An alternative auxiliary field is the electric polarization P of the dielectric,

The vector field P describes the polarization (displacement of charges) occurring in a dielectric when it is inserted between the charged plates of a parallel-plate capacitor. Clearly, the fact that for any insulator εr > 1 (i.e., that D is not simply equal to εrE) has the same physical origin.

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

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

Relation of D to surface charge density σ

Two capacitors at same voltage 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 is easy to visualize. In this example D (the magnitude of vector D, which points from − to + charge) is equal to the true surface charge density   σtrue   (the surface density on the plates of the right-hand capacitor in the figure). 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.

To explain this result for D, 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 Cvac of a capacitor in vacuum,

where we used that Q is σ × A, with A the area of the plates. Clearly, the charge density on the plates increases by a factor εr 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 charge density σp on the side of the negative plate and a negative charge density on the other side. Note, parenthetically, that only the absolute values of the charge densities are indicated and that all vectors are parallel (pointing from − to +). 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) has magnitude P equal to the polarization charge density σp. Indeed,

Tensor character of D

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


(To be continued)