Electric displacement field

From Wikipedia, the free encyclopedia.

In physics, the electric displacement field (also called electrical field/flux density) is a vector field $\mathbf{D}$ that appears in Maxwell's equations. It accounts for the effects of bound charges within materials. "D" stands for "displacement," as in the related concept of displacement current in dielectrics.

1 Definition
2 Displacement field in a capacitor
3 Units
4 References

[edit] Definition

In general, D for a linear, homogeneous, isotropic material with instantaneous response to changes in the electric field, is defined by the following relation

$\mathbf{D} = \varepsilon \mathbf{E}$

where E is the electric field intensity and $\varepsilon$ is the permittivity of the material. Permittivity in terms of vacuum permittivity (also called permittivity of free space) is given by

$\varepsilon = \varepsilon_{0} \varepsilon_{r}$

where $\varepsilon_{r}$ is the called the relative permittivity (measured in comparison to the permittivity of free space). The relative permittivity is also given in terms of electric susceptibility, $χ$ , as follows

$\varepsilon_{r} = 1 + \chi$

and hence we can write

$\mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_{0} \varepsilon_{r} \mathbf{E} = \varepsilon_{0}(1 + \chi) \mathbf{E} = \varepsilon_{0} \mathbf{E} + \varepsilon_{0} \chi \mathbf{E} = \varepsilon_{0} \mathbf{E} + \mathbf{P}$

where

$\mathbf{P} = \chi \varepsilon_{0} \mathbf{E},$

is called the polarization density of the material.

In linear isotropic media $\varepsilon$ is a constant. However, in linear anisotropic media it is a matrix. 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, in that 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:

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

where $ω$ is frequency of the applied field (e.g. in radians/s). 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 $\varepsilon$ ; can be neglected.

[edit] Displacement field in a capacitor

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:

$\oint_A \mathbf{D} \cdot d\mathbf{A} = Q$

On the sides of the box, $d\mathbf{A}$ is perpendicular to the field, so that part of the integral is zero, leaving, for the space inside the capacitor where the effect of the two plates adds:

$|\mathbf{D}| = \frac{Q}{P}$

which is the charge density on the plate. Outside the capacitor, the effect of the two plates compensate each other and $|\mathbf{D}| = 0$ .

[edit] Units

In the standard SI system of units D is measured in coulombs per square meter (C/m²).

This choice of units results in one of the simplest forms of the Ampère-Maxwell equation:

$\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$

If one chooses both B and H to be measured in teslas, and E and D to be measured in newtons per coulomb, then the formula is modified to be:

$\nabla \times \mathbf{H} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{D}}{\partial t}$

Therefore it is seen as being preferential to express B & H, and E & D in different sets of units.

Choice of units has differed in history, for instance in the electromagnetic system of scientific units, in which the unit of charge is defined so that $1 / 4\pi\varepsilon_0 = 1$ (dimensionless), E and D are expressed in the same units.

[edit] References

"electric displacement field" at PhysicsForums

This is an extract from Wikipedia, the Free Encyclopedia

From Wikipedia, the free encyclopedia.

Contents

[edit] Definition

[edit] Displacement field in a capacitor

[edit] Units

[edit] References