Electric displacement field

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In physics, the electric displacement field or electric induction^{[citation needed]} is a vector field <math>\mathbf{D}</math> that appears in Maxwell's equations. It accounts for the effects of unbound charges within materials. "D" stands for "displacement," as in the related concept of displacement current in dielectrics.

[edit] Definition

In general, D is defined by the relation

<math>\mathbf{D} = \varepsilon_{0} \mathbf{E} + \mathbf{P}</math>

where E is the electric field, <math>\varepsilon_{0}</math> is the vacuum permittivity, and P is the polarization density of the material.

If P can be written as a linear function of E, which is the case in most materials, it can be written as

<math>\mathbf{P} = \chi \varepsilon_{0} \mathbf{E},</math>

and by

<math>\mathbf{D} = \varepsilon_{0} \mathbf{E} + \mathbf{P} = \varepsilon_{0}(1 + \chi) \mathbf{E} \equiv \varepsilon \mathbf{E}</math>

it yields

<math>\mathbf{D} = \varepsilon \mathbf{E}</math>

where <math>\varepsilon</math> is the permittivity of the material. In linear isotropic media this will be a constant, and in linear anisotropic media it will be a rank 2 tensor (a matrix). <math>\chi</math> is called the electric susceptibility.

[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 both plates of the capacitor:

<math>\oint_A \mathbf{D} \cdot d\mathbf{A} = Q</math>

On the sides of the box, <math>d\mathbf{A}</math> 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:

<math>|\mathbf{D}| = \frac{Q}{A}</math>

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

[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:

<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} </math>

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:

<math>\nabla \times \mathbf{H} = \mu_0 \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{D}}{\partial t} </math>

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 <math>1 / 4\pi\varepsilon_0 = 1</math> (dimensionless), E and D are expressed in the same units.ca:Desplaçament elèctric cs:Elektrická indukce de:Elektrische Flussdichte fr:Induction électrique it:Campo di spostamento elettrico nl:Elektrische verplaatsing ja:電束密度 pl:Indukcja elektryczna sr:Електрична индукција sv:Elektrisk flödestäthet uk:Вектор електричної індукції