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Electric Displacement Field





DEFINITION


In general, D is defined by the relation

:\mathbf{D} = arepsilon_{0} \mathbf{E} + \mathbf{P}

where E is the Electric Field , arepsilon_{0} 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, we can write
:\mathbf{P} = \chi arepsilon_{0} \mathbf{E},
and by
:\mathbf{D} = arepsilon_{0} \mathbf{E} + \mathbf{P} = arepsilon_{0}(1 + \chi) \mathbf{E} \equiv arepsilon \mathbf{E}
we arive at the simple formula
:\mathbf{D} = arepsilon \mathbf{E}
where arepsilon 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 ). \chi is called the electric Susceptibility .


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 the plate of the capacitor:

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

The part of the box inside the capacitor plate has no field, so that part of the integral is zero. On the sides of the box, d\mathbf{A} is perpendicular to the field, so that part of the integral is also zero, leaving:



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:


abla imes \mathbf{H} = \mu_0 \mathbf{J} + rac{1}{c^2} rac{\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 such that 1 / 4\pi arepsilon_0 = 1 (dimensionless), E and '''D''' are expressed in the same units.