Basic Properties of Dielectric Materials
In electromagnetics we classify materials generally into four broad categories:
1. Conductor – free charge moves easily
2. Semi-conductor – free charge moves somewhat
3. Dielectric (insulators) – no free charge, but produces change to electric field
4. Magnetic – produces change to magnetic field
The electric dipole is used to model the effects that a dielectric material produces on an external electric field. Before discussing dielectric materials, we will first quickly review the electric dipole moment model.
Electric Dipole
An electric dipole is formed from two charges of opposite sign and equal magnitude located close together (wrt to observation distance).
The absolute electric potential is the sum
Where
Now, if the observation distance is much greater than d, then (1) can be approximated as
Defining the electric dipole moment as
p = qd
then (2) can be expressed as
This is the absolute potential at point r of an electric dipole of moment p located at coordinates r, where R =|rer|e . Equation (4) is an accurate representation of the potential and electric field produced by the dipole provided the observation distance is approximately 2.5 times d. (See VisualEM, Section 3.6.2 worksheet.)
Bound Charge
When a dielectric material is placed in an external electric field, the dielectric alters this electric field due to bound or polarization charges that are formed in the dielectric. A capacitor is an example of this.
A simplistic model of the atomic conditions that produce this bound charge is the displacement of the electron cloud around a nucleus. In an electric field, the negatively charged electron cloud becomes displaced very slightly from the positively charge nucleus:
The electrostatic effects of this displacement (potential and electric field) are model by an electric dipole moment p:
This is only an approximation, but for charge neutral molecules and relatively “small” electric fields, it is a very, good model. Using this model, a polarized material can be visualized as
These bound charges cannot move. Unlike free charge, bound charge is induced by an external E field and vanishes when the external E field is removed.
Multipole Expansion
A localized distribution of electric charge density p_{e}(r’) is centered at the origin of a coordinate system. The electric potential outside of an imaginary sphere of radius R that fully contains the charge can be written as an expansion in so-called spherical harmonics as (Jackson, 3^{rd} ed., p. 145)
where q_{lm} is the multipole moment and Y_{im} is the spherical harmonic function. After a bit of manipulation, this equation reduces to
The higher order terms in this expression decay as 1/r^{4} or quicker. Hence, the potential produced by these terms is much smaller than the 1/r^{2} term. Further, if the entire distribution of charge is neutral so that Q_{e}=0, then the potential is dominated by the 1/r^{2} term, which is the electric dipole term. That is why we model the bound charge in a material by the dipole term only.
Polarization Vector
Consider a polarized volume with a density of p’s:
A polarization vector P is defined as
where N is the number of molecules in ev . The macroscopic effects of a polarized dielectric material are modeled with P, which really is an average dipole moment per unit volume of the material.
In summary, when a dielectric material is placed in an external electric field, as we saw in the previous lecture, the dielectric alters this electric field due to bound or polarization charges that are formed in the dielectric.
Electric Susceptibility and Permittivity
It is customary in electromagnetics to “bury” the effects of bound polarization effects in materials through the electric flux density vector, D. The polarization effects of a dielectric can be accounted for by defining D as What we desire now is to know P in terms of E. Basically, without knowing P this theory is not very useful.
It has been found through experimentation that for many materials with “small” E that
P =e_{0}e_{e}E
where e_{e }is the electric susceptibility of a material (dimensionless). Substituting (7) into (6) gives
D =e_{0}E +e_{0}e_{e}E =e_{0} (1+ ee ) E
We can rewrite this as
D =eE [C/m^{2}]
This is called a constitutive equation. The constant e is called the permittivity of the material where
e=e_{r}e_{0} = (1+ e_{e} )e_{0} [F/m]
and
e_{r} =1+ e_{e}
is called the relative permittivity of the material (dimensionless). The relative permittivity e_{r} is usually measured for different materials and then tabulated. (A good reference book is A. von Hippel, Dielectric Materials and Applications, Artech House, 1995.) Here are some examples for DC fields:
Definitions
• Linear material means e_{e} e f (E)
• Homogeneous material means e_{e} e f (r )
• Isotropic material means e_{e} e f (a_{E} ) where a_{E} is the direction of E
• Simple material means that it is linear, homogeneous and isotropic.
Gauss’ Law
In the presence of dielectric materials, there is a slight change that must be made to Gauss’ law:
e_{s}Deds = Q_{ free}
where Q_{free} is the net free charge enclosed by s. Applying the divergence theorem to (11) and simplifying gives the point form of Gauss’ law:
eeD = e_{v}
where it is understood that ev is the free charge density only. These two equations (11) and (12) are ALWAYS true. From these two equations, we can deduce that sources of D are free charge only. Conversely, the sources of E are both free and bound charge.