2.2 Electromagnetic Properties of Materials
2.2.1 Basic Electromagnetic Properties
Electromagnetic properties of a material include: magnetic permeability, electrical conductivity, and dielectric permittivity. These properties will be briefly introduced in this section.
Magnetic permeability
When a charge q (which is negative for electrons) is in motion in a magnetic field H (ampere m-1). The charge q would experience a force called the magnetic force, Fm. The characteristics of Fm can be described by defining a vector field quantity, the magnetic flux density B, thus the magnetic force can be expressed as (Cheng, 1989):
B q
Fm = u× [2-1]
On the other hand, as predicted by the Biot-Savart law, moving charges q generates magnetic field, H. The magnetization M (ampere per meter, A m-1) of a material depends on the field H and the magnetic properties of the medium. Therefore, magnetic field or magnetic field intensity H can define as (Cheng, 1989):
B M
H = −
μ0 [2-2]
where the μ0 (= 4π*10-7H m-1) is the permeability of free space. The use of the vector H can write a curl equation relating the magnetic field and the distribution of free current in any medium.
As the magnetic properties of the medium are linear and isotropic, the magnetization is directly proportional to the magnetic field intensity (Cheng, 1989):
H
M =χm [2-3]
where χm is a dimensionless quantity called magnetic susceptibility. Therefore, by substituting Eq. [2-3] into Eq. [2-2] yields (Cheng, 1989):
H H H
B=μ0(1+χm) =μ0μr =μ [2-4]
where μr is another dimensionless quantity called relative permeability of the medium. The parameter μ is the absolute permeability (or sometimes just permeability) of the medium.
For simple media (linear, isotropic, and homogeneous), χm and μr are constants.
The permeability of most materials is very close to μ0. For ferromagnetic materials such
as iron, nickel, and cobalt, μr could be very large. The relative permeability of some selected materials is listed in Table 2-1.
Table 2-1 Relative permeability data for selected materials (modified from Cheng, 1989) Material Relative Permeability (μr)
Nickel 250 Cobalt 600
Iron (pure) 4,000
Aluminum 1.000021 Magnesium 1.000012
Palladium 1.00082 Titanium 1.00018
Bismuth 0.99983 Gold 0.99998
Copper 0.99999
Water 1
Electrical conductivity
Consider N number of charges q across a surface with velocity u, it is convenient to define a vector point function, volume current density, or simply current density, J, in amperes per square meter (Cheng, 1989)
Nqu
J = [2-5]
Since the conduction currents are the results of the drift motion of charges carried under the influence of an applied electric field E (newtons per coulomb, N C−1 or, equivalently, volts per meter, V m−1), it can be justified that for most conducting materials the average drift velocity is directly proportional to the electric field intensity. For metallic conductors (Cheng, 1989):
E
u =−μe [2-6]
where μe is the electron mobility measured in (m2V-1s-1). Therefore, substituting Eq. [2-6]
into [2-5], the current density, J, can be written as:
E
J =−ρeμe [2-7]
where ρe = Nq is the charge density of the drifting electrons and is a negative quantity. Eq.
[2-7] then can be rewritten as:
E
J =σ [2-8]
where the proportionality constant, σ =−ρeμe, is a macroscopic constitutive parameter of the medium called electrical conductivity (siemens per meter, S m-1) (Cheng, 1989). Table 2-2 shows the electrical conductivities of some frequently used materials.
Table 2-2 Electrical conductivities of materials (modified from Cheng, 1989) Material Conductivity, S m-1
Silver 6.17 x 107
Copper 5.80 x 107
Aluminum 3.54 x 107
Iron 107
Seawater 4
Fresh water 10-3
Distilled water 2 x 10-4
Dry soil 10-5
Glass 10-12
Fused quartz 10-17
Dielectric permittivity
Before the discussion of the dielectric permittivity, phenomenon of polarization should be introduced. Polarization arises when a force displaces a charge from some equilibrium position. However, polarization cannot occur instantaneously given that charges possess inertia; therefore, polarization is a dynamic phenomenon with a characteristic time-scale (Santamarina et al., 2001).
In the static case (or zero frequency), when the dielectric properties of the medium are linear and isotropic, the polarization is directly proportional to the electric field intensity, E, and the proportionality constant is independent of the direction of field, thus the polarization vector P can be written as:
E
P=ε0χe [2-9]
where ε0 ( ≒ 1/36π * 10-9 farad per meter, F m-1) is the permittivity of free space, χe is a dimensionless quantity called electric susceptibility. A dielectric medium is linear if χe is independent of E and homogenous if χe is independent of space coordinates.
Therefore, a new fundamental field quantity, called the electric flux density or electric displacement, D, can be defined with polarization vector P as:
E
where εr is a dimensionless quantity known as relative permittivity or dielectric constant of the medium. The coefficient ε = ε0εr is the absolute permittivity of the medium and is
-1
used materials (Cheng, 1989).
For single component and homogenous materials (like fluid) may experience three types of polarization mechanisms: electronic, ionic, and dipolar. Electronic polarization occurs when the externally applied electric field causes a shift in the atom’s positive and negative charges, as shown in Fig. 2-1a. Equilibrium is attained when the internal Coulomb attractive force produced by the charge separation balances the applied force. When a charge separation occurs, it essentially has a microscopic electric dipole. Ionic polarization occurs in molecules composed of positively and negatively charged ions (cations and anions). An externally applied electric field again results in a microscopic separation of charge centers thus resembling a dipole of charge, as shown in Fig. 2-1b. Dipolar polarization, on the other hand, occurs in materials that possess permanent, microscopic separations of charge center. In the absence of an applied electric field, these permanent dipoles are randomly oriented. In the presence of an applied electric field, these permanent dipoles tend to rotate to align with the applied field as shown in Fig. 2-1c (Lin, 1999).
In the dynamic case, polarization mechanisms display one of two characteristic spectra:
resonance or relaxation. Polarization mechanisms that trigger restoring forces, in general, display a resonance spectrum (This is the case of electronic and ionic polarization). The complex permittivity for resonance is described as (Santamarina et al., 2001)
res
where ε'dc and ε'∞ are the real permittivity at frequencies much lower and higher than the resonant frequency ωres, ω is radian frequency, which equals 2πf (f is frequency), j = √-1, and β represents the viscous drag coefficient in the equation of motion of a single degree of
freedom system.
If the polarization does not restore forces or if damping prevails over inertial forces, the material exhibits a relaxation spectrum. This is the case for molecular, spatial, and double layer polarizations. A typical relaxation equation as well-known Debye's equation is
rel
where ωrel is the characteristic relaxation frequency, and
(
/)
2As discussed above, Fig. 2-2 shows frequency response of permittivity and loss factor for a hypothetical dielectric by various contributing phenomena. Polarizations form different mechanisms accumulate towards lower frequency. As a result, the real permittivity ε’r
increases with decreasing frequency, and resonant mechanisms shows a peak near resonance.
After introducing the polarization mechanism and the dielectric permittivity, the effective imaginary permittivity combining polarization losses and conduction losses can be defined as:
0
expressed in terms of the “effective alternating current (AC) conductivity”:
Therefore, it is convenient in electric field analysis to combine the dielectric loss and conductive loss terms. The resulting equivalent complex permittivity becomes
⎟⎟⎠
Table 2-3 Relative permittivity of some often used materials [modified from Cheng, 1989]
Material Relative Permittivity
Air 1.0
Distilled water 80
Dry soil 3 - 4
Teflon 2.1
Sea water 72
+
Direction of applied field-+ (a) Electronic
polarization (resonance)
- +
(b) Ionic polarization (resonance)
(c) Orientation polarization of dipolar molecules (relaxation)
Before After
+
+
Direction of applied field--+ (a) Electronic
polarization (resonance)
- +
(b) Ionic polarization (resonance)
(c) Orientation polarization of dipolar molecules (relaxation)
Before After
Fig. 2-1 Polarization mechanisms in single component materials (modified from Santamarina et al., 2001)
Fig. 2-2 Frequency response of permittivity and loss factor for a hypothetical dielectric showing various contributing phenomena (Ramo et al., 1994)
2.2.2 Dielectric Behavior of Water and Soil Solid