rel="nofollow" href="#fb3_img_img_8b8a0528-4486-512b-a05d-34db9d7a8270.png" alt="images"/> are always positive quantities in a passive lossy medium. The lossy capacitor and the lossy inductor can model the complex relative permittivity and permeability of a medium. This is called the circuit modeling of a material medium. The modeling of the relative permittivity of a medium by the capacitor shows that both the medium and capacitor can store electric energy. Likewise, the relative permeability of medium and the corresponding inductor model both are the magnetic energy storage components.
4.2.6 Static Conductivity of Materials
Figure (4.6a) shows a cylindrical section of the lossy material of conductivity (σ), i.e. resistivity ρ = 1/σ. Its cross‐sectional area and length are A and h, respectively. The lossy material can be modeled as a resistor R, also shown in Fig. (4.6a). The conduction current density Jc flowing through the conductor, due to the free mobile charges, is Jc = Ic/A, where conduction current is Ic = V/R. The electric field intensity in the material is
The above expression (4.2.19b) is Ohm's law for a lossy medium. The following expression of the equivalent resistance of a lossy material, in terms of its resistivity, follows from the above equation (4.2.19c):
(4.2.20)
In general, the Ohm’s law for the anisotropic medium is written in the vector form as
Figure (4.6a) considers h = Δx length of a cylindrical section of conducting material with a cross‐sectional area ΔA. The free charges move in the direction x with a velocity vx on the application of electric field intensity Ex. The conduction current in the x‐direction is the rate of flow of total charge ΔQe contained in a volume, ΔV = ΔA × Δx.
Figure 4.6 Circuit model, parameters of a dielectric medium.
(4.2.21)
In the limiting case,
(4.2.22)
In a material, the charge movement is random due to the scattering and so forth. However, an average motion is assumed, giving the drift of charges in the x‐direction with a drift velocity
where constant μm is called the mobility of a charge. It is noted that μ is also used as a symbol for permeability. On comparing equations (4.2.19b) and (4.2.23b), the following expression for conductivity is obtained:
(4.2.24)
If N is the number of free charges per unit volume, with charge q on each carrier, the charge density is ρe = Nq. The equation (4.2.24) of the conductivity is changed to
(4.2.25)
In the case of a conductor, the charge carrier is electron, i.e. q = qe (the electron charge) and μ = μem (electron mobility). However, for a semiconductor, its conductivity σs is due to both electrons and holes leading to the following expression:
(4.2.26)
where Ne and Nh are numbers of electrons and holes per unit volume. The charges on electron and holes are equal qe = qh = e = 1.6 × 10−19 Coulombs. The electron and hole mobilities, in a semiconductor, are
4.3 Circuit Model of Medium
The circuit model helps to understand the electrical property of a medium. It is further useful for simulating the electrical responses of a medium. The electrical property of a dielectric medium is expressed through relative permittivity that shows the electric energy storage ability of the medium. The capacitor also stores electric energy. Therefore, a dielectric medium is modeled as a capacitor. The loss in a medium is due to the dissipation of energy that is modeled as a resistor. Similarly, the permeability of a medium, such as an inductor, shows its ability to store magnetic energy. Therefore, the permeability of a medium is modeled as an inductor [B.10–B.12].
4.3.1 RC Circuit Model of Lossy Dielectric Medium
Figure (4.6b) shows a parallel‐plate capacitor, containing a lossy dielectric medium with complex relative permittivity
