Anand K. Verma - Introduction To Modern Planar Transmission Lines

(4.1.2)

The above expression is the integral form of Gauss’s law . It can be converted to the differential form by using Gauss’s vector integral identity,

(4.1.3)

Similar to the electric charge distribution, the magnetic charge distribution can be assumed in a volume of the body. The magnetic charge density is expressed as ρ m. The magnetic charge creates a magnetic flux Ψ m. Similar to the case of the electric charge, the elemental magnetic charge in the volume dv is ρ mdv, and the elemental magnetic flux coming out of the surface is where is the magnetic flux density as ds is the elemental surface of the enclosed volume v. It is also called the magnetic displacement vector . The Gauss’s law for the magnetic charge and magnetic flux can be written in the integral form as follows:

(4.1.4)

Figure 4.1 The unit vector is in the direction normal to the surface.

Again, by using Gauss’s vector integral identity, the above expression is written below in the differential form:

(4.1.5)

However, the magnetic charges are not found in nature, i.e. ρ m= 0. Therefore,

(4.1.6)

The amount of charge, or current, is an absolute quantity. It does not dependent on the material medium. Thus, the corresponding flux or the flux density is also not dependent on the surrounding medium. In brief, the charge and current create the electric and magnetic flux field, i.e. the flux densities ; and these are not influenced by a material medium .

Experiments demonstrate that the electrically charged body, or a current‐carrying conductor, interacts with other charged body, or another current‐carrying conductor. Such interaction , i.e. the mutual force , is influenced by the medium surrounding these bodies. Therefore, medium‐independent flux densities cannot explain the interaction between two charged bodies or current‐carrying conductors. The interactions between the charges and current‐carrying conductors take place through the force fields, expressed by the electric field intensity and magnetic field intensity . The field intensities are also responsible for the electromagnetic (EM)‐power transportation through a medium. However, the field intensities are influenced by the electrical and magnetic properties of a medium.

It is noted above that the charge and current create the electric and magnetic flux fields , described by the flux densities They also create the force field described by the electric and magnetic field intensities . The flux density parameters are related to the force field intensity parameters by the following constitutive relations :

(4.1.7)

where ε 0and μ 0are the permittivity and permeability of the free space. The permittivity of any medium is its ability to store electric energy, such as a capacitor. Therefore, it is identified as the capacitance of the free space. Any dielectric material medium can store more electric energy through the mechanism of electric polarization . The electric dipole is created during the process of polarization and the total induced charge is shown as electric flux density . Chapter 6presents a detailed discussion of material polarization. The electric polarization of material under the influence of an external electric field gives a higher value of permittivity as compared to the permittivity of the free space. This is known as the relative permittivity ε rof a medium. It is also known as the dielectric constant of a medium. The permittivity of a medium is ε = ε 0ε r. For an isotropic dielectric medium, ε ris a scalar quantity. However, for an anisotropic medium, it is a tensor quantity.

Likewise, the free space has also an ability to store magnetic energy. It is expressed as its permeability . Magnetic material is magnetized by the process of magnetization under the influence of an external magnetic field. Thus, magnetic material stores more magnetic energy as compared to the free space. The ability of a magnetic material to store magnetic energy is expressed through its relative permeability . The permeability of the medium is μ = μ 0μ r. The permittivity ε 0and permeability μ 0of the free space are the primary physical constants ε 0= 8.854 × 10 −12F/m, μ 0= 4π × 10 −7H/m. Again, for the isotropic magnetic medium, the relative permeability μ ris a scalar quantity, and for an anisotropic medium, it is a tensor quantity.