Electromagnetic Wave Radiation and Scattering

#### **Chapter 4**

## Using Electromagnetic Properties to Identify and Design Superconducting Materials

*Fred Lacy*

#### **Abstract**

Superconductors have a wide array of applications, such as medical imaging, supercomputing, and electric power transmission, but superconducting materials only operate at very cold temperatures. Thus, the quest to engineer room temperature superconductors is currently a hot topic of research. To accomplish this mission, it is important to have a complete understanding of the material properties that are being used to create these superconductors. Understanding the atomic and electromagnetic properties of the prospective materials will provide tremendous insight into the best choice for the materials. Therefore, a theoretical model that incorporates electromagnetic field theory and quantum mechanics principles is utilized to explain the electrical and magnetic characteristics of superconductors. This model can be used to describe the electrical resistance response and why it vanishes at the material's critical temperature. The model can also explain the behavior of magnetic fields and why some superconducting materials completely exclude magnetic fields while other superconductors partially exclude these fields. Thus, this theoretical analysis produces a model that describes the behavior of both type I and type II superconductors. Since there are subtle differences between superconductors and perfect conductors, this model also accounts for this distinction and explains why superconductors behave differently than perfect conductors. Therefore, this theory addresses the major properties associated with superconducting materials and thus will aid researchers in the pursuit of designing room temperature superconductors.

**Keywords:** conductivity, permittivity, permeability, resistivity, resonance, Schrödinger wave equation

#### **1. Introduction**

Superconductors are materials in which electricity can flow indefinitely because electrons can move through the material without losing energy. Superconductivity is a state in which a material's electrical properties or characteristics are altered when the temperature reaches a sufficiently low value. This temperature is known as the material's critical temperature and when the material falls below this temperature, two phenomena will result. One event that occurs is the electrical resistance drops to zero (or electric fields inside the material must vanish). The other outcome which takes place is that magnetic fields diminish inside the material. For

some materials the magnetic field becomes zero (if a field exists in the material prior to the temperature change it becomes zero, and if a field is applied after the temperature change, it will not enter the material) [1–5].

Superconductors are known to exhibit zero electrical resistance or infinite electrical conductivity when the temperature of the superconductor reaches its critical temperature. Electrical conductivity is well described on the macroscale and can be modeled by Ohm's Law. Although macroscale theories usually are not appropriate on a microscale, Ohm's Law remains valid at the atomic level [6]. Bloch's theorem states that for perfect periodic lattices, conduction electrons will lose no energy and thus experience zero resistance [7]. However, since it is impractical to obtain a perfect lattice and since some materials exhibit superconductivity and others do not demonstrate this behavior (e.g., gold, silver, copper), additional theories and/or explanations are needed to explain the phenomenon of superconductivity.

Research is currently underway to develop superconducting materials at room temperature. Electrical engineers, physicists, and material scientists are engaged in this research to understand, identify, and create materials that exhibit superconducting properties. Thus, if superconductivity research can advance this science and develop room temperature materials, many industries will be revolutionized [8–13].

One goal of superconductivity research is to identify and/or create materials to transmit energy or information efficiently and dependably. Because electrical resistance in wires will lead to wasted energy and information loss during transmission, superconductivity research is important in unlocking these mysteries to explain the nature of these materials. Successfully accomplishing the goal in these research areas will revolutionize electrical power transmission and information technology [8–10].

Another major goal in superconductivity research is to develop room temperature superconductors for use in medical imaging. High intensity magnetic fields are needed for MRI and NMR imaging and in order to produce these large fields, superconducting wires are used. The image quality as well as the cost to operate these imaging systems depends upon the use of superconducting magnets. Creating and maintaining the necessary large magnetic fields requires a substantial amount of energy. If this energy is consumed by the resistance in the wires, then the quality and/or operating cost will suffer. Achieving the goal of creating room temperature superconductors will further reduce the cost to operate and transform the medical industry [11–13].

Regarding electrical field properties, electrical resistivity is a fundamental material property that leads to electrical resistance in a material. Electrical resistance is a function of the dimensions of the material that is being utilized or analyzed, whereas electrical resistivity is based on atomic interactions with conduction electrons (thus resistivity is independent of the material dimensions). Electrical resistivity is a function of temperature and a material's resistivity will generally decrease as its temperature decreases [14]. When a material's temperature decreases, conduction electrons will interact less with lattice atoms and therefore they will lose less energy. When a superconductor enters the superconducting state and its electrical resistance vanishes, electrons will flow through that material unimpeded and current flows indefinitely [15].

Regarding magnetic field properties, the Meissner effect is a condition that results in magnetic fields vanishing inside the interior of superconductors (provided that the magnetic field is small). For these small magnetic fields, whether the field is present before or after the material is cooled below its critical temperature and whether the material is a type I or type II superconductor, that magnetic field will

#### *Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

be expelled from the superconductor's interior. If the magnetic field increases and has a value between the upper and lower critical fields, type II superconductors will enter a mixed state in which a portion of the magnetic field will penetrate the superconductor's interior [1–5].

To truly understand why some materials are superconductors and why others are not, and to advance research in superconductivity, theoretical models are needed. Various microscopic or atomic theories have been developed to explain superconductivity [16]. Thermodynamic theories have been developed to explain certain aspects of superconductivity behavior, however, there have been inconsistencies in these theories or explanations [17]. Researchers understand that even though theories may be incorrect or partially correct, those theories can push the science forward and provide some insight into a material's behavior. For example, the BCS theory and the London equations are two of the most successful theories on superconductivity, but they have limitations in certain aspects of their explanations. In particular, the BCS theory was developed several decades before high temperature superconductors were discovered. As a result, it does well in describing the behavior of type I materials, but it is inadequate in explaining how high temperature superconductors operate [18–20]. Furthermore, the BCS theory is imperfect and insufficient in explaining several fundamental properties of superconducting materials such as the Meisner effect [18]. Again, a theory does not have to explain every aspect of a material's behavior, but if it cannot explain the fundamental properties, then it is inadequate. Therefore, a comprehensive theory that explains the electromagnetic properties of all superconducting materials is needed.

There is not a complete theory on superconductivity that explains the electromagnetic properties of type I and type II superconducting materials. Because these previously developed models are incomplete, one model may explain one aspect, but it cannot address another. A given theory can handle certain characteristics of superconductors, but it fails in other areas. So, if scientific research is going to make significant advances, a comprehensive theory is needed.

A general theoretical model has been developed to explain the relationship between conductivity and temperature [21, 22]. This model used atomic analysis and solid state physics principles to develop the theory and explain why electrical conductivity is dependent on temperature. The relationship between conductivity and temperature is first derived and then its accuracy is demonstrated through comparisons to known linear responses from platinum and nickel. Again, a model does not have to be entirely correct in order to be useful and move the science forward. This aforementioned model does not and did not intend to account for superconducting effects. Therefore, the model presented herein will provide missing pieces of the puzzle and demonstrate that it is sufficient to characterize properties of superconductors.

The theory presented here accounts for the electromagnetic properties of both type I and type II superconductors. The theoretical model has been obtained by using quantum mechanics and analyzing conduction electron interactions with atoms in a lattice. This analysis is then used to specify how the electrical resistance of a material will respond. Analysis reveals that certain conditions will allow electrons to move through a material without interference. Under these conditions, the material will act as a superconductor. This theoretical model will then be used to explain the Meisner effect or the response of superconducting materials to external magnetic fields. Since type I and type II superconducting materials display different characteristics, the model will be used to explain this difference. The theory can be used to explain the difference between perfect conductors and superconductors. Finally, the frequency spectrum of a generic material's dielectric response is analyzed to demonstrate the feasibility of this model in explaining superconductivity.

#### **2. Why a new resistance model is needed**

To determine the electrical resistivity of a material (or the electrical resistance since resistivity and resistance are proportional), Matthiessen's rule states that the total resistivity is comprised of the sum of the individual resistivities associated with electron interactions with lattice phonons as well as lattice imperfections [1]. In equation form the resistivity is *ρ* ¼ *ρ<sup>p</sup>* þ *ρ<sup>i</sup>* and thus the equivalent electrical resistance equation is

$$R = R\_p + R\_i \tag{1}$$

where *Rp* is the resistance that arises due to phonons and is a function of temperature, and *Ri* is the resistance that arises due to lattice imperfections (e.g., defects, impurities, grain boundaries, etc.) and is independent of temperature. In essence, Matthiessen's rule is a 'series' approach to electrical resistance, but this approach alone cannot account for the electrical resistance of superconductors since all resistance terms would have to become zero in order for the total electrical resistance to become zero. **Figure 1** shows the individual resistances and the total or sum of these resistance for a typical material. This figure demonstrates that a series approach alone cannot explain electrical resistance of superconductors. As a result, modifications must be made to Matthiessen's rule or a different approach has to be taken to account for superconducting effects in materials. Rather than using a series concept to analyze and explain electrical resistance, a general approach using parallel concepts will be used.

The two-fluid model and the resistively shunted junction have been used to describe superconductivity and it incorporates a parallel approach to phenomenologically describe electrical resistance [23, 24]. These approaches use a 'regular' channel to represent normal or non-superconducting electrons and a 'superconductor' channel to represent superconducting or Cooper-paired electrons. It is presupposed that the Matthiessen series model can be incorporated into the regular channel and therefore the regular channel can model the non-superconducting response of materials. However, since Cooper-paired electrons cannot explain the behavior of high-temperature superconducting materials, these models need to be modified to include or reflect the underlying physics as well as the behavior of all types of superconductors.

#### **Figure 1.**

*(a) Example of electrical resistances for a material associated with conduction electron interaction with lattice atoms (phonons) Rp, as well as conduction electron interaction with lattice imperfections Ri. (b) Example of the total resistance using Matthiessen's rule in which the resistance is the sum of the individual resistances. Using Matthiessen's rule makes it impossible to explain why electrical resistance vanishes in superconductors.*

*Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

#### **3. Electrical resistance analysis**

Information about this model has been developed and published [21, 22, 25, 26]. Since the conduction electrons shown in **Figure 2** travel down two different paths, they will encounter two different electrical resistances. Resistance R1 will represent the resistance that conduction electrons experience if they are not in the space or pathway of atoms. Resistance R2 will represent the resistance that conduction electrons experience if they are in the path of atoms. Since these resistors are in parallel, they have an equivalent resistance represented by R as given in Eq. (2).

$$\frac{1}{R} = \frac{1}{R\_1} + \frac{1}{R\_2} \tag{2}$$

To develop and understand the theoretical model, consider a generic atomic lattice shown in **Figure 2**. The model is a two-dimensional lattice structure with periodic atoms and conduction electrons that travel through the lattice. An electron will either travel through a path that contains atoms, or it will not. Atoms will vibrate at a rate that is a function of temperature. These vibrations will enlarge or expand the space that contains atoms or conversely, as temperature decreases, the space between atoms will increase. A higher proportion of conduction electrons in the space between atoms will lead to higher electrical conductivity. By analyzing a unit cell within the lattice, an equation can be obtained that represents the response for the entire lattice.

This model will yield an equation that describes the relationship between conductivity (or resistivity) and temperature. This model utilizes the concept that when electrons are in the space or path containing atoms, these atoms will always impede the flow of the electrons. In the case of superconducting materials at temperatures below their critical temperature, this will not be the case. So, in order to address superconductivity with this model, additional elements will be factored into the analysis.

When conduction electrons are in the region where they will only directly encounter other electrons, the electrical resistivity in this region will be temperature dependent and will have a response that linearly decreases as temperature decreases. A typical example of this response is shown in **Figure 3** where the

#### **Figure 2.**

*Illustration of the atomic lattice showing electrons (small circles) and atoms (large circles). In a nonsuperconducting state, an electron will experience very little resistance (leftmost electron) if it travels between atoms (resistance R1) and much resistance (rightmost electron) if it travels in the path of atoms (resistance R2). These are the two cases that can occur within a unit cell (dotted box).*

**Figure 3.**

*Example of electrical resistance as a function of temperature for conduction electrons that travel in pathways between atoms and will not directly contact atoms. This graph represents the resistance of resistor R1.*

resistivity *ρ* and resistance R1 generally decreases with decreasing temperature. Because of lattice defects, grain boundaries, and impurities, resistance R1 will not reach zero when the temperature is zero, but it will have a residual value. This represents the resistance in the region where atoms do not exist regardless of whether the material is superconducting or non-superconducting.

Resistance R2 represents electrons that will travel in regions where they will encounter atoms. This resistance will have two different responses depending upon whether the material is a superconductor or not. If the material is a superconductor, then at some critical temperature, the atoms will offer no resistance or they will be invisible to conduction electrons, but above this critical temperature, there will be a non-zero temperature dependent response. An example of this response is shown in **Figure 4a**. For non-superconducting materials, the electrical resistance R2 will not become zero like it does for superconductors at the critical temperature. The resistance will have some temperature dependent response and have some non-zero value when the temperature reaches zero. An example of this response is shown in **Figure 4b**.

This evaluation provides an analysis of the resistances that can (and will) be used in the theoretical model. To fully incorporate superconducting effects into the theoretical model, the interaction between the atoms and conduction electrons must be examined. This will provide details that explain when and why certain materials become superconductors. The analysis of the conduction electron and atom interaction will be accomplished by utilizing the Schrödinger wave equation and quantum mechanics.

#### **Figure 4.**

*Example of electrical resistance as a function of temperature for conduction electrons traveling in the pathway of atoms (a) in a superconducting material [type I] and (b) in a non-superconducting material. These graphs represent the resistance of resistor R2.*

*Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

#### **4. Atomic theory analysis**

Through the wave particle duality theorem, it is understood that objects behave like waves and like particles. Thus, atomic particles will display both particle-like properties as well as wave-like properties. To highlight this theorem, electrons are indeed particles with electrical charge and mass, but they are also waves that travel with a wavelength and frequency [1]. Traveling electrons can be characterized by the Schrödinger wave equation as shown in Eq. (3).

$$-\frac{\hbar^2}{2m}\frac{\partial^2 \psi(\mathbf{x},t)}{\partial \mathbf{x}^2} + V\psi(\mathbf{x},t) = i\hbar \frac{\partial \psi(\mathbf{x},t)}{\partial t} \tag{3}$$

In this equation, *V* represents the potential energy that a traveling electron will encounter at a particular time and location. Because many electrical charges exist in this model, the overall potential energy can be represented by the sum of the potentials associated with the charges that a traveling electron will encounter. So

$$V = \sum \frac{qQ}{4\pi\varepsilon\alpha} \tag{4}$$

where *q* is the charge of a traveling electron, *Q* represents the charge of a nearby object (e.g., nucleus or another electron), *ε* is the permittivity or dielectric constant, and *x* is the distance between the traveling electron and nearby object.

Because there are many charges in the system, the exact solution to the Schrödinger wave equation will be extremely complex. However, the general solution to the wave equation, or Eq. (3) is given by

$$
\psi(\mathbf{x},t) = A e^{j(k\mathbf{x}-\alpha t)} u\_k(\mathbf{x})\tag{5}
$$

where *k* is the wavenumber, *ω* is the frequency of the traveling electron wave, and *uk(x)* is the Bloch function and has the periodicity of the lattice [1]. If the potential energy, *V*, is zero, then the electron will be a free particle and thus the wavenumber, *k,* will be a real number. This means that the wave can be modeled with a sinusoidal response and thus the electron will travel unimpeded.

Thus, if and when the permittivity becomes large enough, the potential energy will become sufficiently small and thus as conduction electrons travel through the lattice, they will not be attenuated. Therefore, if or when the permittivity becomes infinitely large, atoms will offer no resistance to moving electrons.

#### **5. Magnetic field analysis**

The electrical resistance response of a material is a major characteristic to determine if it is a superconductor. However, the response of a material to magnetic fields is also a major factor in determining if the material is a superconductor as well as what type of superconductor the material is. Therefore, magnetic field analysis is needed and must be incorporated into the theoretical model. To integrate a material's magnetic field response into the model, electromagnetic field theory will be considered.

Maxwell's equations can be used to understand how electromagnetic fields interact with matter [27]. Information such as how much of an electromagnetic wave will be transmitted or reflected at an interface between two different materials, and a wave's velocity, frequency, and wavelength in a material can be

determined by analyzing Maxwell's equations. To gain a clear understanding of why the Meissner effect occurs in superconductors, Ampere's law (one of Maxwell's equations), will be used. In differential equation form

$$
\nabla \mathbf{x} \overline{B} = \mu \left( \overline{f} + \varepsilon \frac{d\overline{E}}{dt} \right) \tag{6}
$$

where *B* is the magnetic flux, *μ* is the permeability of the material, *ε* is the permittivity of the material, *J* is the current density (and *J* ¼ *σE*), and *E* is the electric field. Electric fields and magnetic fields are related, and Eq. (6) shows that a magnetic field can be produced from an electric current and from a time changing electric field.

In addition to using Ampere's law as given in Eq. (6), the theoretical model will incorporate other electromagnetic equations and concepts because of the interdependence of the electric and magnetic fields. Specifically, Maxwell-Faraday's equation will be used or equivalently in differential equation form, <sup>∇</sup>*xE* ¼ � *dB dt*, and the force on charged particles due to external fields will be used or equivalently *<sup>F</sup>* <sup>¼</sup> *<sup>q</sup> <sup>E</sup>* <sup>þ</sup> *v x <sup>B</sup>* .

Eq. (6) reveals that the magnetic field is related to the electric field (or the time derivative of the electric field), but this equation also illustrates that this relationship is dependent upon the material's permittivity. Since a material's permittivity will behave as outlined and described previously, the main properties that characterize superconductors, zero electrical resistance and zero interior magnetic field are interrelated and therefore will be coupled.

#### **6. Electrical resistance results**

To demonstrate that the theoretical model functions properly and characterizes the behavior of superconducting materials, the resistance models, the parallel resistor concept, and the Schrödinger wave equation (and its solutions), will be used. Consequently, the model will be validated by demonstrating that it accurately describes the electrical resistance of both superconducting and nonsuperconducting materials. Furthermore, the distinction between the resistance of type I and type II superconductors will be made.

First the case for superconducting materials is considered. Resistances R1 and R2 are displayed on the same graph as a function of temperature as shown in **Figure 5a**. These curves are representative of the two resistances that would occur for superconductors. The theoretical model is represented by parallel resistors, so when these values are combined using the parallel resistor equation, the result is shown in **Figure 5b**. It is seen that the equivalent resistance has a linearly decreasing slope until the temperature reaches the transition temperature TC. When it reaches this temperature, the resistance then abruptly goes zero. This is typical for the electrical resistance response of a superconducting material [1–5].

Next, the case for non-superconducting materials is considered. Resistances R1 and R2 are displayed on the same graph as a function of temperature as shown in **Figure 6a**. These curves are symbolic of the two resistance types that would occur for non-superconductors. Again, the theoretical model is represented by parallel resistors, so when these values are combined using the parallel resistor equation, the result is shown in **Figure 6b**. It is seen that the equivalent resistance resembles resistance *R1*. This result is accurate and expected because resistance *R2* is always much larger than resistance *R1* and since parallel resistors will resemble the smaller

*Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

**Figure 5.**

*Example of electrical resistances for a type I superconducting material. (a) Graph showing the resistance of electrons traveling in the 'gaps' (represented by resistor R1) and the resistance of electrons traveling in the pathway of atoms (represented by resistor R2). (b) Graph showing the total resistance of a superconducting material when the two resistors R1 and R2 are combined in parallel. There is no resistance from the atoms when the material reaches its transition temperature.*

#### **Figure 6.**

*Example of electrical resistances for a non-superconducting material. (a) Graph showing the resistance of electrons traveling in the 'gaps' (represented by resistor R1) and the resistance of electrons in the pathway of atoms (represented by resistor R2). (b) Graph showing the total resistance of a non-superconducting material when the two resistors R1 and R2 are combined in parallel. The resistance of a non-superconductor will not reach zero regardless of the material's temperature.*

resistor value, the total resistance will approximately equal *R1*. As a result, when the resistance of non-superconductors is analyzed over a wide temperature range, there is no abrupt change in resistance because *R2* does not have a transition temperature. Therefore, conduction electrons will never experience zero resistance when they encounter atoms in this material. This is typical for the electrical resistance response of a non-superconducting material [1–5].

The previous analysis focused on revealing the electrical properties of type I superconductor materials. But the same theoretical model can be used to explain the behavior of type II superconductors materials. It is noted that there is a subtle difference between the electrical resistance response of type I materials and type II materials. At the transition temperature, the resistance of type I materials has a sharper transition to zero (i.e., the transition occurs over a small temperature range) whereas type II materials exhibit a slower transition (i.e., the transition occurs over a large temperature range) [28]. This difference in the transition temperature range can be explained using the following analysis.

Resistance *R1* (which is the resistance seen by conduction electrons that will not directly encounter atoms) will be the same for type I and type II materials. However, resistance *R2* (which is the resistance seen by the conduction electrons that directly interact with lattice atoms) for type II materials will be different than the

**Figure 7.**

*Example of electrical resistances for a type II superconducting material. (a) Graph of electrical resistance R2 for electrons traveling in the pathway of atoms and (b) graph of the total electrical resistance for a type II superconducting material. The total electrical resistance is obtained by combining resistances R2 (as shown in this figure) and R1 (as shown in Figure 2) in a parallel manner.*

resistance of type I materials. The difference is that *R2* (or in essence the permittivity of the atoms), will have a response that is not linear near the transition temperature. Whereas type I materials will have a resistance with a sharp response as the temperature approaches the critical temperature, type II materials will have a resistance that gradually approaches zero near the critical temperature. The type II material response is shown in **Figure 7a**. When the resistance shown in **Figure 7a** is used to represent resistance *R2*, the equivalent resistance will have the response shown in **Figure 7b**. This equivalent resistance response is obtained experimentally for typical type II materials [28].

#### **7. Magnetic field results**

The theoretical model will now be validated by demonstrating that it accurately describes the response of both superconducting and non-superconducting materials to applied magnetic fields. When superconducting materials are above their critical temperature they will behave as other materials and thus external magnetic fields can exist on the interior of the superconductor. Upon cooling the superconductor to a temperature below its critical temperature, regardless of whether a magnetic field is applied before or after cooling, type I superconductors will have no interior magnetic field [1–5]. However, the manner in which the magnetic field is excluded from the interior of the superconductor in these two cases is different. These two cases of applying the magnetic field before and after cooling will be analyzed. To simplify this analysis, it is assumed that there is no initial current flowing through the superconductor, and thus current density *J* is zero (it is noted that this assumption will not affect the magnetic field analysis or results).

First, the case of applying the magnetic field after cooling a type I superconductor below its critical temperature is examined. In the area where atoms are located, the permittivity *ε* will be infinite and initially since there are no fields, the electric field is zero and *dE/dt* is zero. Now, when a magnetic field is applied, this field changes in space and time. The change goes from zero to a value that is non-zero and finite. So, ∇*xB* will be non-zero and thus *dE/dt* will become non-zero and finite and as a result, (*ε*)(*dE/dt)* will be infinite. Since Ampere's law as shown in Eq. (6) must be true, (*μ*)(*ε*)(*dE/dt)* must be finite and this will only be true if the permeability *μ* = 0. Since *μ* is zero, there will be no magnetic flux in the material. In essence, conduction electrons on the material's surface will circulate according to

*Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

*F* ¼ *qv x B* and produce magnetic fields that will cancel the external field. Thus, there will be no net magnetic flux inside the material (it is noted that the magnetic field will decay exponentially according to the London penetration depth [1, 2, 29]). When the external magnetic field is removed, the conduction electrons stop circulating and return to their normal motion. The top half of **Figure 8** illustrates the magnetic field's behavior when a type I superconductor is cooled before the magnetic field is applied. Note that if the external magnetic field becomes too large, it will stretch and alter the atoms in the material, the surface current will be affected, and the permittivity *ε* of the atoms will no longer be infinite. As a result, the permeability *μ* will no longer be zero and therefore, the magnetic field will be able to penetrate the material.

Next the case of applying the magnetic field before cooling a type I superconductor below its critical temperature is examined. Initially the permittivity *ε*, the permeability *μ,* and the magnetic flux *B* inside the material will all be non-zero and finite. Then when the material is cooled and reaches the transition or critical temperature, the permittivity will change from finite to infinite. As a result, conduction electrons will circulate to counteract the applied field according to *F* ¼ *qv x B*. These circulating charges will reduce the external magnetic field. As a result of the changing magnetic flux, a changing electric field *E* will be produced, and therefore *dE/dt* will become non-zero. Since ∇*xB* will now be non-zero and finite, then the term (*μ*) (*ε*)(*dE/dt)* must be finite. This will be true if the permeability becomes zero or *μ* = 0. If *μ* is zero, there will be no magnetic flux in the material, so the field will get expelled from the interior of the superconductor. Again, when the external magnetic field is removed, the conduction electrons stop circulating and return to their normal motion. The bottom half of **Figure 8** illustrates the magnetic field's behavior when a type I superconductor is cooled after the magnetic field is applied. Again, note that if the external magnetic field becomes too large, it will stretch and alter the atoms and subsequently surface currents and superconducting properties will be destroyed (thus the electrical resistance will no longer be zero and a magnetic field can penetrate the material).

The previous analysis focused on revealing the magnetic properties or magnetic field response of type I superconductor materials. The model (along with the same analysis) is not only capable of explaining why type II superconductors exhibit their response, but it is able to explain the behavior of type II materials. Type II superconductors have two critical magnetic fields: a lower field TCL and a higher field TCH. If the applied magnetic field is less than TCL, the type II material behaves as a type I material and excludes all of the magnetic flux from its interior. If the applied

#### **Figure 8.**

*Illustration of the response of a superconductor to an external magnetic field. The top half of this figure shows the response of the superconductor if the material is cooled before the field is applied and the bottom half of the figure shows the response if the field is applied before the material is cooled. Once the material is cooled, there will be no magnetic field inside the superconductor.*

magnetic field is greater than TCH, the type II material also behaves like type I materials and will allow all of the magnetic field to penetrate its interior. However, when the applied field is between TCL and TCH only a portion of the external field will penetrate type II materials. This is the mixed state in which superconducting and non-superconducting regions exist in the material [1–5]. Based on the state of the surface current and the state of the permittivity of the atoms in the material, the three different states of type II superconductors (i.e., superconducting, mixed, and normal) can be explained.

In the presence of a magnetic field, the atoms in the material will be stretched and altered. As long as the external magnetic field is smaller than TCL, the atoms are not altered significantly. So, surface currents will exist, and the permittivity of the atoms will be infinite, and the material will exist in the superconducting state. This response is similar to type I materials, so no additional analysis is necessary.

The permittivity of the atoms in the material will vary as a function of the magnetic field (as well as temperature). So, when the external magnetic field exceeds TCL, the permittivity of the atoms will be affected. Thus, the permittivity *ε* will not be infinite and based on Ampere's law, the permeability *μ* will no longer be zero. So, a magnetic field will be able to exist in the material. The surface currents will remain until the magnetic field reaches the upper critical field TCH. The surface currents will block most of the magnetic field but some of the external field will penetrate through the surface. Since the permeability is not zero, the magnetic field will be able to exist in the interior of the material. This creates a pattern of superconducting and non-superconducting regions and forms an Abrikosov lattice [30]. This is the mixed state.

Finally, if the magnetic field is continually increased and reaches TCH, the atoms in the material become stretched to the point where they interfere with all of the surface currents. Since the surface currents will not be able to flow unimpeded and since the permittivity of the atoms will not be infinite, the material moves from the mixed state to a non-superconducting state.

#### **8. Superconductors vs. perfect conductors**

If a model that explains superconductivity is going to be completely correct, it must also be able to explain why superconductors are different from perfect conductors. This theoretical model can explain this difference. The main difference between a perfect conductor and a superconductor is in the response to a magnetic field that is applied before the material is cooled to the transition temperature. A superconductor will exclude the magnetic field when it reaches the transition temperature whereas a perfect conductor will allow the magnetic field to remain [28, 31]. The magnetic field analysis for superconductors has been performed using the theoretical model, and the model explains the behavior. A similar analysis will be utilized to explain the behavior of perfect conductors.

A perfect conductor is defined as a material that has no resistivity or equivalently zero conductivity when the material is below the transition temperature. So, to compare the perfect conductor to a superconductor, *σ* = ∞ for the perfect conductor, but *ε* = ∞ for superconductors. Although the perfect conductor will have infinite conductivity, its other electromagnetic parameters, *ε* and *μ*, remain normal. Since *σ* = ∞ and *J = σ E*, the electric field *E* must vanish in the material to keep the current density *J* finite. As a result, *dE/dt* must be zero. Furthermore, the term (*μ*) (*ε*)(*dE/dt)* must be zero since the first two terms are finite and *dE/dt* will be zero. Ampere's law requires that ∇*xB* and (*μ*)(*ε*)(*dE/dt)* must be equal, so ∇*xB* must be zero and therefore *B* cannot change spatially. Thus, the perfect conductor will

*Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

#### **Figure 9.**

*Illustration of the response of a perfect conductor to an external magnetic field. The top half of this figure shows the response of the perfect conductor if the material is cooled before the field is applied and the bottom half of the figure shows the response if the field is applied before the material is cooled. Regardless of when cooling occurs, the material maintains the magnetic field that it had when the external field is removed.*

contain the same amount of magnetic flux before and after the material cools to its transition temperature.

Thus, if a magnetic field is applied after the perfect conductor is cooled, that field will not be able to penetrate that material (because the material's interior had no magnetic field prior to cooling). The top half of **Figure 9** illustrates the magnetic field's behavior when a perfect conductor is cooled before the magnetic field is applied. Additionally, if a magnetic field is present in a perfect conductor, it will remain in that material after it is cooled to its transition temperature (since the magnetic flux cannot change spatially, it will remain in the material). The bottom half of **Figure 9** illustrates the magnetic field's behavior when a perfect conductor is cooled after the magnetic field is applied. Therefore, this theoretical model can explain the response of perfect conductors to applied magnetic fields.

#### **9. Permittivity examination**

Superconductors exhibit the properties of zero electrical resistance as well as magnetic field exclusion because the material's permittivity becomes infinite. It can be shown that the permittivity of atoms can become infinite under specified conditions [27]. Atoms possess a resonant frequency and if they are excited at this frequency, their permittivity becomes infinitely large. So, if or when the frequency of the conduction electrons aligns with the resonant frequency of the atoms, sufficient conditions will be met (and the permittivity of the material will become infinite).

In general, a graph of a material's permittivity as a function of frequency displays three notable regions. In the microwave region or below 10<sup>9</sup> Hz the permittivity is constant, in the infrared region or at approximately 10<sup>12</sup> Hz the permittivity displays a spike or discontinuity, and in the ultraviolet region or at approximately 10<sup>15</sup> Hz the permittivity displays a second spike or discontinuity. A typical graph of this response is shown in **Figure 10** [32].

These spikes or discontinuities reveal that the permittivity goes to infinity at these frequencies. Therefore, these types of graphs alone indicate that the permittivity can be infinitely large and thus according to the theoretical model, the potential energy vanishes in the Schrödinger wave equation. When the potential energy term becomes zero, conduction electrons experience no impedance and thus superconductivity results. So, if conduction electrons in a material exist with a frequency that matches the frequency at one of these permittivity spikes, then the necessary conditions will exist that lead to superconductivity.

#### **Figure 10.**

*Illustration of the permittivity as a function of frequency for a generic material. These first discontinuity, which occurs around 1012 Hz represents the resonance that occurs for atom-atom interactions. The second discontinuity, which occurs around 10<sup>15</sup> Hz represents the resonance that occurs for electron-atom interactions.*

The frequency of conduction electrons can be determined by using the Planck-Einstein relationship between energy and frequency. In equation form, *E = hf*, where *E* is the energy of the electron, *h* is Planck's constant, and *f* is the frequency of the electron. It is understood that conduction electrons have larger energies compared to other electrons in a material. These energies are approximately equal to the Fermi energy and will depend upon the material. Fermi energies have been determined for many materials and can range from approximately 1 to 10 eV. Using an average energy value of 5 eV, this corresponds to a frequency of 10<sup>15</sup> Hz [1].

Comparing conduction electron frequencies to the discontinuities in the permittivity spectrum, it is seen that these two frequencies can align at 10<sup>15</sup> Hz. Depending upon a material's characteristics, these two frequencies will align for some materials but will not align for others. This general analysis confirms that the mechanisms and analysis associated with the theoretical model are plausible and thus will lead to the phenomena of superconductivity.

#### **10. Summary and discussion**

A theoretical model has been developed that explains why some materials behave like superconductors (and thus display the corresponding electrical and magnetic properties), and why other materials do not. The theoretical model produces results in which electrical resistance is a function of temperature as well as results that explain why magnetic fields can or cannot exist in these materials. These theoretical results are validated by experimental results regardless of whether a material is classified as a type I superconductor, a type II superconductor, or a non-superconductor.

It is reasonable to state that atoms have a temperature dependent frequency response since energy, temperature, and an atom's motion are directly related [33]. The theoretical model and material properties can be used to help design superconducting materials that achieve zero resistance at specified temperatures. Having knowledge of how atoms function in materials will enhance the ability of scientists and engineers to create materials with specific properties. Therefore, by understanding the temperature and frequency relationship of atoms, engineers can manufacture or create materials that operate with zero resistance under desired conditions [34]. Moreover, since type II superconductors are better than type I materials at withstanding higher magnetic fields (before they exhibit nonsuperconducting properties), engineers can also use this knowledge of atomic behavior in materials to create superconductors with desired properties.

The electrical resistance for superconducting materials is linear for temperatures above the critical temperature, zero for temperatures below the critical

#### *Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

temperature, and has a transition region at the transition temperature. For type I materials, the transition occurs within a narrow temperature range (approximately 10<sup>3</sup> K), whereas for type II materials the transition occurs over a large temperature range (approximately 1 K) [28]. The value and slope of the transition to zero resistance can be affected by many factors such as purity and the presence of isotopes [28, 35, 36]. Because the electrical resistance response of type II materials is prone to these lattice irregularities, they can be considered as type I materials that are either impure materials or composite materials. Therefore, it is reasonable to alter the theoretical model for type I materials and modify the characteristics of the atoms to obtain the response for type II materials.

It has been determined that permittivity is the electromagnetic parameter that is of importance in determining whether a material is a superconductor or not. Because permittivity is the governing parameter and not conductivity, ultimately it would probably be better to model the superconducting channel as a capacitor rather than a resistance. These two electrical circuit elements are similar in that they will limit electrical current. However, a capacitor would offer impedance rather than resistance and thus frequency effects could be modeled [37, 38]. Nevertheless, since impedance and resistance are similar concepts, if frequency effects are not important to an application (and in many applications, frequency effects are not important), the main characteristics associated with superconductors can still be modeled quite well using the resistor concept.

Sometimes a classical physics approach to modeling a system can provide accurate results and insight into the behavior of that system [39–41]. When a classical approach is used to analyze atoms in a material, it can be shown that the atoms have a resonant frequency. Furthermore, it can be shown that the atoms have a permittivity that is dependent upon this resonant frequency [27]. If the frequency of conduction electrons matches the resonant frequency of the atoms, the permittivity will become infinitely large and the material will behave as a superconductor.

Based on this resonant frequency analysis, at sufficiently low temperatures, superconducting atoms will have a resonant frequency that aligns with the frequency of conduction electrons. This allows conduction electrons to travel through the material without encountering electrical potentials. However, the conduction electron and atom frequency alignment will not occur at elevated temperatures due to effects of thermal energy on atoms. Conversely, reducing the temperature of non-superconducting materials will not lead to superconductivity because the atoms (or molecules) in non-superconducting materials have a resonant frequency that will not match the frequency of the conduction electrons.

In applying atomic theory to the resistance analysis, note that the resonant frequency of the atoms is normally 'hidden' at elevated temperatures. An additional frequency will be superimposed onto the resonant frequency of the atoms due to thermal energy. As a result, the resonant frequency will be masked if the temperature is above the critical temperature [42–44]. When the temperature of the superconductor is lowered to the threshold or critical value, the atoms in the material will display their resonance and therefore the material will behave as a superconductor.

In addition to verifying the electrical properties, this theoretical model also accounts for the magnetic field properties associated with superconductors. In other words, the model explains why the Meissner effect occurs. A material's permittivity and permeability are linked through Ampere's law. So, for a material that behaves as a superconductor, when its permittivity becomes infinitely large at sufficiently low temperatures, its permeability must go to zero. Therefore, the material must exclude magnetic fields from its interior. Surface currents produce magnetic fields and will account for this cancelation.

This theoretical model can also explain why the mixed state occurs in type II superconductors (i.e., the model explains why the magnetic field can exist in specified regions of the material). A magnetic field can penetrate the material because when a strong field is applied, the atoms are stretched and distorted and as a result the permittivity will no longer be infinite [45]. Perpetual currents will not be able to exist in the interior of the material to shield the external magnetic field. However, surface currents that exist above the atoms will be able to shield some of the external magnetic field. A fraction of the magnetic field will penetrate this surface shield and will exist inside the material.

To further explain the mixed state or why the magnetic field phenomena occurs in type II materials, consider the following. Superconductors have well known relationships between temperature and resistivity as well as between temperature and magnetic field strength. Because of these relationships, when a magnetic field is applied to a superconductor, that field will have an effect on the electrical resistance [46–48]. Since the magnetic field will alter the relationship between the atoms and electrons, if the field becomes large enough, it will alter the resonant frequency of the atoms. As a result, the permittivity *ε* will no longer be infinite, but it will become finite. Thus, the atoms in the bulk of the material will exhibit electrical resistance (and electrical currents will not flow indefinitely) and the atoms will not have the ability to block external magnetic fields.

#### **11. Conclusion**

A theoretical model has been developed based on atomic level analysis and electromagnetic field theory to explain why some materials exhibit superconductor properties and other materials do not. Specifically, the theoretical model addresses electrical resistance and the material's response to magnetic fields. The model explains why the electrical resistance of superconductors becomes zero or why conduction electrons are unimpeded by atoms. The model also explains why superconductors exhibit the Meissner effect and exclude magnetic fields from their interior. Additionally, the theoretical model is general enough such that it can describe the behavior of all materials (whether they are superconductors or not), but the model is specific enough such that it can explain the behavior of both type I or type II superconductors. Furthermore, the theoretical model also describes or distinguishes the characteristics of superconductors and perfect conductors and thus it is able to differentiate the behavior of these two materials.

Because many theories on superconductivity address specific aspects of superconductors and do not (or cannot) address other aspects, those theories have constraints and thus there are restrictions on the information they provide. Based on the approach and analysis that was used to construct this theoretical model (i.e., atomic physics and electromagnetic field theory), it provides insight into the physical mechanisms that cause materials to become superconducting. Therefore, this theoretical model should aid science and engineering researchers in the quest to develop room temperature superconductors that can be used to produce intense magnetic fields needed for medical imaging, as well as zero electrical resistance needed for supercomputing, and electric power transmission.

#### **Conflict of interest**

The author has no conflict of interest.

*Using Electromagnetic Properties to Identify and Design Superconducting Materials DOI: http://dx.doi.org/10.5772/intechopen.97327*

### **Author details**

Fred Lacy Electrical Engineering Department, Southern University and A&M College, Baton Rouge, LA, USA

\*Address all correspondence to: fred\_lacy@subr.edu

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 5**

## Physics of Absorption and Generation of Electromagnetic Radiation

*Sukhmander Singh, Ashish Tyagi and Bhavna Vidhani*

#### **Abstract**

The chapter is divided into two parts. In the first part, the chapter discusses the theory of propagation of electromagnetic waves in different media with the help of Maxwell's equations of electromagnetic fields. The electromagnetic waves with low frequency are suitable for the communication in sea water and are illustrated with numerical examples. The underwater communication have been used for the oil (gas) field monitoring, underwater vehicles, coastline protection, oceanographic data collection, etc. The mathematical expression of penetration depth of electromagnetic waves is derived. The significance of penetration depth (skin depth) and loss angle are clarified with numerical examples. The interaction of electromagnetic waves with human tissue is also discussed. When an electric field is applied to a dielectric, the material takes a finite amount of time to polarize. The imaginary part of the permittivity is corresponds to the absorption length of radiation inside biological tissue. In the second part of the chapter, it has been shown that a high frequency wave can be generated through plasma under the presence of electron beam. The electron beam affects the oscillations of plasma and triggers the instability called as electron beam instability. In this section, we use magnetohydrodynamics theory to obtain the modified dispersion relation under the presence of electron beam with the help of the Poisson's equation. The high frequency instability in plasma grow with the magnetic field, wave length, collision frequency and the beam density. The growth rate linearly increases with collision frequency of electrons but it is decreases with the drift velocity of electrons. The real frequency of the instability increases with magnetic field, azimuthal wave number and beam density. The real frequency is almost independent with the collision frequency of the electrons.

**Keywords:** electromagnetic waves, permittivity, skin depth, loss angle, absorption, Dispersion equations, electron collisions, growth rate, Hall thruster, beam, resistive instability

#### **1. Introduction**

X-rays are used to detect bone fracture and determine the crystals structure. The electromagnetic radiation are also used to guide airplanes and missile systems. Gamma rays are used in radio therapy for the treatment of cancer and tumor Gamma rays are used to produce nuclear reaction. The earth get heat from Infrared waves. It is used to kill microorganism. Ultraviolet rays are used for the sterilizing of


#### **Table 1.**

*The electromagnetic spectrum.*

surgical instruments. It is also used for study molecular structure and in high resolving power microscope. The color of an object is due to the reflection or transmission of different colors of light. For example, a fire truck appears red because it reflects red light and absorbs more green and blue wavelengths. Electromagnetic waves have a huge range of applications in broadcasting, WiFi, cooking, vision, medical imaging, and treating cancer. Sequential arrangement of electromagnetic waves according to their frequencies or wave lengths in the form of distinct of groups having different properties in called electromagnetic spectrum. In this section, we discuss how electromagnetic waves are classified into categories such as radio, infrared, ultraviolet, which are classified in **Table 1**. We also summarize some of the main applications for each range of electromagnetic waves. Radio waves are commonly used for audio communications with wavelengths greater than about 0.1 m. Radio waves are produced from an alternating current flowing in an antenna.

#### **2. Current status of the research**

Underwater communications have been performed by acoustic and optical systems. But the performance of underwater communications is affected by multipath propagation in the shallow water. The optical systems have higher propagation speed than underwater acoustic waves but the strong backscattering due to suspended particles in water always limits the performance of optical systems [1]. UV radiation, free radicals and shock waves generated from electromagnetic fields

#### *Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

are effectively used to sterilize bacteria. Pulsed electromagnetic fields (streamer discharge) in water are employed for the sterilization of bacteria. For biological applications of pulsed electromagnetic field, electroporation is usually used to sterilize bacteria. This technique is commonly applied for sterilization in food processing. The cells in the region of tissue hit by the laser beam (high intensities <sup>10</sup> – 100 W/cm<sup>2</sup> ) usually dies and the resulting region of tissue burn is called a photocoagulation burn. Photocoagulation burns are used to destroy tumors, treat eye conditions and stop bleeding.

Electromagnetic waves in the RF range can also be used for underwater wireless communication systems. The velocity of EM waves in water is more than 4 orders faster than acoustic waves so the channel latency is greatly reduced. In addition, EM waves are less sensitive than acoustic waves to reflection and refraction effects in shallow water. Moreover, suspended particles have very little impact on EM waves. Few underwater communication systems (based on EM waves) have been proposed in reference [2, 3]. The primary limitation of EM wave propagation in water is the high attenuation due to the conductivity of water. For example, it has been shown in [4] that conventional RF propagation works poorly in seawater due to the losses caused by the high conductivity of seawater (typically, 4 S/m). However, fresh water has a typical conductivity of only 0.01 S/m, which is 400 times less than the typical conductivity of seawater. Therefore, EM wave propagation can be more efficient in fresh water than in seawater. Jiang, and Georgakopoulos analyzed the propagation and transmission losses for a plane wave propagating from air to water (frequency range of 23 kHz to 1 GHz). It has been depicted that the propagation loss increases as the depth increases, whereas the transmission loss remains the same for all propagation depths [5]. Mazharimousavi et al. considered variable permeability and permittivity to solve the wave equation in material layers [6]. The Compton and Raman scattering effects are widely employed in the concept of free electron lasers. These nonlinear effects have great importance for fusion physics, laserplasma acceleration and EM-field harmonic generation. Matsko and Rostovtsev investigated the behavior of overdense plasmas in the presence of the Electromagnetic fields, which can lead to the nonlinear effects such as Raman scattering, modulational instability and self-focusing [7]. The increasing relativistic mass of the particles can make plasma transparent in the presence of high intense the electromagnetic field change the properties of plasmas [8]. The models of electromagnetic field generated in a non absorbing anisotropic multilayer used to study the optical properties of liquid crystals and propagation of electromagnetic waves in magneto active plasmas [9]. Pulse power generator based on electromagnetic theory has applications such as water treatment, ozone generation, food processing, exhaust gas treatment, engine ignition, medical treatment and ion implantation. The similar work was reviewed by Akiyama et al. [10]

Applications for environmental fields involving the decomposition of harmful gases, generation of ozone, and water treatment by discharge plasmas in water utilizing pulsed power discharges have been studied [11–14]. High power microwave can be involved to joining of solid materials, to heat a surface of dielectric material and synthesis of nanocomposite powders. Bruce et al. used a high-power millimeter wave beam for joining ceramics tubes with the help of 83-GHz Gyrotron [15]. The use of shock waves to break up urinary calculi without surgery, is called as extracorporeal shock wave lithotripsy. Biofilm removal to inactivation of fungi, gene therapy and oncology are the interesting uses of shock waves lithotripsy. Loske overviewed the biomedical applications (orthopedics, cardiology, traumatology, rehabilitation, esthetic therapy) of shock waves including some current research. [16]. Watts et al. have reported the theory, characterization and fabrications of metamaterial perfect absorbers (MPAs) of electromagnetic waves. The motivation

for studying MPAs comes mainly from their use in potential applications as selective thermal emitters in automotive radar, in local area wireless network at the frequency range of 92–95 GHz and in imaging at frequency 95 and 110 GHz. [17]. Ayala investigated the applications of millimeter waves for radar sensors [18]. Metamaterial perfect absorbers are useful for spectroscopy and imaging, actively integrated photonic circuits and microwave-to-infrared signature control [19–21]. In [22, 23], authors show the importance of THz pulse imaging system for characterizing biological tissues such as skin, muscle and veins. Reference [24] reported the propagation of EM waves on a graphene sheet. The Reference [25] compared the CNT-based nano dipole antenna and GNR-based nano patch antenna. Due to short wavelength, even a minute variations in water contents and biomaterial tissues can be detected by terahertz radiations due to existence of molecular resonances at such frequencies. Consequently, one of the emerging areas of research is analyzing the propagation of terahertz electromagnetic waves through the tissues to develop diagnostic tools for early detection and treatment such as abnormalities in skin tissues as a sign of skin cancer [26]. Shock waves may stimulate osteogenesis and chondrogenesis effects [27], induce analgesic effects [28] and tissue repair mechanisms [29]. Shock waves therapy are also used to treat oncological diseases and other hereditary disorders [27, 30]. Chen et al. proposed a mathematical model for the propagating of electromagnetic waves coupling for deep implants and simulated through COMSOL Multiphysics [31]. Body area networks technological is used to monitor medical sensors implanted or worn on the body, which measure important physical and physiological parameters [32, 33]. Marani and Perri reviewed the aspects of Radio Frequency Identification technology for the realization of miniaturized devices, which are implantable in the human body [34]. Ultrasonic can transport high power and can penetrate to a deeper tissue with better power efficiency. [35, 36]. Ref. [37], discuss the radar-based techniques to detect human motions, wireless implantable devices and the characterization of biological materials. Low frequency can deliver more power with deeper penetrating ability in tissue [38, 39]. Contactless imaging techniques based on electromagnetic waves are under continuous research. Magnetic resonance imaging technology and physiological processes of biological tissues and organisms [40, 41]. The electrical properties of biological mediums are found very useful because it is related to the pathological and physiological state of the tissues [42–44].

#### **3. Interaction of electromagnetic wave fields with biological tissues**

From last decade, researchers are interested about biological effects of electromagnetic energy due to public concern with radiation safety and measures. The electromagnetic energy produces heating effects in the biological tissues by increasing the kinetic energy of the absorbing molecules. Therefore the body tissues absorb strongly in the UV and in the Blue/green portion of the spectrum and transmit reds and IR. A surgeon can select a particular laser to target cells for photovaporization by determining which wavelengths your damaged cell will absorb and what the surrounding tissue will not. The heating of biological tissues depends on dielectric properties of the tissues, tissue geometry and frequency of the source. The tissues of the human body are extremely complex. Biological tissues are composed of the extracellular matrix (ECM), cells and the signaling systems. The signaling systems are encoded by genes in the nuclei of the cells. The cells in the tissues reside in a complex extracellular matrix environment of proteins, carbohydrates and intracellular fluid composed of several salt ions, polar water molecules and polar protein molecules. The dielectric constant of tissues decreases as the

*Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

frequency is increased to GHz level. The effective conductivity, rises with frequency. The tissues of brain, muscle, liver, kidney and heart have larger dielectric constant and conductivity as compared to tissues of fat, bone and lung. The action of electromagnetic fields on the tissues produce the rotation of dipole molecules at the frequency of the applied electromagnetic energy which in turn affects the displacement current through the medium with an associated dielectric loss due to viscosity. The electromagnetic field also produce the oscillation of the free charges, which in turn gives rise to conduction currents with an associated energy loss due to electrical resistance of the medium. The interaction of electromagnetic wave fields with biological tissues is related to dielectric properties. Johnson and Guy reviewed the absorption and scattering effects of light in biological tissues [45]. In ref. [46], the method of warming of human blood from refrigerated (bank blood storage temperature � 4 to 6°C) has been discussed with the help of microwave.

#### **4. Complex dielectric permittivity**

The dielectric permittivity of a material is a complex number containing both real and imaginary components. It describes a material's ability to permit an electric field. It dependent on the frequency, temperature and the properties of the material. This can be expressed by

$$
\varepsilon\_{\varepsilon} = \varepsilon\_0 (\varepsilon' - j\varepsilon'') \tag{1}
$$

where *ε*<sup>0</sup> is the dielectric constant of the medium. The *ε*<sup>00</sup> is called the loss factor of the medium and related with the effective conductivity such that *<sup>ε</sup>*<sup>00</sup> <sup>¼</sup> *<sup>σ</sup> <sup>ε</sup>*0*<sup>ω</sup>*. These coefficients are related through by loss tangent tan *<sup>δ</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>00</sup> *<sup>ε</sup>*<sup>0</sup> . In other words loss factor is the product of loss tangent and dielectric constant, that is *ε*<sup>00</sup> ¼ *ε*<sup>0</sup> tan *δ*. The loss tangent depends on frequency, moisture content and temperature. If all energy is dissipated and there is no charging current then the loss tangent would tend to infinity and if no energy is dissipated, the loss tangent is zero [45, 47–49]. The high power electromagnetic waves are used to generate plasma through laser plasma interaction. Gaseous particles are ionized to bring it in the form of plasma through injection of high frequency microwaves. The electrical permittivity in plasma is affected by the plasma density [50]. If the microwave electric field (*E*~) and the velocity (~*v*) are assumed to be varying with *e<sup>i</sup>ω<sup>t</sup>* , the plasma dielectric constant can be read as,

$$\varepsilon = \varepsilon\_0 \left( 1 - \frac{\alpha\_{pe}^2}{\alpha^2} \right) \tag{2}$$

Where; the *ωpe* is the electron plasma frequency and given by the relation,

$$
\rho\_{p\epsilon} = \sqrt{\frac{n\_{\epsilon}e^2}{c\_0 m\_{\epsilon}}} \tag{3}
$$

Recently many researchers have studied the plasma instabilities in a crossed field devices called Hall thrusters (space propulsion technology). The dispersion relations for the low and high frequency electrostatic and electromagnetic waves are derived in the magnetized plasma. The dispersion relations for the resistive and Rayliegh Taylor instabilities has been derived for the propagation of waves in a magnetized plasma under the effects of various parameters [51–61].

#### **5. Propagation of EM fields (waves) in conductors**

The behavior of EM waves in a conductor is quite different from that in a source-free medium. The conduction current in a conductor is the cause of the difference. We shall analyze the source terms in the Maxwell's equations to simplify Maxwell's equations in a conductor. From this set of equations, we can derive a diffusion equation and investigate the skin effects.

#### **5.1 Gauss' law for electric field**

The Electric flux *φ<sup>E</sup>* through a closed surface A is proportional to the net charge *q* enclosed within that surface.

$$
\rho\_E = \oint \vec{E} \cdot \hat{n} dA = \frac{q}{\varepsilon\_0} = \frac{1}{\varepsilon\_0} \int\_V \rho dV \tag{4}
$$

$$\text{Differential form, } \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\varepsilon\_0} \tag{5}$$

#### **5.2 Faraday's law**

The electromagnetic force induced in a closed loop, is proportional to the negative of the rate of change of the magnetic flux, *φ<sup>B</sup>* through the closed loop,

$$
\oint \overrightarrow{E} \cdot d\overrightarrow{l} \,\, ^\circ = \frac{\partial \rho\_B}{\partial t} = \frac{\partial}{\partial t} \oint \overrightarrow{B} \cdot dA \,\,\tag{6}
$$

Faraday's law in differential form,

$$
\overrightarrow{\nabla} \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t} \tag{7}
$$

#### **5.3 Magnetic Gauss's law for magnetic field**

The Magnetic flux *φ<sup>B</sup>* through a closed surface, A is equal to zero.

$$
\rho\_B = \oint \overline{B} \cdot d\mathbf{A} = \mathbf{0} \tag{8}
$$

In the differential form

$$
\overrightarrow{\nabla} \cdot \overrightarrow{B} = \mathbf{0} \tag{9}
$$

#### **5.4 Ampere's law**

The path integral of the magnetic field around any closed loop, is proportional to the current enclosed by the loop plus the displacement current enclosed by the loop.

$$
\oint \overrightarrow{B} \cdot d\overrightarrow{l} = \mu\_0 I + \mu\_0 \varepsilon\_0 \frac{\partial \rho\_E}{\partial t} \tag{10}
$$

Ampere's law in differential form

$$
\overrightarrow{\nabla} \times \overrightarrow{B} = \mu \sigma \overrightarrow{E} + \mu \varepsilon \frac{\partial \overrightarrow{E}}{\partial t} \tag{11}
$$

#### **6. Properties of plane wave (monochromatic) in vacuum**

Let us assume that the wave equations (fields) has the solution in the form of *E* ! *B* � �! ¼ *E* ! <sup>0</sup> *B* ! 0 � �*e*�*i kz* ð Þ �*ω<sup>t</sup>* , then the vector operators can be written as <sup>∇</sup> ! �*ik* and *∂ <sup>∂</sup><sup>t</sup>* ! *iω*.

a. The vector *k* and fields *E* ! *B* � �! are perpendicular From Gauss's law *k* � *E* ¼ 0

b. The field *B* ! is perpendicular to the vector *k* and field *E* !

From Faraday's law �*ik* ! � *E* ! ¼ �*iωB* !

$$
\Rightarrow \overrightarrow{B} = \frac{\overrightarrow{k} \times \overrightarrow{E}}{\alpha} = \frac{k\hat{k} \times \overrightarrow{E}}{\alpha} = \frac{\hat{k} \times \overrightarrow{E}}{\alpha} \tag{12}
$$

Where we have used *<sup>ω</sup>* <sup>¼</sup> *ck* and unit vector ^ *k* ¼ *<sup>k</sup>* ! *=<sup>k</sup>*. This implies that all three vectors are perpendicular to one another (**Figure 1**).

Let us apply curl operator to the 2nd equation.

Maxwell's equation:

$$\overrightarrow{\nabla} \times \left(\overrightarrow{\nabla} \times \overrightarrow{E}\right) = -\overrightarrow{\nabla} \times \left(\frac{\partial \overrightarrow{B}}{\partial t}\right) = -\frac{\partial}{\partial t} \left(\overrightarrow{\nabla} \times \overrightarrow{B}\right) = -\frac{\partial}{\partial t} \left(\mu \overrightarrow{\sigma E} + \mu \varepsilon \frac{\partial \overrightarrow{E}}{\partial t}\right) \tag{13}$$

So

$$-\nabla^2 \overrightarrow{E} = -\mu \sigma \frac{\partial \overrightarrow{E}}{\partial t} - \mu \varepsilon \frac{\partial^2 \overrightarrow{E}}{\partial t^2} \tag{14}$$

Similar, the magnetic field satisfy the same equation

$$-\nabla^2 \overrightarrow{B} = -\mu \sigma \frac{\partial \overrightarrow{B}}{\partial t} - \mu \varepsilon \frac{\partial^2 \overrightarrow{B}}{\partial t^2} \tag{15}$$

**Figure 1.** *Orientations of electric field, magnetic field and wave vector.*

#### **6.1 Skin depth**

Suppose we have a plane wave field. It comes from the �*z* direction and reaches a large conductor. Surface at *<sup>z</sup>* <sup>¼</sup> 0 outside of a conductor: *<sup>E</sup>* <sup>¼</sup> *<sup>E</sup>*0*e*�*iω<sup>t</sup> ex* at *z* ¼ 0. Looking for the wave like solution of electric (magnetic) fields by assuming the wave inside the conductor has the form, where *k* is an unknown constant. Suppose, the waves are traveling only in the z direction (no x or y components). These waves are called plane waves, because the fields are uniform over every plane perpendicular to the direction of propagation. We are interested, then, in fields of the form

$$
\overrightarrow{E}\left(\overrightarrow{B}\right) = \overrightarrow{E}\_0 \left(\overrightarrow{B}\_0\right) e^{-i(kx - at)}\tag{16}
$$

for the waves of the above type, we find from the diffusion equation

$$
\hbar^2 E = -i\alpha\mu\sigma E + \mu\epsilon\omicron\sigma^2 E \tag{17}
$$

$$\begin{cases} \text{Or } \left(k^2 + i\alpha\mu\sigma - \mu\epsilon\omega^2\right)E = 0\\\\ \text{For non-trivial solution } k^2 + i\alpha\mu\sigma - \mu\epsilon\omega^2 = 0 \end{cases} \tag{18}$$

The presence of imaginary term due to conductivity of the medium gives different dispersion relation from the dielectric medium. From Eq. (18) we can expect the wave vector to have complex form.

Let us write

$$
\overrightarrow{\dot{k}} = \overrightarrow{a} - \overrightarrow{i}\overrightarrow{\beta}\tag{19}
$$

Here the real part *α* ! determine the wavelength, refractive index and the phase velocity of the wave in a conductor. The imaginary part *β* ! corresponds to the skin depth in a conductor. The solutions of Eqs. (18) and (19), gives the real and imaginary part of wave vector k in terms of materials' properties.

$$a = a\sqrt{\frac{e\mu}{2}} \left[ \sqrt{1 + \frac{\sigma^2}{\varepsilon^2 \alpha^2}} + 1 \right]^{\frac{1}{2}} \tag{20}$$

$$\text{And } \beta = \frac{a\mu\sigma}{2a} \tag{21}$$

$$\text{Or } \beta = \alpha \sqrt{\frac{\varepsilon\mu}{2}} \left[ \sqrt{1 + \frac{\sigma^2}{\varepsilon^2 \alpha^2}} - 1 \right]^{\frac{1}{2}} \tag{22}$$

If we use complex wave vector *k* ! ¼ *α* ! � *iβ* ! into Eq. (16), then the wave equation for a conducting medium can be written as

$$
\overrightarrow{E} = \overrightarrow{E}\_0 \mathbf{e}^{-\beta \mathbf{z}} \mathbf{e}^{-i(\alpha \mathbf{z} - \alpha \mathbf{t})} \tag{23}
$$

It is clear from the above equation that the conductivity of the medium affects the wavelength for a fixed frequency. The first exponential factor *e*�*β<sup>z</sup>* gives an exponential decay in the amplitude (with increasing z) of the wave as shown in **Figure 2**. The cause of the decay of the amplitude of the wave can be explained in a very precise way in terms of conservation of energy. Whenever the incoming

*Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

electromagnetic radiation interacts with a conducting material, it produces current in the in the conductor. The current produces Joule heating effect which must be compensated from the energy of the wave. Therefore we can expect the decay in the amplitude of the wave. The second factor *e*�*i*ð Þ *<sup>α</sup>z*�*ω<sup>t</sup>* gives the plane wave variations with space and time.

#### **7. Alternating magnetic field in a conducting media**

From Faraday's law, the both fields are related by

$$
\overrightarrow{\dot{k}} \times \overrightarrow{E}\_0 = a\overrightarrow{B}\_0 \tag{24}
$$

$$\text{Or } \overrightarrow{B}\_0 = \frac{\overrightarrow{k} \times \overrightarrow{E}\_0}{\alpha} \tag{25}$$

Thus as in dielectric case, both fields are perpendicular to each other and also perpendicular to the direction of motion with same phase angle.

#### **7.1 Phase change in fields in a conducting media**

The complex wave vector k, gives the phase angle between the fields in a conducting medium. Let us assume that E is polarized along the x direction

$$
\overrightarrow{E} = \hat{i}\overrightarrow{E}\_0 e^{-\beta \mathbf{z}} e^{-i(\alpha \mathbf{z} - \alpha \mathbf{t})} \tag{26}
$$

And the magnetic field results from Eq. (25) is given by

$$
\overrightarrow{B} = \hat{j}\frac{|k|}{\alpha}\overrightarrow{E}\_0 e^{-\beta x} e^{-i(ax-\alpha t)}\tag{27}
$$

From Eq. (19), the complex number *k* can be written as

$$|k| = \sqrt{a^2 + \beta^2}e^{i\rho} = \text{Re}^{i\rho} \tag{28}$$

$$\text{Thus } R = \sqrt{a^2 + \beta^2} = a\sqrt{\epsilon\mu} \left[ \sqrt{1 + \frac{\sigma^2}{\epsilon^2 \alpha^2}} \right]^{\frac{1}{4}} \tag{29}$$

$$\text{And the phase angle } \varphi = \tan^{-1} \frac{\beta}{a} \tag{30}$$

Further if the initial phases of the fields are *φ<sup>E</sup>* and *φB*, then the amplitude are given by

$$
\overrightarrow{E}\_0 = E\_0 e^{i\varphi\_E} \tag{31}
$$

$$
\overrightarrow{B}\_0 = B\_0 e^{i\varphi\_B} \tag{32}
$$

From Eq. (27)

$$B\_0 e^{i\varphi\_\oplus} = \frac{\text{Re}^{i\varphi}}{\alpha} E\_0 e^{i\varphi\_\to} \tag{33}$$

Therefore, the both fields are out of phase with angle

$$
\rho = \varphi\_B - \varphi\_E \tag{34}
$$

From Eq. (33) the ratio of the magnetic field to the electric field is

$$\frac{B\_0}{E\_0} = \frac{R}{\alpha} \tag{35}$$

By using Eq. (29)

$$\frac{B\_0}{E\_0} = \sqrt{\varepsilon\mu} \left[ \mathbf{1} + \frac{\sigma^2}{\varepsilon^2 \alpha^2} \right]^{\frac{1}{4}} \tag{36}$$

In other words, we can say that the magnetic field advanced from electric field by the phase angle *φ*. In terms of sinusoidal form, these fields follow the following expressions.

$$
\overrightarrow{E} = \hat{i}\overrightarrow{E}\_0\varepsilon^{-\beta\varepsilon}\cos\left(\alpha t - \alpha \mathbf{z} + \varphi\_E\right) \tag{37}
$$

And the magnetic field results from Eq. (32) is given by

$$\overrightarrow{B} = \hat{j}\overrightarrow{B}\_0\varepsilon^{-\beta\varepsilon}\cos\left(\alpha t - a\varepsilon + \varphi\_E + \varphi\right) \tag{38}$$

The above equations direct that the amplitude of an electromagnetic wave propagating (through a conductor) decays exponentially on a characteristic length scale, d, that is known as the skin-depth [48].

#### **7.2 Skin depth**

Skin depth measure the distance that the wave travels before it's amplitude falls to 1/e of its original value [48]. From Eq. (37), the amplitude of the wave falls by a

factor 1/e in a distance *<sup>z</sup>* <sup>¼</sup> <sup>1</sup> *<sup>β</sup>*. In other words it is a measure of how far the wave penetrates into the conductor. Mathematically skin depth is denoted by *δ*, therefore

$$
\delta = \frac{1}{\beta} \tag{39}
$$

If we study poor conductor, which satisfies the inequality *σ* < <*εω*, then Eqs. (20) and (21) leads to

$$a \approx a\sqrt{\epsilon\mu} \tag{40}$$

$$\text{And we know that } \beta = \frac{a\mu\sigma}{2a} \tag{41}$$

Substitute the value of *α* into Eq. (32), we get

$$\text{Or } \beta \approx \alpha \frac{\mu \sigma}{2\alpha \sqrt{\epsilon \mu}} \tag{42}$$

$$\text{Or } \beta \approx \frac{\sigma}{2} \sqrt{\frac{\mu}{e}} \tag{43}$$

$$\text{The phase velocity } V\_{ph} = \frac{\alpha}{a} \approx \frac{1}{\sqrt{\varepsilon\mu}} \tag{44}$$

$$\text{Skin depth in power conductor } \delta = \frac{1}{\beta} = \frac{2}{\sigma} \sqrt{\frac{\varepsilon}{\mu}} \tag{45}$$

So, it is independent from the frequency.

The Eq. (41) state that at higher frequency, the absorbing parameter lost its significance that is *β* < <*α*. We can conclude that at higher frequency the wavelength does not decay very fast in a poor conductor. Moreover both the fields are also in same phase by the relation *ωB*<sup>0</sup> ¼ *αE*0. Also the phase velocity is independent from the frequency [47].

#### **8. Wave propagation in perfect conductors**

The transmission lines and communication systems are made up with silver, copper and aluminum. In most cases these conductors satisfies the inequality *σ* > >*εω*, then Eqs. (20) and (21) leads to

$$
\alpha \approx \sqrt{\frac{a\mu\sigma}{2}}\tag{46}
$$

$$\text{And } \beta \approx \sqrt{\frac{a\mu\sigma}{2}} \tag{47}$$

$$\text{Therefore } \beta \approx a \tag{48}$$

$$\text{Skin depth } \delta = \frac{1}{\beta} = \frac{1}{a} = \sqrt{\frac{2}{a\mu\sigma}}\tag{49}$$

The wave decays significantly within one wavelength. Since *δ*∝ ffiffiffiffiffiffiffiffiffiffiffi 1*=ωσ* p , the deep penetration occurs, when the inequality*σ* < <*εω* is satisfied (at Low frequency in a Poor conductor).

#### **9. Electromagnetic wave propagation into water**

EM wave propagation can be more efficient in fresh water than in seawater. The radiofrequency wave propagation works poorly in seawater due to the losses caused by the high conductivity of seawater. The limitation of EM wave propagation in water is the high attenuation due to the conductivity of water (typically, 4 S/m), however fresh water has a conductivity of 0.01 S/m. These properties are used to construct underwater sensor network based on electromagnetic waves to trace out the natural resources buried underwater, where the conventional optical water sensors are difficult to utilize in an underwater environment due to backscatter and absorptions [47].

Example: For sea water,

*<sup>μ</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> <sup>¼</sup> <sup>4</sup>*<sup>π</sup>* � <sup>10</sup>�7*N=A*<sup>2</sup> , *<sup>ε</sup>* ffi <sup>81</sup>*ε*<sup>0</sup> and *<sup>σ</sup>* <sup>≈</sup>5ð Þ <sup>Ω</sup>*:<sup>m</sup>* �<sup>1</sup> . The skin depth in poor conductor

$$\delta = \frac{2}{\sigma} \sqrt{\frac{\varepsilon}{\mu}} = \frac{2}{\sigma} \sqrt{\frac{8 \mathbf{1} \varepsilon\_0}{\mu\_0}} \tag{50}$$

$$\eta = \frac{2\sqrt{81}}{\sigma Z} = \frac{18}{5 \times 377} \approx 0.96 \text{cm.} \tag{51}$$

If the sea water satisfies the inequality *σ* < < *εω*, of poor conductor, which require

$$f = \frac{\alpha}{2\pi} > > \frac{\sigma}{2\pi\varepsilon} = 10^9 \text{Hz} \tag{52}$$

Therefore at 109*Hz* or *λ*< < 30 *cm*, sea water behave as poor conductor. On the other hand at the radio frequency range *f* < <10<sup>9</sup> Hz, the inequality *σ* > >*εω*, can be satisfied, the skin depth *<sup>δ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*=*ð Þ *ωμσ* <sup>p</sup> is quite short. To reach a depth <sup>δ</sup> = 10 m, for communication with submarines,

$$f = \frac{\alpha}{2\pi} = \frac{1}{\pi\sigma\mu\delta^2} \approx 500 \text{Hz} \tag{53}$$

The wavelength in the air is about

$$
\lambda = \frac{c}{f} = \frac{3 \times 10^8}{500} = 600 \text{km} \tag{54}
$$

The skin depth at different frequency in sea water are 277 m at 1 Hz, 8.76 m at 1KHz, 0.277 m at 1 MHz and 0.015 at 1GHz if the conductivity of sea-water is taken to about *σ* ¼ 3*=*Ω*m* and *ε<sup>r</sup>* ¼ 80. These effects leads to severe restrictions for radio communication with submerged submarines. To overcome this, the communication must be performed with extremely low frequency waves generated by gigantic antennas [47].

#### **9.1 Short wave communications**

At 60 km to 100 km height from the earth, ionosphere plasma has a typical density of 1013/m3 , which gives the plasma frequency of order 28 MHz. the waves below this frequency shows reflections from the layer of ionosphere to reach the receiver's end. The conductivity of the earth is 10�<sup>2</sup> S/m, Earth behave as a good conductor, if the inequality *σ* > > *εω* is satisfied. In other word

$$f < < \frac{\sigma}{2\pi\epsilon} = 180\tag{55}$$

MHz, therefore below 20 MHz, the earth is good conductor. Example: skin depth at *f* ¼ 60 Hz for copper.

$$\delta = \sqrt{\frac{2}{2\pi \times 60 \times 4\pi \times 10^{-7} \times 6 \times 10^{7}}} = 8 \text{mm} \tag{56}$$

The frequency dependent skin-depth in Copper (*<sup>σ</sup>* <sup>¼</sup> <sup>6</sup>*:*<sup>25</sup> � <sup>10</sup><sup>7</sup> *=*Ω*m*) can be expressed as *<sup>d</sup>* <sup>¼</sup> <sup>6</sup>ffiffiffiffiffiffiffiffiffiffi *f Hz* ð Þ <sup>p</sup> cm. It says that the skin-depth is about 6 cm at 1 Hz and it reduced to 2 mm at 1 kHz. In other words it conclude that an oscillating electromagnetic signal of high frequency, transmits along the surface of the wire or on narrow layer of thickness of the order the skin-depth in a conductor. In the visible region (*<sup>ω</sup>* � <sup>10</sup>15/s) of the spectrum, the skin depth for metals is on the order of 10*A*0. The skin depth is related with wavelength *λ* (inside conductor) as

$$
\lambda = \frac{2\pi}{a} = 2\pi \sqrt{\frac{2}{a\mu\sigma}}\tag{57}
$$

$$\text{The phase velocity } V\_{ph} = \frac{\alpha}{a} = \frac{\alpha \lambda}{2\pi} \approx \sqrt{\frac{2\alpha}{\mu \sigma}} \tag{58}$$

Therefore for a very good conductor, the real and imaginary part of the wave vector attain the same values. In this case the amplitude of the wave decays very fast with frequency as compared to bad conductor. The phase velocity of the wave in a good conductor depends on the frequency of the electromagnetic light. Consequently, an electromagnetic wave cannot penetrate more than a few skin-depths into a conducting medium. The skin-depth is smaller at higher frequencies. This implies that high frequency waves penetrate a shorter distance into a conductor than low frequency waves.

**Question:** Find the skin depths for silver at a frequency of 10<sup>10</sup> Hz.

$$\text{Skin depth } \delta = \sqrt{\frac{2}{a\mu\sigma}}\tag{59}$$

$$\delta = \sqrt{\frac{2}{2\pi \times 10^{10} \times 4\pi \times 10^{-7} \times 6.25 \times 10^{7}}} = 6.4 \times 10^{-4} \text{mm} \tag{60}$$

Therefore, in microwave experiment, the field do not penetrate much beyond .00064 mm, so no point it's coating making further thicker. There is no advantage to construct AC transmission lines using wires with a radius much larger than the skin depth because the current flows mainly in the outer part of the conductor.

**Question:** wavelength and propagation speed in copper for radio waves at 1 MHz. compare the corresponding values in air (or vacuum). *<sup>μ</sup>*<sup>0</sup> <sup>¼</sup> <sup>4</sup>*<sup>π</sup>* � <sup>10</sup>�<sup>7</sup> H/m. From Eq. (40),

$$
\lambda\_{\text{Cu}} = \frac{2\pi}{a\_{\text{Cu}}} \text{ and } a\_{\text{Cu}} \approx \sqrt{\frac{a\mu\sigma\_{\text{Cu}}}{2}} \tag{61}
$$

Therefore, *λCu* ¼ 2*π* ffiffiffiffiffiffiffiffiffi <sup>2</sup> *ωμσCu* <sup>q</sup>

$$\lambda\_{\text{Cu}} = 2\pi \sqrt{\frac{2}{2\pi \times 10^6 \times 4\pi \times 10^{-7} \times 6.25 \times 10^7}} = 0.4 \text{mm} \tag{62}$$

The propagating velocity in copper *Vph* <sup>¼</sup> *<sup>ω</sup> <sup>α</sup>* <sup>¼</sup> *ωλ* 2*π*

$$V\_{ph} = 0.4 \times 10^{-3} \times 10^6 = 400 \text{m/s} \tag{63}$$

The above parameters are quite different in vacuum as follow

$$
\lambda\_{\text{Vacuum}} = \frac{c}{\nu} = \frac{3 \times 10^8}{10^6} = \mathbf{300m} \tag{64}
$$

There is no advantage to construct AC transmission lines using wires with a radius much larger than the skin depth because the current flows mainly in the outer part of the conductor.

#### **10. Complex permittivity of bread dough and depth of penetration**

After baking for few minutes, the relative permittivity of bread dough at frequency 600 MHz is *εcr* ¼ 23*:*1 � *j*11*:*85. Calculate the depth of penetration of microwave.

Solution: the loss tangent of bread dough is

$$
\tan \delta = \frac{\mathbf{11.85}}{\mathbf{23.1}} = \mathbf{0.513} \tag{65}
$$

The depth of penetration is given as

$$d \approx \frac{c\sqrt{2}}{2\pi \xi' e\_r' \sqrt{\left(\sqrt{1 + \tan \delta^2} - 1\right)}}\tag{66}$$

After substituting all the parameters, we get

$$d \approx \frac{\sqrt{2} \times 3 \times 10^8}{2\pi \times 600 \times 10^6} \frac{1}{23.1 \sqrt{\left(\sqrt{1 + \left(0.513\right)^2} - 1\right)}} \approx 6.65 \text{cm} \tag{67}$$

It is worthy to note that the depth of penetration decreases with frequency.

#### **11. The AC and DC conduction in plasma**

Let the collision frequency of electrons with ions and ω the frequency of the EM waves in the conductor. The equation of motion for electrons is:

*Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

$$m\frac{dv}{dt} = -eE - m\nu v\tag{68}$$

Assume *<sup>υ</sup>* <sup>¼</sup> *<sup>υ</sup>*0*e*�*iω<sup>t</sup>* and use *<sup>∂</sup>=∂<sup>t</sup>* ! �*iω*, we obtain

$$-i\alpha m\nu = -eE - m\nu\nu \to \nu = \frac{-e}{m(\nu - i\alpha)}E\tag{69}$$

the current density is expressed by *j* ¼ �*enυ*

$$j\_f = \frac{-ne^2}{m(\nu - i\nu)}E\tag{70}$$

Therefore, the AC conductivity can be read as

$$\sigma(\alpha) = \frac{1}{\left(\nu - i\alpha\right)} \frac{n e^2}{m} \tag{71}$$

In infrared range *<sup>ω</sup>*< <*<sup>ν</sup>* � <sup>10</sup><sup>14</sup>ð Þ <sup>1</sup>*<sup>=</sup>* sec , so the DC conductivity

$$
\sigma = \frac{n\varepsilon^2}{mv} \tag{72}
$$

can be taken.

Let us now compare the magnitude of conduction current with that of the displacement current.

Assume *<sup>E</sup>* <sup>¼</sup> *<sup>E</sup>*0*e*�*iω<sup>t</sup>* . Then

$$\left|\frac{\dot{j}\_f}{\varepsilon \frac{\partial E}{\partial t}}\right| = \frac{\sigma E}{\varepsilon oE} = \frac{\sigma}{\varepsilon o} \tag{73}$$

In copper, *<sup>σ</sup>* <sup>¼</sup> <sup>6</sup> � <sup>10</sup><sup>7</sup> ð Þ *<sup>s</sup>=<sup>m</sup>* . The condition for *<sup>j</sup> <sup>f</sup>* <sup>≈</sup> *<sup>ε</sup> <sup>∂</sup><sup>E</sup> ∂t* , or *<sup>σ</sup> εω* ≈1 leads to

$$
\omega = \frac{\sigma}{\varepsilon} = \frac{6 \times 10^7}{8.85 \times 10^{-12}} \sim 7 \times 10^{19} (rad/\text{sec})\tag{74}
$$

At frequencies *<sup>ω</sup>* <sup>&</sup>lt;10<sup>12</sup>ð Þ *rad<sup>=</sup>* sec (communication wave frequency), *σ εω* > > 1 or *j <sup>f</sup>* > > *<sup>ε</sup> <sup>∂</sup><sup>E</sup> ∂t* .

#### **12. Electromagnetic pulse and high power microwave overview**

Several nations and terrorists have a capability to use electromagnetic pulse (EMP) as a weapon to disrupt the critical infrastructures. Electromagnetic pulse is an intense and direct energy field that can interrupt sensitive electrical and electronic equipment over a very wide area, depending on power of the nuclear device and altitude of the burst. An explosion exploded at few heights in the atmosphere can produce EMP and known as high altitude EMP or HEMP. High power microwave (HPM) can be produced with the help of powerful batteries by electrical equipment that transforms battery power into intense microwaves which may be harmful electronics equipments [62–71]. The high- power electromagnetic (HPEM) term describes a set of transient electromagnetic environments with intense electric and magnetic fields. High- power electromagnetic field may be produced by electrostatic discharge, radar system, lightning strikes, etc. The nuclear bursts can lead to the production of electromagnetic pulse which may be used against the enemy country's military satellites. Therefore the sources derived from lasers, nuclear events are vulnerable and called laser and microwave threats. Microwave weapons do not rely on exact knowledge of the enemy system. These weapons can leave persisting and lasting effects in the enemy targets through damage and destruction of electronic circuits, components. Actually HEMP or HPM energy fields, as they instantly spread outward, may also affect nearby hospital equipment or personal medical devices, such as pacemakers. These may damage critical electronic systems throughout other parts of the surrounding civilian infrastructure. HEMP or HPM may damage to petroleum, natural gas infrastructure, transportation systems, food production, communication systems and financial systems [62–71].

#### **13. Generation of high - frequency instability through plasma environment**

The beams of ions and electrons are a source of free energy which can be transferred to high power waves. If conditions are favorable, the resonant interaction of the waves in plasma can lead to nonlinear instabilities, in which all the waves grow faster than exponentially and attain enormously large amplitudes. These instabilities are referred to as explosive instabilities. Such instabilities could be of considerable practical interest, as these seem to offer a mechanism for rapid dissipation of coherent wave energy into thermal motion, and hence may be effective for plasma heating [72, 73]. A consistent theory of explosive instability shows that in the three-wave approximation the amplitudes of all the waves tend to infinity over a finite time called explosion time [74, 75]. In ref. [74], an explosive- generated – plasma is discovered for low and high frequency instabilities. The solution of dispersion equation is found numerically for the possibility of wave triplet and synchronism conditions. The instabilities is observed to propagate whose wave number.

#### **14. Electron beam plasma model and theoretical calculation**

Here we considers ions, electrons and negatively charged electron beam are immersed in a Hall thruster plasma channel [51–55]. The magnetic field is consider as Β ! ¼ Β^*z* so that electrons are magnetized while ions remains un-magnetized and electrons rotates with cyclotron frequency <sup>Ω</sup> <sup>¼</sup> *eB me* , whereas the gyro-radius for ions is larger so that they cannot rotate and simply ejects out by providing thrust to the device. The axial electric field Ε ! ¼ Ε*x*^ (along the x - axis) which accelerates the particles. It causes electrons have a Ε ! � Β ! drift in the azimuthal direction (y-axis) whereas the movement of ions is restricted along x-axis. Similar to previous studies, here, we consider the motion of all the species i.e. for ions (density *ni*, mass*mi*, velocity *vi*) for electrons (density *ne*, mass *me*, velocity *ve*), for electron beam (density *nb*, mass *mb*, velocity *vb*) and collision frequency for the excitation of instability. The basic fluid equations are given as follows:

$$\frac{\partial \mathfrak{n}\_i}{\partial t} + \vec{\nabla} \cdot \left(\overrightarrow{\nu}\_i \mathfrak{n}\_i\right) = \mathbf{0} \tag{75}$$

*Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

$$m\_i \left(\frac{\partial}{\partial t} + \left(\overrightarrow{\nu}\_i, \overrightarrow{\nabla}\right)\right) \overrightarrow{\nu}\_i = e\overrightarrow{E} \tag{76}$$

$$\frac{\partial \mathfrak{n}\_{\epsilon}}{\partial t} + \overrightarrow{\nabla} \cdot \left(\overrightarrow{\nu}\_{\epsilon} \mathfrak{n}\_{\epsilon}\right) = \mathbf{0} \tag{77}$$

$$m\_{\varepsilon} \left( \frac{\partial}{\partial t} + \left( \overrightarrow{\boldsymbol{\nu}}\_{\varepsilon} \cdot \overrightarrow{\boldsymbol{\nabla}} \right) + \boldsymbol{\nu} \right) \overrightarrow{\boldsymbol{\nu}}\_{\varepsilon} = -e \left( \overrightarrow{\boldsymbol{E}} + \overrightarrow{\boldsymbol{\nu}}\_{\varepsilon} \times \overrightarrow{\boldsymbol{B}} \right) \tag{78}$$

$$\frac{\partial \mathfrak{n}\_b}{\partial t} + \vec{\nabla} \cdot \left(\overrightarrow{\nu}\_b \mathfrak{n}\_b\right) = \mathbf{0} \tag{79}$$

$$m\_b \left(\frac{\partial}{\partial t} + \left(\overrightarrow{v}\_b \cdot \overrightarrow{\nabla}\right)\right) \overrightarrow{v}\_b = -en\_b \overrightarrow{E} \tag{80}$$

$$
\varepsilon\_0 \nabla^2 \rho\_1 = e(n\_{\epsilon 1} - n\_{i1} + n\_{b1}) \tag{81}
$$

Since the larmor radius of ions are larger than the length of the channel (6 cm), therefore ions are considered as unmagnetized in the channel and are accelerated along the axial direction of the chamber. We consider ions initial drift in the positive x – direction (*υ* !*i*<sup>0</sup> ¼ *υ<sup>i</sup>*0*x*^) with neglecting motion in both azimuthal and radial directions [51–55]. Electron has motion in the x-direction (*υ* ! *<sup>b</sup>* ¼ *υbx*^) since electrons are affected by magnetic field and get magnetized, we takes their *E* ! � *B* ! initial drift in the y – direction (*υ* !*<sup>e</sup>* ¼ *υe*^*y*).

To find the oscillations by the solutions of the above equations we take the quantities varied as the *A r*ð Þ¼ , *<sup>t</sup> <sup>A</sup>*0*ei k*ð Þ *:r*�*ω<sup>t</sup>* for first order perturb quantities *ni*1, *ne*1, *nb*1, *υi*1, *υe*1, *υ<sup>b</sup>*<sup>1</sup> and *E* ! <sup>1</sup> together with *ω* as a frequency of oscillations and the *k* is the wave propagation vector within plane of (x, y) . On remarking the magnetic fields are large enough in Hall thruster and condition Ω > > *ω*, *kyυe*0, *v* is satisfied [51–56]. By solving the equation of motion and the equation of continuity for electrons, we get the perturbed density of electrons in terms of oscillating potential *φ*<sup>1</sup> in the following way

$$m\_{\epsilon1} = \frac{e n\_{\epsilon0} \hat{\alpha} k^2 \rho\_1}{m\_{\epsilon} \Omega^2 \left(\rho - k\_{\mathcal{Y}} \nu\_{\epsilon0}\right)}\tag{82}$$

Let us consider, *<sup>ω</sup>*^ <sup>¼</sup> *<sup>ω</sup>* � *kyυ<sup>e</sup>*<sup>0</sup> � *iv*, the cyclotron frequency <sup>Ω</sup> <sup>¼</sup> *eB me* and *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *y*.

Similarly, on solving equation for ions we get the ion density term as

$$m\_{i1} = \frac{e k^2 n\_{i0} \rho\_1}{m\_i (\rho - k\_\chi \nu\_{i0})^2} \tag{83}$$

Similarly for electron beam density given as

$$m\_{b1} = -\frac{ek^2 n\_{b0} \rho\_1}{m\_b \left(\rho - k\_\text{x} \nu\_{b0}\right)^2} \tag{84}$$

By putting these density values in the Poisson's equations

$$-k^2 \rho\_1 = \frac{e^2 n\_{\varepsilon 0} \hat{\alpha} k^2 \rho\_1}{m\_{\varepsilon} \varepsilon\_0 \Omega^2 \left(\rho - k\_\gamma \nu\_{\rm c0}\right)} - \frac{e^2 k^2 n\_{i0} \rho\_1}{m\_i \varepsilon\_0 \left(\rho - k\_\mathbf{x} \nu\_{i0}\right)^2} - \frac{e^2 k^2 n\_{b \, 0} \rho\_1}{m\_b \varepsilon\_0 \left(\rho - k\_\mathbf{x} \nu\_{b \, 0}\right)^2} \tag{85}$$

On taking the plasma frequencies as; *ωpe* ¼ ffiffiffiffiffiffiffi *e*<sup>2</sup>*neo meε*<sup>0</sup> q , *ωpi* ¼ ffiffiffiffiffiffiffi *e*<sup>2</sup>*nio miε*<sup>0</sup> q , and *ωpb* ¼ ffiffiffiffiffiffiffi *e*<sup>2</sup>*nbo mbε*<sup>0</sup> q . Then the above equation reduces in the form as

$$-k^{2}\boldsymbol{\rho}\_{1} = \frac{\boldsymbol{\alpha}\_{p\boldsymbol{\epsilon}}^{2}\boldsymbol{\hat{\alpha}}\boldsymbol{k}^{2}\boldsymbol{\rho}\_{1}}{\boldsymbol{\Omega}^{2}\left(\boldsymbol{\omega} - \boldsymbol{k}\_{\boldsymbol{\gamma}}\boldsymbol{\nu}\_{0}\right)} - \frac{\boldsymbol{\alpha}\_{p\boldsymbol{i}}^{2}\boldsymbol{k}^{2}\boldsymbol{\rho}\_{1}}{\left(\boldsymbol{\omega} - \boldsymbol{k}\_{\boldsymbol{\varepsilon}}\boldsymbol{\nu}\_{0}\right)^{2}} - \frac{\boldsymbol{\alpha}\_{p\boldsymbol{b}}^{2}\boldsymbol{k}^{2}\boldsymbol{\varphi}\_{1}}{\left(\boldsymbol{\omega} - \boldsymbol{k}\_{\boldsymbol{\varepsilon}}\boldsymbol{\nu}\_{00}\right)^{2}}\tag{86}$$

Since the perturbed potential is not zero i.e. *φ*<sup>1</sup> 6¼ 0 then we get

$$\frac{\alpha\_{pe}^{2}\hat{o}\nu}{\Omega^{2}\left(\boldsymbol{\omega}-k\_{\text{y}}\nu\_{e0}\right)}-\frac{\alpha\_{pi}^{2}}{\left(\boldsymbol{\omega}-k\_{\text{x}}\nu\_{i0}\right)^{2}}-\frac{\alpha\_{pb}^{2}}{\left(\boldsymbol{\omega}-k\_{\text{x}}\nu\_{b0}\right)^{2}}+\mathbf{1}=\mathbf{0}\tag{87}$$

This is the modified dispersion relation for the lower-hybrid waves under the effects of collisions and electrons beam density.

#### **15. Analytical solutions under the limitations**

Consider now waves propagating along the ^*y* direction, so that *kx* ¼ 0, which, in real thruster geometry, corresponds to azimuthally propagating, waves. We discuss below its limiting cases through Litvak and Fisch [78].

$$
\omega \ll \left| k\_{\gamma} v\_{\epsilon 0} \right|, \tag{88}
$$

The solutions for the dispersion relation (57) can be obtained as follows:

$$\rho \alpha^2 \approx \frac{\left(\alpha\_{pi}^2 + \alpha\_{pb}^2\right) \Omega^2}{\left(\Omega^2 + \alpha\_{pe}^2\right) \left[1 + \frac{i\nu\_{e}\alpha\_{pr}^2}{\left(\Omega^2 + \alpha\_{pc}^2\right)\nu\_{\gamma}\nu\_{\ell 0}}\right]} \tag{89}$$

Since the last terms in the second square brackets of the denominator in the right-hand side of (89) are small, we obtain the following

$$\rho \approx \pm \sqrt{\frac{\Omega^2 \left(\boldsymbol{\alpha}\_{pi}^2 + \boldsymbol{\alpha}\_{pb}^2\right)}{\left(\Omega^2 + \boldsymbol{\alpha}\_{p\epsilon}^2\right)}} \left[1 - \frac{i\nu\_\epsilon \boldsymbol{\alpha}\_{p\epsilon}^2}{2k\_\jmath \nu\_{\epsilon 0} \left(\Omega^2 + \boldsymbol{\alpha}\_{p\epsilon}^2\right)}\right] \tag{90}$$

Finally, the growth rate *γ* of the resistive instability is calculated from (90) as follow

$$\gamma \approx \frac{\nu\_{\epsilon} \alpha\_{p\epsilon}^{2}}{2k\_{\jmath} \nu\_{\epsilon 0} \left(\Omega^{2} + \alpha\_{p\epsilon}^{2}\right)} \times \sqrt{\frac{\Omega^{2} \left(\alpha\_{pi}^{2} + \alpha\_{pb}^{2}\right)}{\left(\Omega^{2} + \alpha\_{p\epsilon}^{2}\right)}}\tag{91}$$

The corresponding real frequency *ωr*ð Þ *ω* � *ω<sup>r</sup>* � *iγ* is obtained as

$$\alpha\_{r} \approx \sqrt{\frac{\Omega^2 \left(\alpha\_{pi}^2 + \alpha\_{pb}^2\right)}{\left(\Omega^2 + \alpha\_{pe}^2\right)}}\tag{92}$$

The Eqs. (91) show that the growth of the high frequency instability depends on collision frequency, electron density, ion density, beam density, azimuthal wave

*Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*


**Table 2.** *Plasma parameters.*

number, initial drift and on the applied magnetic field. On the other hand, the real frequency of the wave depends only on the magnetic field, electron plasma density, ion density and beam density. By tuning these parameters one can control the frequency of the generating wave. In the below **Table 2**, the different parameters of a Hall thruster are given [51–56].

#### **16. Results and discussion**

The Eqs. (91) and (92) are solved with MATLAB by using appropriate parameters given in **Table 2**. We plot various figures for investigating the variation of growth rate and real frequency of the instability with magnetic field B0 and density of beam nb, initial drift, collision frequency ν and wave number. For these sets of parameters, only one dominated mode of the dispersion relation is plotted in the figures. **Figure 3** shows the variation of growth rate and real frequency for different values of magnetic field. The reason for the enhanced growth rate as well as real frequency can be understood based on Lorentz force and the electron collisions. Since the electrons have their drift in the y-direction, they experience the Lorentz force due to the magnetic field in the negative of x-direction, i.e., in the direction opposite to the ions drift. The higher Lorentz force helps these transverse oscillations to grow relatively at a faster rate owing to an enhancement in the frequency. On the other hand, this is quite plausible that larger cyclotron frequency of the electrons leads to stronger effects of the collisions because of which the resistive coupling becomes more significant and hence the wave grows at its higher rate. Opposite effect of the magnetic field was observed by Alcock and Keen in case of a drift dissipative instability that occurred in afterglow plasma [76]. Similarly studied are also investigated by Sing and Malik in magnetized plasma [51–56].

In **Figure 4**, we have plotted the variation of growth rate γ and real frequency with the azimuthal wavenumber in order to examine the growth of these waves, when the oscillations are of smaller or relatively longer wavelengths. Here, the oscillations of larger wave numbers (or smaller wavelengths) are found to have lower growth. The faster decay that is observed on the larger side of k is probably due to the stronger Landau damping. The growth rate shows parabolic nature but the real frequency is almost increases linearly with respect to azimuthal wave number. It means that oscillations of smaller wavelengths are most unstable. Kapulkin et al. have theoretically observed the growth rate of instability to directly

**Figure 3.** *Variation of growth rate and real frequency with the magnetic field.*

proportional to the azimuthal wavenumber [77]. Litvak and Fisch have also shown that the rate of growth of instability is inversely proportional to the azimuthal wave number [78].

On the other hand, the variation of growth rate γ and real frequency with the collision frequency is depicted in **Figure 5**. The wave grow at faster rates in the presence of more electron collisions. This is due to the resistive coupling, which get much stronger in the presence of more collisions. In the present case, the growth rate grows at a much faster rate and real frequency is constant, and graph shows that the growth rate is directly proportional to the collision frequency. During the simulation studies of resistive instability, Fernandez *et al.* also observed the growth

**Figure 4.** *Variation of growth rate γ and real frequency with azimuthal wavenumber.*

#### *Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

rate to be directly proportional to the square root of the collision frequency [79]. In **Figure 6,** we show the dependence of the growth rate on the electron drift velocity. it is observed that the growth rate is reduced in the presence of larger electron drift velocity. In this case the resistive coupling of the oscillations to the electrons' drift would be weaker due to the enhanced velocity of the electrons. The reduced growth under the effect of stronger magnetic field is attributed to the weaker coupling of the oscillations to the electrons closed drift. The variation of growth rate γ and real frequency with beam density are shown in **Figure 7**. The growth shows asymmetric Gaussian type behavior but the real frequency varies linearly with beam density of electrons. This is due to the increased collisional effect with the large plasma density.

**Figure 5.** *Variation of growth rate γ and real frequency with collision frequency.*

**Figure 6.** *Variation of growth rate γ and real frequency with electron drift velocity.*

**Figure 7.** *Variation of growth rate γ and real frequency with beam density.*

#### **17. Conclusions**

The present chapter discuss the properties of electromagnetic waves propagating through different media. In the first part of the chapter, the dispersion relation for the electromagnetic waves in conducting medium is derived. It has been experienced that the penetration of the electromagnetic field depend on the frequency of the source as well as the electrical properties of the medium. The significance of skin depth for biological and conducting media are explained through numerical examples. In the second part of the chapter, the generation of high frequency instability in plasma is discussed which grow with the magnetic field, wave length, collision frequency and the beam density. The growth rate linearly increases with collision frequency of electrons but it is decreases with the drift velocity of electrons. The real frequency of the instability increases with magnetic field, azimuthal wave number and beam density. The real frequency is almost independent with the collision frequency of the electrons.

#### **Acknowledgements**

The University Grants Commission (UGC), New Delhi, India is thankfully acknowledged for providing the startup Grant (No. F. 30-356/2017/BSR).

*Physics of Absorption and Generation of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.99037*

#### **Author details**

Sukhmander Singh<sup>1</sup> \*, Ashish Tyagi2 and Bhavna Vidhani<sup>3</sup>

1 Plasma Waves and Electric Propulsion Laboratory, Department of Physics, Central University of Rajasthan, Ajmer, Kishangarh, India

2 Physics Department, Swami Shraddhanand College, University of Delhi, Delhi, India

3 Department of Physics and Electronics, Hansraj College, University of Delhi, Delhi, India

\*Address all correspondence to: sukhmandersingh@curaj.ac.in

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Section 3
