Electromagnetics for Engineering Students Part I

The electromagnetic field theory is a fundamental subject for the electrical engineering students. The student should be acquainted with a quite knowledge in differential and integral calculus, differential equations, vector analysis, matrix algebra, and complex numbers in order to understand the subject of electromagnetics and be able to analyze the behavior of electromagnetic fields in various media. The knowledge of the vector analysis is essential in the analysis of electromagnetic problems. This first chapter of the book is focusing on the important concepts and techniques in the vector analysis that are required to understand the topics of the electromagnetic fields in the later chapters of the book. The chapter covers the topics and concepts in the vector analysis including vector algebra, orthogonal coordinate systems, vector differentiation operators, and vector integration. The chapter also introduces the scalar and vector field concepts, in addition to the divergence and Stokes’ theorems. The chapter includes numerous illustrative examples covering each topic in the chapter, in addition to miscellaneous solved problems supported by illustrative figures to enhance the student problems solving skills. The chapter is ended by numerous homework problems covering the topics of the chapter.

The field concept is fundamental to describe and understand the electrostatic phenomenon and its interaction with surrounding objects. Understanding the concepts and laws related to the electrostatic field is necessary before introducing the more advanced topics in electromagnetics. On the other hand, there is a wide range of applications in industry, medicine, and electronic devices that are based on electrostatics. This makes the subject of electrostatics an important topic as a prerequisite for other subjects or as a basic technology in many industrial applications. This chapter introduces the principles of electrostatic fields. The main topics that are covered in the chapter include Coulomb’s law and electrostatic force between the objects, the concept of electric field intensity, Gauss’s law and its applications, the concept of electric potential, the electrostatic energy and the energy stored in the electrostatic field, and the electric dipole. All topics of the chapter are supported by illustrative examples and figures that make the topics understandable easily. To enhance the student's problem solving skills, the chapter is provided by miscellaneous solved problems and numerous homework problems covering the topics of the chapter.

This chapter focuses on the conducting and dielectric materials and their properties under static fields condition. Conducting materials conduct electric current efficiently, while dielectrics possess high insulation capabilities, in addition to its ability to store electric energy. Conducting and dielectric materials are essential in all electrical and electronics systems and equipment. The ability of the material to conduct the electric current is called the conductivity of the material. On the other hand, the interaction between a dielectric material and electrostatic field leads to the formation of dipole moments in the atoms of the material, which is known as polarization. This polarization ability of a material is quantitatively described by a constant known as the permittivity of the material. The permittivity of a dielectric material relative to that of free space is known as the relative permittivity or dielectric constant of the material. Both the conductivity and dielectric constant depend on the intrinsic properties of the material. This chapter presents a detailed derivation for the conductivity and permittivity under static field conduction. In addition to the properties of the conducting and dielectric materials, the chapter discusses the concept of conservation of charge and relaxation time; the boundary conditions between different media; the resistance; the capacitance; the stored energy in a capacitor. The topics of the chapter are supported by numerous illustrative examples and figures in addition to solved problems and homework problems at the end of the chapter.

This chapter deals with more advanced techniques than those discussed in Chapter 2 to solve electrostatic problems. The techniques presented in the chapter include the method of images, the multi-pole expansion, analytical methods, and some numerical methods, which may be invoked as an alternative techniques or when the analytical solution for a particular problem is not available. The method of images and multi-pole expansion techniques are applied to analyze the problems involving charges near a perfectly electric conducting or dielectric interfaces. For the media that are subject to certain boundary conditions, the electrostatic problem is modeled by Poisson's or Laplace's equation then solved analytically or numerically. Analytical solutions for the Laplace's equation in different coordinate systems are presented in details. The method of moments and finite difference method are presented as examples of numerical techniques for the solution of Poisson's or Laplace's equation. The discussions of the topics in the chapter are supported by illustrative examples, figures, solved problems, and computer programs in addition to homework problems at the end of the chapter.

The magnetostatic phenomenon is identically like the electrostatic one in its possession of momentum and energy, and interaction with surrounding objects. Consequently, the field concept is also fundamental to describe it and understand its behavior. The sources of the magnetostatic fields are the permanent magnets and constant electric currents. The essential laws of the magnetostatic fields are Biot- Savart’s law and Ampere’s law. Biot-Savart’s law is general and applicable to all current sources, while Ampere’s law is a special case of Biot-Savart’s law and applies only to specialized problems. In this chapter, Ampere’s law has been introduced first since it is easier to apply, while Biot-Savart’s law is discussed after the introduction of the concept of the magnetic vector potential, which is used to derive it. Additionally, this chapter discusses the concept of magnetic scalar potential, Gauss’s law for magnetostatic fields, the magnetic force, the magnetic energy, the magnetic dipole, and the magnetic torque. Moreover, the multi-pole expansion technique is applied to find the magnetic vector potential and general magnetic fields for some current sources as an alternative technique. The chapter includes more than twenty examples, ten solved problems, and numerous illustrative figures, in addition to numerous homework problems at the end of the chapter.

The first part of this chapter deals with the magnetic materials under static field conditions and their properties. In this part, a detailed description is given for the interaction between a magnetic material and magnetostatic field and how the magnetization process takes place when the material is immersed in the field. The concepts of magnetic susceptibility, permeability, and relative permeability, which are related to the magnetization process, are discussed. Further topics that are related to magnetostatic such as magnetostatic energy, boundary conditions for magnetostatic fields, inductance, and the magnetic circuit are also discussed in this part. The second part of the chapter discusses the dispersive materials and introduces metamaterials and their classifications. Using mathematical analysis it is shown that under a static fields conditions materials are characterized by a constant permittivity and permeability, while under time-harmonic fields the induced polarization and magnetization may result in a permittivity and permeability that are frequency-dependent. Different material models that describe the behavior of the permittivity and permeability with frequency in addition to the classification of metamaterial are presented. The topics of the chapter are supported by numerous illustrative examples, figures, and solved problems in addition to homework problems at the end of the chapter.

Maxwell’s equations are essential mathematical tools in the analysis of electromagnetics and antennas problems. In this chapter, Maxwell's equation for general time-varying electromagnetic fields are derived from the basic laws of electromagnetics, and presented in integral and point forms. The special cases for timeharmonic and static fields are obtained from the general equations. The vector potential concept that has been introduced in Chapter 5 for static fields, is generalized in this chapter for time-varying fields. The electric and magnetic vector potentials are important quantities in determining the electromagnetic fields radiated from electric and magnetic current sources. By solving Helmholtz equations, general formulations for the electric and magnetic vector potentials are presented in terms of electric and magnetic current sources respectively. The chapter includes also the application of multi-pole expansion technique to obtain the vector potential for some time-varying fields problems, derivation of boundary conditions for time-varying fields, derivation of Poynting vector, and discussion of electromagnetic power flow. The topics of the chapter are supported by numerous illustrative examples and figures in addition to solved problems and homework problems at the end of the chapter.