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Cambridge University Press - Electromagnetic Field Theory Fundamentals, Second Edition Bhag Singh Guru and Huseyin R. Hiziroglu Excerpt. Download Electromagnetic Field Theory Fundamentals, 2nd Edition by Bhag Singh Guru and Hüseyin R. Hiziroglu. Electromagnetic Field Theory Fundamentals (2nd ed.) by Bhag Singh Guru. Read online, or download in secure PDF or secure EPUB format.
How do electromagnetic fields propagate in space? What really happens when electromagnetic energy travels from one end of a hollow pipe waveguide to the other?
Electromagnetic Field Theory Fundamentals (2nd ed.)
The primary purpose of this text is to answer some of these questions pertaining to electromagnetic fields. In this chapter we intend to show that the study of electromagnetic field theory is vital to understanding many phenomena that take place in electrical engineering.
To do so we make use of some of the concepts and equations of other areas of electrical engineering. We aim to shed light on the origin of these concepts and equations using electromagnetic field theory. Before we proceed any further, however, we mention that the devel- opment of science depends upon some quantities that cannot be defined precisely. For example, what is time? When did time begin? Likewise, what is temperature? What is hot or cold? We do have some intuitive feelings about these quantities but lack precise definitions.
To measure and express each of these quantities, we need to define a system of units. In the International System of Units SI for short , we have adopted the units of kilogram kg for mass, meter m for length, second s for time, coulomb C for charge, and kelvin K for temperature. Units for all other quantities of interest are then defined in terms of these fundamental units. Therefore, the ampere is a derived unit. Hiziroglu Excerpt More information 2 1 Electromagnetic field theory Table 1.
Electromagnetic Field Theory Fundamentals
Unit conversion factors From Multiply by To obtain gilbert 0. Since English units are still being used in the industry to express some field quantities, it is necessary to convert from one unit system to the other. Table 1. Hiziroglu Excerpt More information 3 1.
Prior to undertaking the study of electromagnetic fields we must define the concept of a field. When we define the behavior of a quantity in a given region in terms of a set of values, one for each point in that region, we refer to this behavior of the quantity as a field. The value at each point of a field can be either measured experimentally or predicted by carrying out certain mathematical operations on some other quantities.
From the study of other branches of science, we know that there are both scalar and vector fields. Some of the field variables we use in this text are given in Table 1.
There also exist definite relationships between these field quantities, and some of these are given in Table 1.
Hiziroglu Excerpt More information 4 1 Electromagnetic field theory From the equations listed in Table 1. Vector analysis is the language used in the study of electromagnetic fields. Without the use of vectors, the field equations would be quite unwieldy to write and onerous to remember. When expressed in scalar form, this equation yields a set of three scalar equtions. In addition, the appearance of these scalar equations depends upon the coordinate system.
In the rectangular coordinate system, the previous equation is a concise version of the following three equations: Moreover, the vector representation is independent of the coordinate system.
Electromagnetic Field Theory Fundamentals
Thus, vector analysis helps us to simplify and unify field equations. The student may be competent to perform such vector operations as the gradient, divergence, and curl, but may not be able to describe the sig- nificance of each operation.
The knowledge of each vector operation is essential to appreciate the development of electromagnetic field theory. Quite often, a student does not know that a the unit vector that transforms a scalar surface to a vector surface is always normal to the surface, b a thin sheet negligible thickness of paper has two surfaces, c the direction of the line integral along the boundary of a surface depends upon the direction of the unit normal to that surface, and d there is a difference between an open surface and a closed surface.
These concepts are important, and the student must comprehend the significance of each.
There are two schools of thought on the study of vector analysis. Hiziroglu Excerpt More information 5 1. We prefer the latter approach and for this reason have devoted Chapter 2 to the study of vectors.
Quite often a student does not understand why we present the same idea in two different forms: It must be pointed out that the integral form is useful to explain the significance of an equation, whereas the differential form is convenient for performing mathematical operations.
This equation states that the divergence of current density at a point is equal to the rate at which the charge density is changing at that point. The usefulness of this equation lies in the fact that we can use it to calculate the rate at which the charge density is changing at a point when the current density is known at that point. However, to highlight the physical significance of this equation, we have to enclose the charge in a volume v and perform volume integration.
In other words, we have to express 1. We can also interchange the operations of integration and differentiation on the right-hand side of equation 1. The integral on the left-hand side represents the net outward current I through the closed surface s bounding volume v. The integral on the right-hand side yields the charge q inside the volume v. This equation, therefore, states that the net outward current through a closed surface bounding a region is equal to the rate at which the charge inside the region is decreasing with time.
Hiziroglu Excerpt More information 6 1 Electromagnetic field theory The details of the preceding development are given in Chapter 4. We used this example at this time just to show that 1. Once again we face the dilemma of how to begin the presentation of electromagnetic field theory. We, however, think that the field theory should always be developed by making maximum possible use of the concepts previously discussed in earlier courses in physics. For this reason we first discuss static fields.
In the study of electrostatics, or static electric fields, we assume that a all charges are fixed in space, b all charge densities are constant in time, and c the charge is the source of the electric field.
Our interest is to determine a the electric field intensity at any point, b the potential distribution, c the forces exerted by the charges on other charges, and d the electric energy distribution in the region.
We will also explore how a capacitor stores energy. We will show that the electric field at any point is perpendicular to an equipotential surface and emphasize its ramifica- tions. Some of the equations pertaining to electrostatic fields are given in Table 1. Hiziroglu Excerpt More information 7 1. Magnetostatic field equations Force equation: If the movement of the charge is restricted in such a way that the resulting current is constant in time, the field thus created is called a magnetic field.
Since the current is constant in time, the magnetic field is also constant in time. The branch of science relating to constant magnetic fields is called magnetostatics, or static magnetic fields. In this case, we are interested in the determination of a magnetic field intensity, b magnetic flux density, c magnetic flux, and d the energy stored in the magnetic field. From time to time we will also stress the correlation between the static electric and magnetic fields.
In simple words, a timevarying magnetic field gives rise to a time-varying electric fie1d and vice versa.
The modification of Ampere's law can also be viewed as a consequence of the equation of continuity or conservation of charge. This equation is also given in Table 1.
When a particle having a charge q is moving with a velocity ii in a region where there exista time-varying electric field E and a magnetic 9 1. With the help of the four Maxwell equations, the equation of continuity, and the Lorentz force equation we can now explain ali the effects of electromagnetism.
This selection of topics is due to the fact that the solution of Maxwell's equations always leads to waves. The nature of the wave depends upon the medium, the type of excitation source , and the boundary conditions.
The propagation of a wave may either be in an unbounded region fields exist in an infinite cross section, such as free space or in a bounded region fields eKist in a finite cross section, such as a waveguide ora coaxial transmission line. Although most of the fields transmitted are in the form of spherical waves, they may be considered as plane waves in a region far away from the transmitter radiating element, such as an antenna.
How far "far away" is depends upon the wavelength distance traveled to complete one cycle of the fields. Using plane waves as an approximation, we will derive wave equations from Maxwell's equations in terms of electric and magnetic fields.
The solution of these wave equations will describe the behavior of a plane wave in an unbounded medi um. We will simplify the analysis by imposing restrictions such that a the wave is a uniform plane wave, b there are no sources of currents and charges in the medium, and c the fields vary sinusoidally in time. We will then determine i the expressions for the fields, ii the velocity with which they travei in a region, and iii the energy associated with them.
The intrinsic impedance of free space is approximately Q. Our discussion of uniform plane waves will also include the effect of interface between two media.
Here we will discuss a how much of the energy o f the incoming wave is transmitted in to the second medi um or reftected back into the first medi um, b how the incoming wave and reftected wave combine to forma standing wave, and c the condition necessary for total reftection. We will show that when one end ofthe transmission tine is excited by a time-varying source, the transmission of energy is in the fonn of a wave. The wave equations in this case will be in terms of the voltage and the current at any point along the transmission line.
The solution of these wave equations will tell us that a finite time is needed for the wave to reach the other end, and for practical transmission lines, the wave attenuates exponentially with the distance. The attenuation is due to the resistance and conductance of the transmission line. This results in a loss in energy along the entire length of the transmission line.
However, at power frequencies 50 or 60 Hz there is a negligible loss in energy due to radiation because the spacing between the conductors is extremely small in comparison with the wavelength.
As the frequency increases so does the loss of signal along the length of the transmission line. At high frequencies, the energy is transmitted from one point to another via waveguides. We will examine the necessary conditions that must be satisfied for the fields to exist, obtain field expressions, and compute the energy at any point inside the waveguide.
The analysis involves the solution of the wave equation inside the waveguide subjected to externai boundary conditions. The analysis is complex; thus, we will confine our discussion to a rectangular waveguide. Although the resulting equations appear to be quite involved and difficult to remember, we must not forget that they are obtained by simply applying the boundary conditions to a general solution of the wave equation.
A transmission line can be used to transfer energy from very low frequencies even de to reasonably high frequencies. TI1e waveguide, on the other hand, has a lower limit on the frequency called the cuto.
The cutoff frequency depends upon the dimensions of the waveguide. Signals below the cutoff frequency cannot propagate inside the waveguide. Another major difference between a transmission line and a waveguide is that the transmission line can support the trar1sverse electromagnetic TEM mode.
In practice, both coaxial and parallel wire transmission tines use the TEM mode. However, such a mode cannot exist inside the waveguide. Why this is so will be explained in Chapter I O. The waveguide can support two different medes, the transverse electric mode and the transverse magnetic mode. The conditions for the existence.
The last application of Maxwell's equations that we will discuss in this text deals with electromagnetic radiation produced by time-varying sources of finite dimensions.
The very presence of these sources adds 11 1. However, if we develop the wave equations in terms of scalar and vector potentials, the solution of either potential function is relatively less involved. By simple algebraic manipulations, we can obtain expressions for the electric and magnetic fields.
The power radiated by the sources can then be computed.
We will examine the fields produced and the power radiated by straight-wire and loop antennas. We will also study how the radiation field patterns can be modified by using antenna arrays. For instance, a to determine the electric field intensity within a parallel-plate capcitor we usually assume that the plates are of infinite extent so that we can apply Gauss's law, b to calculate the magnetic field intensity due to a long current-carrying conductor using Ampere's law we imagine that the conductor is of infinite extent, c to obtain the propagation characteristics and the nature of electromagnetic fields in a source-free region we visualize the fields in the form of a uniform plane wave, d to leam about the radiation pattern of a smalllinear antenna we presume that the length of the antenna is so small that the current distribution is uniform, etc.
Each assumption gives rise to a special situation and the analytical solution thus obtained is precise. In electrostatics we determined the capacitance of an isolated sphere using Gauss's law by exploiting the spherical symmetry. However, the problem becomes very complex when we try to determine the capacitance of an isolated cube. In magnetostatics, we obtained an answer for the magnetic field intensity on the axis of a circular current-carrying conductor using the Biot-Savart law.
Can we follow the same technique to determine the magnetic field intensity when the current-carrying conductor has an arbitrary shape? The answer, of course, is "no" because of the nature of the integral formulation. The need for a numerical solution, which is often approximate, should be clearly evident.
It must be bome in mind that each numerical solution is simply an approximation o f the exact differential 12 1 Electromagnetic field theory or integral equation.
The higher the accuracy, the more refined the numerical method must be. The accuracy of the solution further hinges on the numerical method used and the computing capability of a system. The three methods we discuss in this text are the finite-difference method, the finite-element method, and the method of moments. This information is essential not only to arouse some interest in this area but also to understand more complex developments.Both static electric and magnetic fields are used in the design of many devices.
Author s. However, at power frequencies 50 or 60 Hz there is a negligible loss in energy due to radiation because the spacing between the conductors is extremely small in comparison with the wavelength. In electrostatics we determined the capacitance of an isolated sphere using Gauss's law by exploiting the spherical symmetry.
We will simplify the analysis by imposing restrictions such that a the wave is a uniform plane wave, b there are no sources of currents and charges in the medium, and c the fields vary sinusoidally in time.
Hiziroglu Excerpt More information 3 1. Unit conversion factors From Multiply by To obtain gilbert 0. In addition, the appearance of these scalar equations depends upon the coordinate system.
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