Introduction

The phenomenon of electromagnetism in any medium is completely described by a set of four first order partial differential equations called Maxwell's equations. Maxwell's equations are the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in vacuum or in matter. The equations represent one of the most elegant and concise way to describe the fundamentals of electromagnetism. Maxwell's equations are a combination of the works of Gauss, Faraday, Ampère, Biot, Savart, and others. Remarkably, Maxwell's equations are perfectly consistent with the transformation equations of the special theory of relativity. To be more exact, these equations constitute a complete description of the behavior of electric and magnetic fields separately or jointly in any medium.

Vector Calculus

In vector calculus, the spatial derivatives of one types of vector and scalar fields give other types of vector or scalar fields. Depending upon the requirement, the first order differential operator (see Eq. 5.2) may be applied to a scalar function to obtain a vector function or vice versa. It may also give rise to one type of vector field from another type of vector field! The concept of vector calculus was fully exploited by James Clerk Maxwell in a simple way in discovering the missing link between the electric field and the magnetic field, thus establishing the electromagnetic nature of light. Therefore, for complete appreciation of complexities of electromagnetism, at least a brief explanation of vector calculus would be highly beneficial.

The field, in vector calculus, is defined as a region within which every physical quantity can be expressed as a continuous function of the position of a point in the region. The corresponding function is called a point function. Broadly speaking, fields are of two types – scalar fields and vector fields. All the quantities in a scalar field are scalars and all the quantities in a vector field are vectors. All quantities in both scalar and vector fields are functions of positions and times. The vectors in vector fields and the scalars in scalar fields may change with respect to positions and times. First, we shall discuss what are line integrals, surface integrals and volume integrals.

Line integrals

The integration of a vector along a curve in a vector field is called a line integral.

Science in general may be described as organized common sense. In the real world of science, nothing prevails except rationality and logics. Science does not believe in miracles. Clear understanding of the basic principles of science is essential for technological and social development. Once upon a time, the base of engineering was mainly empirical; however, now it is completely scientific. Physics is a fundamental aspect of science on which all engineering sciences have been built upon. Nowadays, more stress is given to the understanding of the basic principles rather than on remembering specific procedures. The fundamental concepts of physics have paved the way for the development of technologies. All modern technological advances from laser micro surgery to television, from computers to dishwashers to mobile phones, from remote controlled toys to space vehicles, trace back directly to the principles of physics. Accordingly, the syllabus of engineering courses includes physics as an essential ingredient.

This book, entitled Principles of Engineering Physics 1, is designed as a textbook keeping in view the engineering physics course curricula prescribed by most technical universities of India. The present book begins with oscillations and waves and ends with holography, containing altogether fourteen chapters. This book is written in a logical and coherent manner for easy understanding. The concepts of physics are mathematized without losing the beauty of the physical ideas involved. Emphasis has been given to an understanding of the basic concepts and their applications to a number of engineering problems. Each topic has been discussed in detail, both conceptually and mathematically, so that students do not face any kind of difficulties. All the derivations and solutions of numerical examples are given in detail. Each chapter contains a large number of solved numerical examples, unsolved numerical problems with answers, practical applications, theoretical questions, and multiple choice questions with answers. Certain topics and derivations that are not included directly in the syllabi have also been included in the book for the sake of continuity and completeness. The scope of the book thus has been expanded beyond the basic needs of undergraduate engineering students. We hope this book will be of immense help not only to the students but also to the teachers.

The authors sincerely request the readers for their constructive criticisms via emails mdnkhan1964@yahoo.com and spanigrahi@nitrkl.ac.in for future modification of the book.

Introduction

Many physical properties of solids can be understood with help of the free electron model. According to this model, the valance electrons of the constituent atoms become conduction electrons and move freely throughout the solid. The aggregates of free electrons constitute an electron gas or electron cloud. The interpretation of properties of solids by the free electron model was developed long before the advent of quantum physics, a versatile tool of physical science. The classical theory has several conspicuous successes. However, as it is natural, the simplicity of the theory puts limitation on its success. Many properties of solids like specific heats, magnetic susceptibility of solids, superconductivity and the like cannot be explained by the classical theory of free electron model. With the advent of quantum physics, almost all the properties of solids could be explained in minute detail. Here the electron gas is subjected to Pauli's exclusion principle and Fermi–Dirac statistics instead of Maxwell–Boltzmann statistics as in classical theory. This new formulation constitutes the quantum theory of free electrons.

The free electron model of metals gives us a good insight into the heat capacity, thermal and electrical conductivity, susceptibility and electrodynamics of metals. However, the model fails to explain the distinction between metals, semi-metals (materials with a very small overlap between the bottom of the conduction band and the top of the valence band), semiconductors and insulators; the occurrence of positive values of Hall coefficient; the relation between conduction electrons in the metal and the valence electrons of free atoms and many transport properties of solids. The band theory of solids, a simple theory, then was developed to account for all these facts. We shall apply these theories to explain a few properties of solids in detail as far as the scope of this book permits.

Free Electron Theory of Metals

Metals are generally characterized by high electrical conductivity and thermal conductivity. Many of the special properties of metals are attributed to valence electrons. Valence electrons are free to move throughout the solid and constitute the electron gas or electron cloud or simply Fermi gas. In the free electron theory, the fundamental postulate is that potential field due to ion cores is uniform throughout the solid. The potential energy value is negative since there is a force of attraction between the positive ion cores and electrons.

From time immemorial, mankind has manipulated specific properties of materials for specific self-benefits. A clear understanding of the basic principles of materials science is essential for technological development. The rapid development of materials science resulted in the invention of miniature electronic devices. All modern technologically advanced devices are directly related to an understanding of materials at the atomic and sub-atomic levels. Accordingly, the technical universities throughout the world include materials science as an essential ingredient in their course curricula.

Materials science is an interdisciplinary subject relying heavily on basic principles of physics and chemistry. Electrical and thermal conductivity, dielectric constant, magnetization, optical reflection and refraction, strength and toughness etc. are properties that originate from the internal structures of the materials. The present book, entitled Principle of Engineering Physics 2, contains chapters mostly related to materials science. It is designed as a textbook, keeping in view the engineering physics and materials science course curricula prescribed by most technical universities of India. It begins with ‘Crystal Structure’ and ends with ‘Nano Structure & Thin Films’, containing altogether thirteen chapters. The book is written in a logical and coherent manner for easy understanding by students. It presumes a working knowledge of quantum mechanics, optics, electricity and magnetism. Emphasis has been given to an understanding of the basic concepts and their applications to a number of engineering problems. Each topic is discussed in detail both conceptually and mathematically, so that students will not face comprehension difficulties. Derivations and solutions of numerical examples are also provided in detail. Each chapter contains a large number of solved numerical examples, unsolved numerical problems with answers, practical applications, theoretical questions, and multiple choice questions with answers. Certain topics and derivations which are not present in university syllabi have been included in the book for the sake of continuity and completeness. The scope of the book has thus been expanded beyond the basic needs of undergraduate engineering students. We hope, this book will be helpful not only to the students but also to the teachers.

In spite of utmost care, some typographical errors might have inadvertently crept into the book. Readers would be highly appreciated if they convey these errors to the authors. The authors sincerely request the readers for their constructive criticisms via emails mdnkhan1964@yahoo.com and spanigrahi@nitrkl.ac.in for future modification of the book.

Introduction

It is correctly told that mathematics is the queen of all sciences; in the same spirit, quantum physics or quantum mechanics may be called the king of all sciences. Our knowledge in any field of science is incomplete as long as we remain unacquainted with quantum physics. The concepts of quantum physics form the basis for our present understanding of physical phenomena on an atomic and microscopic scale. The concepts of quantum physics can be applied to most fields of science and engineering starting from biology to quantum computers to cosmology. Within engineering, important subjects of practical significance include semiconductor transistors, lasers, quantum optics, and molecular devices where quantum physics plays the most vital role. As technology advances, quantum concepts give birth to an increasing number of new electronic and opto-electronic devices. Their fabrications and functions can only be understood by using quantum physics. Within the next few years, fundamentally quantum devices such as single-electron memory cells and photonic signal processing systems may be available commercially. As nano-and atomicscale devices become easier to manufacture, these sophisticated manufacturing units will require an increasing number of individuals with sound knowledge of quantum physics. Therefore, all universities in the world have included quantum physics as a subject in their technical course curricula. Quantum physics is no longer a theoretical subject with mathematical complexities but an engineering subject!

Need for Quantum Physics

Two time-tested proverbs are, ‘Failure is the pillar of success’ and ‘Necessity is the mother of invention’. Classical physics based on Newtonian laws, thermodynamical laws and classical laws of electromagnetism explained successfully the macroscopic world. The macroscopic world is directly observable or can be made observable by relatively simple devices. However, classical physics failed seriously in explaining the phenomena in the realm of atoms, nucleons and elementary particles. These failures gave birth to a new branch in physics called quantum physics. In the following, we mention a few examples of the failures of classical concepts, though the list is endless.

An accelerated charge emits energy and the electron revolving around a nucleus should emit energy [its energy then should go to zero] resulting in the collapse of the atom; but atom is a stable entity! According to classical theory, the excited hydrogen atom should emit electromagnetic radiations of all the wavelengths continuously.

My dear Peter Giles, I am almost ashamed to be sending you after nearly a year this little book about the Utopian commonwealth, which I'm sure you expected in less than six weeks. For, as you were well aware, I faced no problem in finding my materials, and had no reason to ponder the arrangement of them. All I had to do was repeat what you and I together heard Raphael relate. Hence there was no occasion for me to labour over the style, since what he said, being extempore and informal, couldn't be couched in fancy terms. And besides, as you know, he is a man not so well versed in Latin as in Greek; so that my language would be nearer the truth, the closer it approached to his casual simplicity. Truth in fact is the only thing at which I should aim and do aim in writing this book.

I confess, my dear Peter, that having all these materials ready to hand left hardly anything at all for me to do. Otherwise, thinking through this topic from the beginning and disposing it in proper order might have demanded no little time and work, even if one were not entirely deficient in talent and learning. And then if the matter had to be set forth with eloquence, not just factually, there is no way I could have done that, however hard I worked, for however long a time. But now when I was relieved of all these concerns, over which I could have sweated forever, there was nothing for me to do but simply write down what I had heard. Well, little as it was, that task was rendered almost impossible by my many other obligations. Most of my day is given to the law – pleading some cases, hearing others, arbitrating others, and deciding still others. I pay a courtesy call to one man and visit another on business; and so almost all day I'm out dealing with other people, and the rest of the day I give over to my family and household; and then for myself – that is, my studies – there's nothing left.

THOMAS MORE TO HIS FRIEND PETER GILES, WARMEST GREETINGS

My dear Peter, I was absolutely delighted with the judgement of that very sharp fellow you recall, who posed this dilemma regarding my Utopia: if the story is offered as fact (says he) then I see a number of absurdities in it; but if it is fiction, then I think More's usual good judgement is wanting in some matters. I'm very much obliged to this man, whoever he may be (I suspect he is learned, and I see he's a friend). His frank judgement gratified me more than any other reaction I've seen since my book appeared. First of all, led on by fondness either for me or for the work itself, he did not give up in the middle, but read my book all the way through. And he didn't read carelessly or quickly, as priests read the divine office – those who read it at all – but slowly and carefully in order to consider the different points thoughtfully. Then, having selected certain elements to criticise, and not very many of them, he says that he approves, not rashly but deliberately, of all the rest. Finally, he implies in his very words of criticism higher praise than those who set out to compliment the book on purpose. For he shows clearly how well he thinks of me when he expresses disappointment in a passage that is not as precise as it should be – whereas I would think myself lucky if I had been able to set down just a few things out of many that were not altogether absurd.

Still, if I in my turn can deal as frankly with him as he with me, I don't see why he should think himself so acute (or, as the Greeks say, so ‘sharp-sighted’) just because he has noted some absurdities in the institutions of the Utopians, or caught me putting forth some not sufficiently practical ideas about the constitution of a republic. Aren't there any absurdities elsewhere in the world? And did any one of all the philosophers who have offered a pattern of a society, a ruler, or a private household set down everything so well that nothing ought to be changed?

Overview

So far, our discussion of syntactic structure has tacitly assumed that all constituents in a given structure are overt (in the sense that they have audible phonetic features, as well as carrying grammatical and semantic features). However, in this chapter we will see that syntactic structures may also contain null constituents (also known as empty categories) – i.e. constituents which have grammatical and semantic features but lack audible phonetic features (and so are ‘silent’, ‘inaudible’ or ‘unpronounced’).

Null subjects

We are already familiar with one kind of null constituent from the discussion of the Null Subject Parameter in §1.7. There, we saw that alongside finite clauses like that produced by Speaker A in the dialogue in (1) below with an overt subject like Maria, Italian also has finite clauses like that produced by Speaker B, with a null subject pronoun conventionally designated as pro (and referred to affectionately as ‘little pro’):

(1)

(The subscripts in the glosses mark the following properties: 3 = third person; Sg = singular number; F = feminine gender.) One reason for positing pro in 1B) is that it captures the intuition that the sentence has an ‘understood’ subject (as is clear from the fact that its English translation contains the subject pronoun she). A second reason relates to the agreement morphology carried by the auxiliary è ‘is’ and the participle tornata ‘returned’ in (1). Just as the form of the (third person singular) auxiliary è ‘is’ and the (feminine singular) participle tornata ‘returned’ is determined via agreement with the overt (third person feminine singular) subject Maria in (1A), so too the auxiliary and participle agree in exactly the same way with the null pro subject in (1B), since pro (as used here) is third person feminine singular by virtue of referring to Maria. If the sentence in (1B) were subjectless, it is not obvious how we could account for the relevant agreement facts. Since pro is found in all types of finite clause in Italian (e.g. both indicative and subjunctive clauses), we can refer to it as a finite null subject.

The island of the Utopians is two hundred miles across in the middle part, where it is widest, and nowhere much narrower than this except towards the two ends, where it gradually tapers. These ends, curved round as if completing a circle five hundred miles in circumference, make the island crescent-shaped, like a new moon. Between the horns of the crescent, which are about eleven miles apart, the sea enters and spreads into a broad bay. Being sheltered from the wind by the surrounding land, the bay is not rough, but placid and smooth instead, like a big lake. Thus nearly the whole inner coast is one great harbour, across which ships pass in every direction, to the great advantage of the people. What with shallows on one side and rocks on the other, the mouth of the bay is perilous. Near mid-channel, there is one reef that rises above the water, and so presents no danger in itself; a tower has been built on top of it, and a garrison is kept there. Since the other rocks lie under the water, they are very dangerous. The channels are known only to the Utopians, so hardly any strangers enter the bay without one of their pilots; and even they themselves could not enter safely if they did not direct their course by some landmarks on the coast. Should these landmarks be shifted about, the Utopians could easily lure to destruction an enemy fleet, however big it was.

On the outer side of the island, harbours are found not infrequently; but everywhere the coast is rugged by nature, and so well fortified that a few defenders could beat off the attack of a strong force. They say (and the appearance of the place confirms this) that their land was not always surrounded by the sea. But Utopus, who conquered the country and gave it his name (for it had previously been called Abraxa), and who brought its rude, uncouth inhabitants to such a high level of culture and humanity that they now surpass almost every other people, also changed its geography. After winning the victory at his first assault, he had a channel cut fifteen miles wide where the land joined the continent, and thus caused the sea to flow around the country.