5A Saha ionization equation

At a high enough temperature and/or density, the atoms in a gas suffer collisions due to their high thermal energy, and some of the atoms get ionized, making an ionized gas. In this process, a number of electrons that are normally bound to the atom in orbits around the atomic nucleus become free and thus form an independent electron gas cloud coexisting with the surrounding gas of atomic ions and neutral atoms. These ionized atoms and electrons generate an electric field that causes motion of the charges, and a current is generated in the gaseous medium. This current produces a localized magnetic field. The state of matter thus created is called plasma. In thermal equilibrium, the ionization state of such a gaseous system is related to the ionization potential, temperature, and pressure of the system. Thus, the Saha ionization equation. expresses how the state of ionization of any particular element in a star changes with varying temperatures and pressures. This equation takes into account the combined ideas of quantum mechanics and statistical mechanics for its derivation and is used to explain the spectral classification of stars. This equation was developed by the Indian astrophysicist Prof. Meghnad Saha in 1920. 5A.1 Derivation of Saha ionization equation According to Prof. M. N. Saha, the temperatures in the interior of stars are extremely high, and the elements present there are mostly in the atomic state. Saha argued that under the prevailing conditions inside the stars, atoms move very rapidly and undergo frequent collisions. In the process of such collisions, valence electrons are stripped off from their orbits. This is referred to as thermal ionization and is accompanied by electron recapture to form neutral atoms. The degree of such thermal ionization depends on the temperature of the star. Using the Saha ionization equation, a general relation between the degree of thermal ionization and the temperature can be obtained from the statistical description of plasma in thermodynamic equilibrium.

9.1 Introduction to Classification

We rely on machine learning (ML) to make critical decisions or predictions in the modern world. It is very important to understand how computers by using ML make these predictions. Usually, the predictions made by ML models are classified into two types, i.e., classification and regression. The ML models use various techniques to predict the outcome of an event by analyzing already available data. As machines learn from data, the type of training or input data plays a crucial role in deciding the machine's capability to make accurate decisions and predictions. Usually, this data is available in two forms, i.e., labeled and unlabeled. In label data, we know the value of the output attribute for the sample input attributes, while in unlabeled data, we do not have the output attribute value.

For analyzing labeled data, supervised learning is used. Classification and regression are the two types of supervised learning techniques used to predict the outcome of an unknown instance by analyzing the available labeled input instances. Classification is applied when the outcome is finite or discrete, while the regression model is applied when the outcome is infinite or continuous. For example, a classification model is used to predict whether a customer will buy a product or not. Here the outcome is finite, i.e., buying the product or not buying. In this case, the regression model predicts the number of products that the customer may buy. Here the outcome is infinite, i.e., all possible numbers, since the term quantity refers to a set of continuous numbers.

3A Temperature in Thermodynamics

For an infinitesimal reversible process, a combination of first and second laws of thermodynamics results

where 𝑌 𝑑𝑋 denotes the generalized expression for work done by the system, 𝑑𝑆 is the change in entropy, and 𝑑𝑈 is the change in internal energy of the system. Equation (16) leads to the definition of temperature 𝑇 as

Thus, equation (17) indicates that the temperature at any point depends on the slope of the 𝑆 − 𝑈 curve. If the slope of this curve (point 𝐴 in Figure 3A.1) is positive, the temperature will be positive. On the other hand, the temperature will be negative for the negative slope of the curve (point 𝐶 in Figure 3A.1).

The book Heat and Thermodynamics: Theory, Problems, and Solutions is an informal, readable introduction to the basic ideas of thermal physics. It is aimed at making the reader comfortable with this text as a first course in Heat and Thermodynamics. The basic principles and phenomenological aspects required for the development of the subject are discussed at length. In particular, the extremum principles of entropy and free energies are presented elaborately to make the content of the book comprehensive. The book provides a succinct presentation of the material so that the student can more easily determine the major objective of each section of a particular chapter. In fact, thermal physics is not the subject in physics that starts with its epigrammatic equations—Newton’s, Maxwell’s, or Schrodinger’s, which provide accessibility and direction. Instead, it (thermodynamics) can be regarded as a subject formed by the set of rules and constraints governing interconversion and dissipation of energy in macroscopic systems. Further, the syllabus of statistical mechanics for graduate students has changed significantly with the introduction of National Education Policy 2020.

Thermal physics has established the principles and procedures needed to understand and explain the properties of systems consisting of macroscopically large numbers of particles, typically of the order of 1023 or so. Examples of such collections of systems include the molecules in a closed vessel, the air in a balloon, the water in a lake, the electrons in a piece of metal, and the photons (electromagnetic wave packets) emitted by the Sun. By developing the macroscopic classical thermodynamic descriptions, the book Heat and Thermodynamics: Theory, Problems, and Solutions provides insights into basic concepts and relationships at an advanced undergraduate level. This book is updated throughout, providing a highly detailed, profoundly thorough, and comprehensive introduction to the subject. The laws of probability are used to predict the bulk properties like stiffness, heat capacity, and the physics of phase transition, and magnetization of such systems.

8.1 INTRODUCTION [LO1]

In simple words, the application of a torque on a prismatic member causes twisting or torsion. This causes shear stress if a torque alone is applied. However, this is rarely true in practical cases. A circular bar, used to transmit torque between a prime mover and a machine, is a typical example of a torsion member. However, in many applications, a torque along with a bending moment and axial loading are applied, and there we need to combine these effects and find the principal stresses. A typical example of such combined stresses is a propeller shaft. Torsional problems are important in many applications both in industry and in our daily life. Therefore, we consider torsion alone in this chapter in some detail.

Torsional problems for circular members are generally solved assuming that plane sections normal to the axis of the bar remain plane even after twisting. This assumption was first made by Coulomb intuitively in 1784, and he came up with a correct usable equation for members with circular sections. However, this assumption does not apply to bars with a noncircular cross-section. Navier attempted to solve torsional problems with noncircular sections using Coulomb's assumption and came up with an erroneous solution. The correct solution was provided by St. Venant in 1853 using a warping function. Much later, in 1903, Prandtl came up with a membrane analogy method that could solve problems with any complicated cross-section. First, we shall consider torsional problems with circular cross-sections.

8.2 TORSION OF MEMBERS WITH CIRCULAR CROSS-SECTION [LO2]

The torsion analysis of members with a circular cross-section starts with simplified assumptions made by Coulomb. In order to establish a relation between the applied torque and shear stress developed and the angle of twist in such cases, the following assumptions are made:

• 1. Material is homogeneous and isotropic.

• 2. Plane sections perpendicular to the axis of a circular member remain plane after twisting. No warping or distortion of the parallel planes occurs.

11.1 INTRODUCTION [LO1]

Stresses developed at the contact between two loaded elastic bodies are generally localized and most machine parts or structures are designed based on the stresses in the main body. However, there are many important machine members where the localized stresses developed at the contact between curved surfaces with initially limited contact area play an important role in their design. Ball or roller bearings, gears, cams, and valve tappets of internal combustion engines are some of the examples of machine parts where contact stresses must be taken into account in order to predict their failure probability.

The localized contact stresses that develop between two curved bodies as they are loaded with small deformations are often referred to as Hertzian stresses, following the work of H. Hertz (1881), who first solved these contact problems elegantly more than a century ago. Since then the topic has received a good deal of attention by the researchers due to its importance in engineering practice and science. Much work has been done on the stress distribution at the Hertzian contact surfaces and sub-surfaces. Ball bearings and gear teeth often fail by pitting. Hertzian stress analysis can precisely locate the depth at which maximum shear stress occurs where cracks may initiate and propagate leading to failure. Thus, a remedy to such failures may be prescribed in terms of limiting stresses. In many rolling contact problems, failure occurs with the initiation of a tiny crack that eventually grows due to repeated contacts. Analysis of crack initiation and growth is often based on Hertzian stress analysis. In this chapter, we shall consider the basics and application of contact stress analysis, beginning with some basic elasticity theory necessary for such analyses.

2.1 MATHEMATICAL PRELIMINARIES [LO1]

In any scientific or engineering field of study, knowledge of some mathematical techniques and methods are essential. Solid mechanics is no exception. To develop proper formulation methods and solution techniques for elasticity problems, it is necessary to have an appropriate mathematical background. In this chapter, we shall discuss Cartesian tensors, which have a special significance in the discussion of stress, strain, and displacement fields, and their manipulation. Other mathematical details will be discussed as and when they are required in solving different problems.

Tensors may be defined in a number of ways. One simple definition is that a tensor is a physical quantity that is governed by certain transformation laws when the coordinate system is changed. A tensor is invariant under any change of coordinate system, but its components along the coordinate axes change with the changed coordinate system. Tensors of order zero are called scalars. Common examples of scalars are temperature, density, Young's modulus, or Poisson's ratio. They have a single magnitude at each point in space, and they are invariant with coordinate transformations. A typical example of scalars is often taken as temperature T at a point in space with coordinates (x, y, z) represented as T(x, y, z). Temperature at the same point does not change if we choose a different coordinate system (x′, y′, z′) represented as T′(x′, y′, z′) and we may say

T=T′. (2.1.1)

Tensors of first order are vectors, and we know that a vector has a magnitude and a direction. A typical example of a vector is a velocity vector V. It is sometimes taken as a convention to represent a vector by a bold letter. Consider the velocity vector V in (x, y, z) coordinate system.

Solid mechanics, compared to mechanics of materials or strength of materials, is generally considered to be a higher level course. It is usually offered in higher semester to senior students. There are many textbooks available on solid mechanics, but they generally include a large part of theory of elasticity with in depth mathematical formulations. The usual prerequisites are one or two semester course on elementary strength of materials and a thorough mathematical background, including scalar, vector, and tensor field theory and cartesian and curvilinear index notation. The difference in levels between these books and elementary texts on strength of materials is generally formidable. However, in our experience of teaching this course for many years at premier institutes like IIT Kharagpur and Jadavpur University, despite its complexity, senior students generally cope well with the course using the readily available textbooks.

However, there is a vast student population pursuing mechanical, civil, or allied engineering disciplines across the country in colleges where AICTE curriculum is followed. Through several years of interaction with this group of students, we have found that there is no suitable textbook that suits their requirements. The book is primarily aimed at this group of students, attempting to bridge the gap between complex formulations in the theory of elasticity and elementary strength of materials in a simplified manner for better understanding. Index notations have been avoided, and the mathematical derivations are restricted to second-order differential equations, their solution methodologies, and only a few special functions, such as stress function and Laplacian operators.

The text follows more or less the AICTE guidelines and consists of twelve chapters. The first five chapters introduce the engineering aspects of solid mechanics and establish the basic theorems of elasticity, governing equations, and their solution methodologies. The next four chapters discuss thick cylinders, rotating disks, torsion of members with both circular and noncircular cross-sections, and stress concentration in some depth using the elasticity approaches. Thermoelasticity is an important issue in the design of high-speed machinery and many other engineering applications. This is dealt with in some detail in the tenth chapter. Problems on contact between curved bodies in two-dimensional and three-dimensional situations can be challenging, and they have wide applications in mechanical engineering such as in bearing and gear technology.

3.1 STATE OF STRESS AT A POINT [LO1]

When a body is subjected to external forces, its behavior depends on the magnitude and distribution of forces and properties of the body material. Depending on these factors, the body may deform elastically or plastically, or it may fracture. The body may also fail by fatigue when subjected to repetitive loading. Here we are primarily interested in elastic deformation of materials.

In order to establish the concept of stress and stress at a point, let us consider a straight bar of uniform cross-section of area A and subjected to uniaxial force F as shown in Figure 3.1. Stress at a typical section A - A′ is normally given as σ = F/A. This is true only if the force is uniformly distributed over the area A, but this is rarely true. Therefore, definition of stress must be considered by progressively reducing the area until it is small enough such that the force may be considered to be uniformly distributed.

To understand this, consider a body subjected to external forces P1, P2, P3, and P4 as shown in Figure 3.2. If we now cut the body in two pieces,

Internal forces f1, f2, f3, etc. are developed to keep the pieces in equilibrium. Now consider an infinitesimal element of area ΔA Dat the cut section and let the resultant force on the element be Δf.

Introduction

All metals and alloys exhibit a reduction in electrical resistance as they cool. As the temperature drops, atoms’ thermal vibrations become less intense, and conduction electrons scatter less frequently. The resistivity should decrease toward zero as the temperature approaches zero Kelvin for a perfect pure metal, where the only thing standing in the way of an electron's travel is the thermal vibrations of the lattice. This zero resistance, which a hypothetical perfect specimen would acquire if it could be cooled to absolute zero, is the phenomenon of superconductivity. Any real specimen of metal cannot be perfectly pure and will contain some impurities. As a result, in addition to being scattered by the thermal vibrations of the lattice atoms, the electrons are also dispersed by impurities, and this impurity scattering is largely temperature independent. As a result, at the lowest temperature, there will be some residual resistance. The residual resistivity of a metal increases with the degree of impurity.

The phenomenon of superconductivity was first discovered by Dutch physicist H. Kamerling Onnes of Leiden University in 1911 during the investigation of the variation of electrical resistance of mercury in the newly available range of low temperatures, in the neighborhood of temperature of liquid helium (or 4.2 K). He observed that the resistance of mercury suddenly falls from 0.08 ohm at about 4 K to less than 3 × 10−6 ohm over a very small temperature of 0.01 K.

Introduction

The nonconducting materials such as paper, wood, glass, ceramics, polymers and so on do not have free charge carriers, that is, electrons or holes. Therefore, they prevent the flow of electrical current and heat through them.

When the main function of nonconducting materials is to provide electrical isolation then they are called insulators.

When the main function of nonconducting materials is for charge storage then it is called dielectric.

The dielectrics are polarized under the influence of an external electric field.

Dielectric Constant

Let us consider two parallel plates separated by a distance “d” connected with a dc supply of voltage V, as shown in Figure 6.1(a). Now the circuit is disconnected, and the dielectric is inserted between the plates, as shown in Figure 6.1(b).

Then, the voltage across the capacitor is reduced from V to V′. The change in voltage across the plates can be related by a factor as

Since V < V , the relative permittivity or dielectric constant ɛr 1 >.

The capacitance without dielectric is given as

The capacitance with dielectric is given as

Now, put the value of C and C¢ in equation (6.1), the relative permittivity or dielectric constant is

If “A” is the area of the plates, then

where ɛ is the permittivity of the material.

Introduction

In the early days, an operational amplifier (op-amp) was the only linear integrated circuit (IC) that was used in the design of linear IC circuits and systems. Typical applications of the op-amps were mathematical operations, such as summation, subtraction, integration, small signal amplification, and generating oscillations. Over the years, other devices, such as operational transconductance amplifiers, current conveyors, and so on, have also come into common use; still, it has not reduced the importance and areas of application of op-amps. Rather, it became possible to realize many more advanced functions with linear ICs and many applications coming under the domain of nonlinear applications with advances in the process technology and increased level of integration. Some of the more common nonlinear applications are precision rectifiers, voltage-level detectors, and Schmitt trigger circuits. The Schmitt trigger circuit itself is very popular in generating varieties of pulses and other waveforms like triangular waveforms. Some other nonlinear applications such as log and antilog amplifiers, analog multiplier, charge amplifier, and isolation amplifiers are discussed in brief; phase lock loop and its basic function are also included.

Precision Rectifiers

Conventional rectifiers work well for converting alternating supply to a pulsating one. Filters are normally used to remove ripples in the pulsating voltage to obtain dc. It is observed that these rectifiers have some limitations. One of the main limitations is that when a diode conducts during rectification, it has a voltage drop across its terminals, which is approximately 0.7 V. Hence, the ac voltage available for conversion to dc is reduced by that amount.

Wave optics is the branch of modern physics in which the nature of light and its propagation are studied.

Interference

When two waves of the same frequency, having a constant phase difference between them, and traveling in the same medium are allowed to superimpose each other, there is a modification in the intensity pattern. This phenomenon is known as interference of light.

When the resultant amplitude at certain points is the sum of the amplitudes of the two waves, this interference is known as constructive interference.

When the resultant amplitude at certain points is the difference of the amplitudes of the two waves, this interference is known as destructive interference, as shown in Figure 11.1.

COHERENT SOURCES

Two sources are said to be coherent if the waves emitted from them have a constant phase difference with time.

THEORY OF INTERFERENCE

Let us consider two coherent sources S1 and S2 that are equidistant from source S. Let a1 and a2 be the amplitudes of the waves originated from source S1 and S2, respectively, as shown in Figure 11.2. Then the displacement y1 from the source S is given by

where δ is the phase difference between the two waves.

Now, according to the law of superposition, the resultant wave is given by

Band Theory ofSolids

The band theory of solids is different from the others because the atoms are arranged very close to each other such that the energy levels of the outermost orbital electrons are affected. But the energy level of the innermost electrons is not affected by the neighboring atoms.

In general, if there is n number of atoms, then there will be n discrete energy levels in each energy band. In such a system of n number of atoms, the molecular orbitals are called energy bands shown in Figure 7.1.

CLASSIFICATION OF SOLIDS ON THE BASIS OF BAND THEORY

The solids can be classified on the basis of band theory. The parameter that differentiates the solids among insulator, conductor, and semiconductor is known as energy band gap and represented by (Eg), as shown in Figure 7.2. When the energy band gap (Eg) between conduction band and valence band is greater than 5 eV (electron-volt) then the solid is classified as insulator. When the energy band gap (E g)between conduction band and valence band is 0 eV (electron-volt), that is, overlapping of bands occurs then the solid is classified as conductor. When the energy band gap (Eg) between conduction band and valence band is approximately equals to 1 eV (electron-volt) then the solid is classified as semiconductors.

Introduction

Statistical mechanics bridges the gaps between the laws of thermodynamics and the internal structure of the matter. Some examples are as follows:

• 1. Assembly of atoms in gaseous or liquid helium.

• 2. Assembly of water molecules in solid, liquid, or vapor state.

• 3. Assembly of free electrons in metal.

The behavior of all these abovementioned assemblies is totally different in different phases. Therefore, it is most significant to relate the macroscopic behavior of the system to its microscopic structure.

In this mechanics, most probable behavior of assembly are studied instead of individual particle interactions or behavior.

The behavior of assembly that is repeated a maximum time is known as most probable behavior.

hase Space

Six coordinates can fully characterize the state of any system:

• 1. Three for describing the position x, y, z and three for momentum Px, Py, Pz.

• 2. The combined position and momentum space (x, y, z, Px, Py, Pz) is called phase space.

• 3. The momentum space represents the energy of state,

For a system of N particles, there exists 3N position coordinates and 3N momentum coordinates. A single particle in phase space is known as a phase point, and the space occupied by it is known as µ-space.

olume Element ofµ-Space

4. Consider a particle having the position and momentum coordinates in the range.