Today, Python is known to be one of the most in-demand programming languages. As per the stats of GitHub (a provider of Internet hosting for software development), Python is the second most popular programming language, following JavaScript, as shown in Figure 2.1, and soon it may be on the top of the chart. Python surpassed Java, PHP, and other prominent languages in 2019.

Python is easy and versatile. So it is acclaimed as the major programming language to work on many new-age technologies like machine learning (ML), artificial intelligence, data science, and natural language processing. The creator of Python, Guido van Rossum, in 1991, stated that Python is a high-level programming language, and its core design philosophy is about code readability and syntax, which allows programmers to express concepts in a few lines of code. Interestingly, the name Python is inspired by Guido's favorite television show Monty Python's Flying Circus.

In this chapter, we will discuss various programming constructs of Python so that you can easily implement ML algorithms by using it. Before writing the actual code in Python, let us focus on the features of Python that make it so popular and unique.

2.1 Features of Python

Features offered by Python can be visualized in Figure 2.2. Talking about them profoundly, the main features of Python are as follows:

• • Beginner's Language: Python is not only just easy to code and learn, but also fast to grasp, and hence it is a suitable choice for any novice user who wants to learn to program. This is why nowadays this language is introduced to students in schools.

• • Interpreted: Unlike other programming languages such as C or C++, Python does not require you to compile programs before executing them. It is an interpreted language, i.e., the code written in Python gets processed in real-time line by line.

• • Interactive: The interactive feature of Python enables real-time feedback, allowing programmers to experiment, debug, and make adjustments on the go.

5.1 Introduction to Simple Linear Regression

As discussed in earlier chapters, regression predicts a continuous value or real-valued output. This chapter will discuss how regression works (from a mathematical aspect) to predict the continuous value for the given dataset. Our first learning algorithm is simple linear regression. In this section, we will discuss the fundamental concepts and mathematical modeling of simple linear regression.

We usually have a dependent variable having a continuous value whose value we wish to predict based on one or more independent variables. If we have only one independent or input variable, this situation is known as simple linear regression (also called univariate regression). If we have multiple independent or input variables, it is known as multiple linear regression or multivariate regression.

Linear regression could be used for studying patterns in different real-life scenarios. Consider a research lab where a researcher wants to understand how the stipend is effected by the years of experience, or, in simple words, we wish to predict the stipend based on the years of experience of the researcher. Machine learning (ML) is about learning from past experiences or data. Thus, to predict the researcher's stipend, we have to collect some data about past researchers, specifically their stipend and experience.

In the supervised learning models, we need a dataset called a training set. We will use the dataset as given in Table 5.1 for training the model, and our job will be to build the ML model that learns from this data and hence predicts the stipend of a researcher based on his experience. Here, the stipend will be considered the dependent or output variable because it depends on the researcher's years of experience. Thus, years of experience will be considered an independent or input variable. So, we will use simple linear regression to build the ML model. For proceeding with this problem, we will use a dataset of researchers’ stipends with their corresponding years of experience, as shown in Table 5.1.

Learning Outcomes After reading this chapter, the reader will be able to

• Know various types of thermodynamic systems such as open, closed, and isolated, and the surroundings

• Classify between intensive and extensive thermodynamic variables

• Understand various types of equilibrium conditions satisfied by a thermodynamic system

• State the zeroth law of thermodynamics and highlight its physical significance

• Comprehend the idea of temperature from the zeroth law of thermodynamics

• Solve numerical problems and multiple choice questions on thermodynamic equilibrium and the zeroth law of thermodynamics

7.1 Introduction

Heat is a form of energy. It can be transformed from one form to another as well as can be transferred between various objects maintained at suitable temperatures. For example, in an electric motor, heat is transformed into mechanical energy by the turbine to power the motor. This mechanical energy is then transformed into electrical energy by the engine to illuminate light bulbs. “Thermodynamics” is a branch of physics that deals with heat and the transformation of heat from one form to another, work, temperature, and their relation to energy, entropy, and other physical properties of matter and radiation. It establishes the relation between heat and various forms of energy and describes the transformations that occur in thermal energy from one energy state to another and how this transformation affects matter. A thermodynamic system is described within a framework based on the four laws of thermodynamics that facilitate a quantitative description of the average macroscopic properties of the system in equilibrium. Macroscopic matter refers to large objects that consist of many atoms and molecules. The average properties of such macroscopic systems are determined by the physical quantities such as volume, pressure, and temperature that do not depend upon the detailed microscopic positions and velocities of the atoms and the molecules comprising the macroscopic system. In the equilibrium state of a thermodynamic system, these average properties also do not change with time. These physical quantities are called thermodynamic coordinates, variables, or parameters. If a subset of these properties are experimentally measured, the rest of them can be calculated using thermodynamic relations. Thermodynamics not only gives the exact description of the state of equilibrium but also provides an approximate description (to a very high degree of precision!) of relatively slow processes. This branch of physics can be successfully applied to a wide variety of topics in science, such as physics, physical chemistry, biochemistry, chemical engineering, and mechanical engineering, but also in other complex fields, such as meteorology.

5.1 INTRODUCTION [LO1]

In any three-dimensional (3D) elasticity problem, there are 15 unknown parameters: 6 stress components, 6 strain components, and 3 displacements. There are 15 related equations: 3 equations of equilibrium, 6 compatibility equations, and 6 constitutive equations. Solutions to a particular elasticity problem require evaluation of these 15 unknown parameters using 15 equations, satisfying all the boundary conditions. As discussed in the earlier chapters, there may be displacement or stress, or mixed boundary conditions. In many cases, solutions to 3D problems are not easy analytically. Even numerical solutions may be difficult.

There are mainly three methods of simplification of solution techniques:

• (a) If the boundary conditions are in terms of stresses, stress function approach may be made as discussed in the earlier chapter. This makes the solution simpler.

• (b) Assumptions of plane stress and plane strain reduce 3D problems to two-dimensional (2D) ones and this also makes the solution simpler.

• (c) Use of St. Venant's principle and superposition principle also makes the solution of elasticity problems simpler.

An introduction to stress function approach has been discussed in Chapter 4. We therefore start our discussion on plane stress and plane strain approaches.

5.2 PLANE STRESS AND PLANE STRAIN PROBLEMS [LO2]

The idealizations of both plane stress and plane strain states are suitable for certain classes of problems that are made to reduce the complexity of solutions. We shall consider the plane stress state first.

Learning Outcomes

After reading this chapter, the reader will be able to

• Learn the basic concept of the theory of probability

• List the assumptions used in the derivation of Maxwell's speed distribution law

• Derive Maxwell's speed distribution law and test its validity experimentally

• Calculate average, root mean square and most probable speed, energy, and momentum in one, two, and three dimensions, respectively

• State and prove the law of equipartition of energy

• Calculate the specific heat of gases

• Solve numerical problems and multiple choice questions on the distribution of molecular speed, energy, and momentum

3.1 Introduction

In Chapter 2, various characteristic features of a gaseous system based on the model of the kinetic theory of gases (KTG) have been discussed elaborately. Macroscopic properties and various relations among the thermodynamic variables have been explained in terms of this kinetic model.

According to the assumptions used in this model, a gaseous system is composed of a large number of particles (atoms or molecules) with practically no volume occupied by them. Most of the times, these molecules move randomly through empty space at temperatures above absolute zero, and such motions remain unaffected by the presence of other particles. This motion of the molecules is extremely chaotic and is characterized by straight-line trajectories interrupted by collisions with other molecules or with a physical boundary. In such a collision, the transfer of kinetic energy with a change in direction takes place depending on the nature of the relative kinetic energies of the particles. Any individual molecule collides with others at a huge rate, typically of the order of a billion times per second. This chapter is focused to present a comprehensive and quantitative discussion on the distributions of velocities, energies, and momenta of these molecules in various dimensions.

Measurement of the velocities of the molecules at a given time leads to a large distribution of values; some molecules may move very slowly and others very quickly. As these molecules move constantly in different directions, the velocity could be momentarily equal to zero

Learning Outcomes

After reading this chapter, the reader will be able to

• Gather knowledge about state function and its properties

• Understand the meaning of internal energy and its significance in formulating the first law of thermodynamics

• Formulate the first law of thermodynamics and apply it to various thermodynamic processes

• Grasp the idea of various thermodynamic processes and the related work done in these processes

• Find a relation between specific heats at constant volume and constant pressure for ideal and real gases

• Find expression for isothermal compressibility and volume expansion coefficient for ideal and real gases

• Understand how the temperature of air varies with height assuming an adiabatic process

• Solve numerical problems and multiple choice questions on the first law of Thermodynamics

8.1 Introduction

Systems, surroundings, and interactions between them play a vital role in the development of the subject—thermodynamics. To extract out the physical properties of a thermodynamic system, it is essential to have knowledge about the fundamental laws and concepts of thermodynamics. For example, heat and work are two interrelated concepts. Heat is the transfer of thermal energy between two bodies that are at different temperatures and are not equal to thermal energy. Work is the external physical parameter used to transfer energy between a system and its surroundings. Further, work is needed to create heat and to transfer the thermal energy. Thus, work and heat together allow systems to exchange energy. The relationship between these two physical quantities, heat and work, can be analyzed through the laws and concepts of thermodynamics. The interaction between heat and other types of energy is primarily focused on the topic of thermodynamics. To understand the relationship between heat and work, it is required to have an idea about a third linking factor, known as the change in internal energy

Chapter Objectives

• To understand the process of implementation of the artificial neural network (ANN).

• To understand the role of keras and its different modules in building the ANN.

• To understand the syntax for adding input layer, hidden layers, and output layer to ANN.

• To perform a compilation of the ANN model.

• To fit the ANN model on the training dataset.

• To make predictions with a trained ANN model.

• To evaluate the performance of the ANN classifier by using confusion matrix, precision, and recall.

17.1 Building Artificial Neural Network for Cancer Detection

Machine learning (ML) can play a crucial role in cancer detection. In this chapter, we will build a neural network for cancer detection by using a breast cancer dataset.

You can download this dataset by using the following link.

https://www.kaggle.com/datasets/uciml/breast-cancer-wisconsin-data

The acquired data contains the records of cancer patients in the United States. These records were created by Dr William H. Wolberg and others at the University of Wisconsin, USA. The whole data has 32 columns along with 569 rows. The prominent attributes are the radius, texture, perimeter, smoothness, concavity, symmetry, area, compactness, concave points, and the fractal dimension of the tumor. A snapshot of the dataset is depicted in Figure 17.1.

The dataset has a diagnosis column used as an output variable, while the remaining variables will be used as input data. The class attribute diagnosis has two classes, i.e., malignant identified as M and benign identified as B. Thus, it will be a binary classifier.

The code and dataset used in this chapter are also available at the following link.

https://github.com/bhatiaparteek/ml_with_python/tree/main/Chapter_17_ANN

To build ANN over this dataset, the whole procedure can be divided into three sub-parts below.

• i. Loading the dataset and performing pre-processing of data

• ii. Building the artificial neural network (ANN)

• iii. Making predictions and performing the validations

Let us perform all these operations by following a step-by-step approach.

17.2 Loading the Dataset and Pre-processing

In this step, we will perform tasks of loading the dataset and pre-processing.

17.2.1 Step 1: Importing the Libraries

To perform this task, we need to import two libraries, i.e., Pandas and NumPy, as shown in code snippet 1. NumPy facilitates mathematical operations, while Pandas specializes in loading and extracting datasets.

7A Planck's units

In 1899, Max Planck first proposed his radical theory of energy quantization. He proposed to build a system of “natural units” from a few of the more important constants in physics. These important constants include the speed of light, the universal gravitational constant, the Planck constant, and the Boltzmann constant. These quantities have great significance in various fields in physics. Combining these fundamental constant quantities, Planck generated various expressions with units of mass, length, time, and temperature. These units are known as Planck units. These four units signify us something different about the nature of reality. Before describing these units, a brief description about the four fundamental physical constants is presented below.

7A.1 The speed of light 𝑐

The speed of light c in a vacuum is a natural unit for speed and has magnitude 𝑐 = 299, 792, 458 ms−1. According to the special theory of relativity, this speed is the upper limit for the speed at which conventional matter or energy can travel through space. It is the universal “speed limit”. It was recognized during the information age that the photons of electromagnetic radiation and material objects are used as the carriers of information. The speed of light is then a restriction on the speed at which information may travel. The speed of light can be used in time of flight measurements to measure large distances to extremely high precision. In a paper published in 1865, James Clerk Maxwell proposed that light is an electromagnetic wave and travels at speed c. In 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame of reference is a constant and is independent of the motion of the light source.

7A.2 Universal gravitational constant𝐺

The gravitational constant, denoted by the capital letter G, is an empirical physical constant with magnitude 𝐺 = 6.67428 × 10−11 N m2 kg−2. This universal gravitational constant is involved in the calculation of gravitational effects in Newton's law of universal gravitation and in Einstein's theory of general relativity.

15.1 Building Association Mining Model

In this chapter, we will implement the Apriori algorithm in Python to solve a business problem. Association rule mining is one of the most popular machine learning applications, which is often used by supermarket chains and retail outlets to find the relation between the sales of item X and item Y. It is often called a “market basket” analysis. Discovering associations among attributes can lead to fact-based marketing strategies for store floor plans, special discounts, coupon offerings, product clustering, and catalog design to identify items that need to be put in combo packs.

Let us solve the problem of one such retail store by implementing the Apriori algorithm in Python.

Problem statement: Consider a supermarket store selling 16 products. To better understand the association between the sales of different items, the store manager decides to perform a market-basket analysis through the Apriori algorithm. The goal is to find association mining rules to improve the store's sales. The list of 16 items in-store is shown in Table 15.1, and the manager is analyzing 25 transactions of the sale of these items as given in Table 15.2. The number of items and transactions in a real application will be larger. In Table 15.2, each row in the table represents one transaction, i.e., the items bought by one customer.

In this dataset, the manager is interested in finding association rules that should have a minimum of 25% support and 70% confidence.

The implementation of association mining can be broken down into multiple steps. These steps are described below:

• Step 1: Importing libraries and loading the dataset—We must import the required libraries for model building into the environment, and then we must load the required dataset.

• Step 2: Making transactions—Apriori takes an input as a set of transactions in the desired structure; thus, we need to prepare a set of transactions containing the combination of items.

• Step 3: Building the model—An Apriori model will be trained on the lists of transactions, and rules will be generated based on some key parameters (support and confidence).

10.1 INTRODUCTION [LO1]

There are many applications where structures or machine parts are subjected to significant changes in temperature, for example, turbine blades, high-speed rotating machinery, and boilers in thermal power plants. Large thermal stresses may be developed in such applications, and sometimes such stresses may exceed the yield limit. It is therefore necessary to make provisions in the design of components to avoid failure due to thermal stresses. If the ends of a rod or any other machine parts are rigidly fixed such that the expansion or compression is prevented and the temperature is changed, tensile or compressive stress would be set up, and in simple terms, these stresses are called thermal stresses. In a long steam pipe, expansion joints are sometimes inserted, and in bridges, one end may be rigidly fastened to the main structure while the other end rests on rollers to avoid thermal stresses. In simple terms, this may be demonstrated considering a rod of length l and cross-sectional area A fixed at both ends (Figure 10.1) and temperature is raised by ΔT. This would produce a thermal strain ∈t in the rod such that

where a is the coefficient of thermal expansion. Since the rod is not free to expand, a compressive stress σc would be developed in the rod, and this is given by

If the rise in temperature is significant, the rod may buckle, which is of serious consideration in the design of machine parts or structures. To avoid this, we need to find the critical force Pcr for buckling. Using the basic buckling criterion, this can be given for a column pin ended at both ends, by

where I is the least moment of inertia of the constant cross-section column (rod) and A is the cross-sectional area of the column.

Machine learning (ML) algorithms work on cleaned data. Usually, the data we collect for building ML models suffers from noise, missing values, inconsistent data types, and different data scales. This makes pre-processing of data a very important phase in preparing the data for building ML models. Pre-processing is when we apply transformations over the data before feeding it to the ML algorithm. In short, data pre-processing symbolizes a set of procedures applied to the data to make it fit for ML algorithms. It generally involves the following steps:

• Step 1—Importing libraries: It involves importing the necessary libraries that are required to carry out the subsequent data manipulation and cleaning tasks.

• Step 2—Loading the dataset: The dataset that needs to be pre-processed must be loaded.

• Step 3—Handling the missing values: Dataset often contains missing or null values; these values need to be handled appropriately.

• Step 4—Handling the categorical data: In the data pre-processing phase, it is crucial to address categorical attributes that often contain multiple categories. Handling categorical data becomes an important step to ensure proper treatment and transformation of these attributes.

• Step 5—Splitting the dataset into training and testing datasets: Training and testing is the most important part of ML; thus, we need to split the dataset into training and testing subsets before building the ML models.

• Step 6—Feature scaling: In datasets, the range of data often varies, or data is often of different scales. Thus, feature scaling needs to be done to ensure uniformity in results.

It is important to note that it is not necessary to apply all of these steps to pre-process the data. However, based on the nature of the dataset, some of these steps may be skipped for building the model. In the coming sections, we will discuss the importance or need of these steps and discuss how to perform these steps in Python.

6A Radiation as a photon gas

In the nineteenth century, physicists applied classical electromagnetic theory to explain the experimental results of black body radiation but were unable to provide an adequate explanation. This was a major problem to the physicists as classical theory predicted an infinite amount of energy of the radiation emitted from a black body. This gross disagreement was called the “ultraviolet catastrophe”. Max Planck presented a paper on December 14, 1900, in which he guessed the answer to this problem of black body radiation. This guess marked the very beginning of quantum mechanics.

The electromagnetic radiation inside a black body chamber exists as patterns of standing waves or modes. A single mode is like a standing wave on a guitar string and is characterized by a frequency. Planck made a bold hypothesis that the energy of such a mode is quantized, ithat is, only certain energies of these oscillation modes are allowed. Thus, the energy 𝐸𝑛 of 𝑛th mode is given by 𝐸𝑛 = (𝑛 + 12 ) ℏ𝜔 = (𝑛 + 12 ) ℎ𝜈, where 𝜈 is the oscillation frequency, ℎ is Planck's constant, and 𝑛 is an integer starting with 0. Thus, each oscillation mode can exist in any one of an infinite number of energy states whose energies are equally separated by the energy ℏ𝜔. In the discussion of the properties of the photon gas, for the time being, we shall ignore 12 in the expression for 𝐸𝑛 as it has no effect on the results we seek. Therefore, we take 𝐸𝑛 = 𝑛ℏ𝜔 as the energy of the 𝑛th mode whose (angular) frequency is 𝜔. When the energy of a mode is 𝐸𝑛, we say that there are 𝑛 photons in the mode. Each photon has energy equal to ℏ𝜔. Thus, according to the prescription of Max Planck, a black body radiation chamber consists of a number of photons in various energy states with different amounts of energy.

20.1 Neural Networks and Human Brain

The inspiration to build artificial neural networks (ANNs) has come from the deep desire to simulate the working of the human brain. As our understanding of the human brain is getting enriched and improved daily due to the ongoing research on this topic, researchers are also improving the ANNs accordingly. One such recent addition is the recurrent neural networks (RNNs) and the long short-term memory (LSTM). In this chapter, we will discuss these developments in a simplified manner so that our readers can understand this amazing technology and use it for their development projects.

Sections 20.1 and 20.2 draw their inspiration from the blog titled “The Ultimate Guide to Recurrent Neural Networks (RNN)” published by SuperDataScience Team. Most of the figures used in these sections are adapted from this blog.

For detailed information on the blog, please refer to the “Additional Resources” section of this chapter.

Research on the human brain gives us the idea that the human brain has three main parts: cerebrum, cerebellum, and brainstem, as shown in Figure 20.1. Let us briefly discuss the main functions of these parts and their components to understand the association between neural networks and the human brain.

Cerebrum

From Figure 20.1, we can understand that the cerebrum has four lobes. These are as follows.

Welcome to the World of Machine Learning!

In this ever-evolving era of technology, the field of machine learning (ML) stands at the forefront of innovation, promising unprecedented insights and solutions to complex problems. This textbook provides a comprehensive understanding of the theoretical foundations of ML algorithms, coupled with practical implementation using Python, designed to be a companion for both beginners venturing into the field and seasoned practitioners seeking to deepen their understanding.

Why Machine Learning?

ML is not just a technical endeavor; it is a transformational force shaping the future of how we interact with data. From enhancing search engines and revolutionizing social media to powering self-driving cars and advancing artificial intelligence (AI), the applications of ML are boundless. ML has paved the way for developing sophisticated chatbots, generative AI models, and AI copilots. This book aims to demystify the intricate concepts of ML, making them accessible to learners of all backgrounds.

What This Book Offers

• - Foundations: We begin with the fundamentals, laying a solid groundwork to help you grasp the core concepts of ML.

• - “Learning by Doing” Approach: This book adopts a “learning by doing” approach, offering step-by-step coding instructions for various ML techniques. The aim is to empower you with the knowledge and skills to implement these principles effectively.

• - Real-World Applications: Understanding theory is essential, but applying it to real-world scenarios is where the true power of ML unfolds. This book bridges theory and application, ensuring a holistic learning experience.

For Whom This Book Is Intended

Whether you are a student exploring the realms of ML for the first time, a data professional aiming to expand your skill set, or a business leader seeking insights into how ML can elevate your organization, this book is crafted with you in mind.

How to Use This Book

Feel free to navigate the chapters based on your familiarity with the subject. If you’re a beginner, start from the beginning, and if you’re seeking advanced knowledge, dive into specific sections of interest. Explore GitHub resources that provide access to datasets, sample code, and examples used in this book for hands-on learning. For enhanced visual understanding, this book is complemented by an online video course available at learncompscience.com, allowing you to reinforce your knowledge through engaging video sessions.

19.1 Building Image Classifier with CNN

The convolutional neural network (CNN) is the perfect choice for building an image classifier system. In this chapter, we will implement various CNN classifiers in Python by considering various case studies.

19.2 Dog–Cat Classifier

Consider an image classification problem that involves identifying photos as either containing a cat or dog. Our task is to develop a CNN model that can identify the object in the image as a dog or cat. The CNN model will be trained on a dataset of images containing a dog or cat, and then this trained model will be used for classification. Once the model is built, it can be used as a template to build other models for predicting the class of any image on the trained dataset. It means we can use the same model to classify the tumors by training them on medical images.

Preparing the Dataset

To build a dog–cat classifier, we will use 11000 manually annotated photos of dogs and cats, where 5500 photos are of dogs, and the remaining ones are of cats. This is the partial dataset that has been created from the dogs-vs-cats dataset available at Kaggle. The link for this dataset is given below.

https://www.kaggle.com/competitions/dogs-vs-cats/data

This full dataset contains 37500 images (24000 in the train and 13500 in the test folders), which must be manually segregated into dog and cat folders. Since the dataset is huge and segregation is time-consuming, we took 10000 images (5000 images of dogs and 5000 images of cats) in the train folder and 1000 images in the test folder (500 images of dogs and 500 images of cats).

12.1 Introduction to Clustering

In machine learning (ML), labeling the data is one of the crucial tasks. But sometimes, we do not have the labeled data. Even though the data is not labeled, we can still analyze it using clustering techniques. As you know, algorithms in ML are broadly classified as supervised and unsupervised techniques. In the case of supervised learning, the input data points/examples are labeled, while in unsupervised learning, the input data points/examples are not labeled. Cluster analysis, also called clustering, comes under unsupervised learning. Here, the input data points are not labeled. Clustering is the most popular technique in unsupervised learning.

Clustering is defined as grouping the input data points into various clusters/groups based on their similarity.

A cluster contains objects that are more similar to each other. In other words, during cluster analysis, the data is grouped into classes or clusters, so that records within a cluster (intra-cluster) have high similarity with one another but have high dissimilarities in comparison to objects in other clusters (inter-cluster).

The clustering algorithm aims to minimize the intra-cluster distance and maximize the inter-cluster distance, as shown in Figure 12.1.

An example of clustering is shown in Figure 12.2. Here, records in the input have different shapes. Here, we only have the input data without any label of shape. After applying the clustering algorithm, they were classified into three types of clusters. Here, the clustering algorithm considers the dimensions of the object and its color as the input features. Records whose features are highly similar are gathered to form a single cluster. In this case, we get three clusters representing three types of records, i.e., it clusters rhombus, circle, and triangle separately, as shown in Figure 12.2.

Learning Outcomes

After reading this chapter, the reader will be able to

• Understand the distribution of energy density in a black body radiation as a function of wavelength and temperature

• Derive classical laws of black body radiation such as Wien distribution law and Rayleigh–Jeans law

• Get an idea about the development of quantum theory of radiation

• Understand Planck's quanta postulates and explain the black body radiation spectrum

• Derive Planck's law of black body radiation

• Verify Planck's law of black body radiation experimentally

• Derive Wien distribution law and Rayleigh–Jeans law and explain ultraviolet catastrophe from Planck's law of black body radiation

• Determine the temperature of cosmic microwave background radiation using Planck's law of black body radiation

• Solve numerical problems and multiple choice questions on black body Radiation

11.1 Introduction

Figure 11.1 The whole electromagnetic spectrum. Thermal radiation ranges in frequency from the shortest infrared rays through the visible-light spectrum to the longest ultraviolet rays.

Radiation emitted from the surface of a heated source is known as thermal radiation. In this process, thermal energy is spread out in all directions in the form of electromagnetic radiation and travels directly to its point of absorption at the speed of light. It does not require an intervening medium for its propagation. The wavelength of thermal radiation ranges from the longest infrared rays through the visible-light spectrum to the shortest ultraviolet rays. Such electromagnetic spectrum is shown in Figure 11.1 as a function of frequency. The distribution of radiant energy with their corresponding intensities within various ranges of wavelengths is governed by the temperature of the emitting surface .