## Sunday, April 12, 2009

### 1. Fundamental electromagnetic field and wave theory

1. Introduction

1.1 Prologue

I have read quite a number of textbooks on the subject of electromagnetism and my favourite ones are 'Electromagnetics for Engineer' by Ulaby and 'Field and Wave Electromagnetics' by D. K. Cheng. Both textbooks have pros and cons in the way they presented this subject to the public but they complement each other well.

The book by Ulaby has a more practical approach and therefore emphasized the applied side of electromagnetism. This is usually more suitable for the general engineers - you don't ask (too much) why, as long as it works. The book by Cheng is more mathematically rigorous and it could be quite tough for students without strong mathematical foundations. However, it looks at the Maxwell's equations as a mathematical subject, which is what it should be. At higher levels, I reckon, students will appreciate Cheng's book more. Nevertheless, both are good introductory books to electromagnetism and I highly recommend them for undergraduate students.

I have always wanted to write a book on Electromagnetism myself. And I will attempt to do so here in my blog. I hope to distinguish my little online 'book' on electromagnetism from the other textbooks by trying to explain the world of electromagnetism in as little jargon and mathematics as possible. This is no doubt daunting because mathematics is behind everything in Maxwell's equations.

Also, most of the textbooks chose to explain the equations first, then the observations (as a consequence of the equations). To explain electromagnetism in this order is understandable, because it would be easier. However, giving explanation in that order will give little information on how the equations itself emerged and creates a false impression that the equations emerged before the observations. This 'book' will attempt to cover this gap.

1.2 Objective

My online 'book' would, of course, be unable to replace the actual textbooks. I am obviously unable to give a detail account of the subject using many mathematical equations and graphs as the textbooks are able to. But what I will attempt to do here is to provide an account of the subject from my perspective and attempt to explain the theories of electromagnetism in a layman manner. I feel that many of the textbooks explained electromagnetism from a very mathematical point of view, and many undergraduate students ended up memorising Maxwell's equations without really knowing what they mean physically. Upon completion of a course in electromagnetism, the students can apply Stoke's and Gauss's theorem effortlessly and know that the divergence of electric flux density equals to the total charge density. But what does all these means in the physical world? This will usually elude the general students. I hope this book will be able to give more insight into the physical meaning of the governing equations of electromagnetism.

Also, there is a need for students to know that the process of scientific inference began not from equations, but from experimental observations, i.e. experiments are carried out in order to find the laws of the nature and mathematical models are proposed to explain the observations of the experiments. Not the other way around. All modern scientific theories take a similar path. First, there is the law of nature. Then, there are the observations. Mathematical models are proposed to explain the observations. Finally, the models are used to predict other results or to engineer new products. Therefore, for example, to explain why one of the Maxwell's equations says that the divergence of magnetic flux is equal to zero? It is simply because from experimental observations, magnetic flux always close upon themselves. In mathematical form, it is denoted as '∇.B=0' . There is no point further asking why to this, because there is really no reason apart from the fact that no other experimental observations deviated from this mathematical equation (of course, this changed with the emergence of quantum mechanics, but we shall leave that to another time).

The equations make more sense if you look at it as a 'model' to explain the experimental results. Therefore, in order to fully appreciate Maxwell's equation it is necessary to understand the major experimental observations from late 18th century and follow their historical development that led to Maxwell's famous equations. As you may have guessed now, Maxwell did not perform the experiments himself. He merely came up with the 4 equations that succinctly describe all the phenomenon observed in the experiments. The word 'merely' may be an understatement here, because this was not an ordinary feat.

It is worthwhile to point out that with the emergence of quantum mechanics in the early 20th century, the Maxwell's view of electromagnetism is not entirely accurate, especially at the atomic level. This is why some textbooks emphasize the word 'field theory' when they explain electromagnetism using Maxwell's equations. As we shall see later, Maxwell's equations worked on the assumption that electric charges produced 'invisible' fields into the space and exerted forces along these field lines. Nevertheless, on the macroscopic level (up to about 100nm), Maxwell's equations work perfectly well.

1.3 Pre-requisite

One of the most annoying part of being an undergraduate student is finding a suitable textbook. Some books are too difficult and some books are just too easy. There's never a book that is 'just nice'! And I'm sure, as students, we constantly find ourselves in the situation of reading a textbook only to find that it contained some weird symbol or equation that you have never came across before. You will have to pick up another few books, just so that you could understand them. Only to realise that the few other books that you picked up, also contained some symbol or equation that you do not understand. And the search goes on...

With the advent of internet, things become much easier. But nevertheless annoying. Therefore, it is important to write a book that is self-containing and has a clear pre-requisite in order for the reader to fully appreciate it.

For this book, I assume that the reader would have a sound understanding of A-level physics and elementary mathematics (further mathematics not required). The reader would also be familiar with basic matrix and vector operation as well as some knowledge in vector calculus. All other equations used in this book will be derived from this basic understanding or other equations that have been derived earlier in the book. Therefore, with this pre-requisites, I hope this book will be self-contained.

1.4 Structure

I shall structure the book into the following chapters: chapter 1 is introduction (this chapter); chapter 2 on electrostatic; chapter 3 on magnetostatic; chapter 4 on ... (I will add/modify this as I write more). I will put a tag called 'EM book' on all the entries so that when you filtered my blog using this tag, you will get the 'book' in the order of the chapters described here.

These chapters shall be written in a modular format so that readers can jump to a particular chapter without any loss of continuity if they wish to.

Sometimes, I will digress from the main topic to provide additional detail. Whenever this happens, I will denote it with a long asterisk line like this:

*********************************
digressing from main topic
and additional detail here
but you can choose to skip
*********************************

In order for the reader to have a continuous flow of ideas, the reader could choose to skip those paragraphs contained within the two asterisk lines.

The book begins now... and if you like what I am doing, tell your other engineering undergraduate friends and drop me a comment or two so that I know.

#### 1 comment: Agree, too much mathematical expression, too little explanation of physical meaning.

In the end, we just know to solve standard question, but can't apply in real problem.