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MEFware*: Open-Access Materials for Teaching and Learning Fundamentals of Electromagnetics (FEM), using the Maxwell's Equations First (MEF) Approach


Nannapaneni Narayana Rao
Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign (UIUC), USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, India

(Updated December 18, 2009)


Photo of Everitt Lab     Group Photo in India

"Fill your heart with love and express it in everything you do." – Amma Mata Amritanandamayi Devi, Chancellor, Amrita Vishwa Vidyapeetham

"Our duty to others means helping others; doing good to the world. Why should we do good to the world? Apparently to help the world, but really to help ourselves.... Do not stand on a high pedestal and take five cents in your hand and say, ‘Here, my poor man,’ but be grateful that the poor man is there, so that by making a gift to him, you are able to help yourself. It is not the receiver that is blessed, but it is the giver. Be thankful that you are allowed to exercise your power of benevolence and mercy in the world, and thus become pure and perfect." – Swami Vivekananda

"Take up one idea.  Make that one idea your life – think of it, dream of it, live on that idea.  Let the brain, muscles, nerves, every part of your body, be full of that idea, and just leave every other idea alone.  This is the way to success." – Swami Vivekananda

Let the one idea be “our duty to others.” 

The materials provided here through the links can be downloaded without prior permission, by anyone and anywhere in the world, for the purpose of teaching and learning Electromagnetics. When used for lectures or for presentations to audiences, an acknowledgment of the source will be appreciated. Thank you – Narayana Rao

(*The term "MEFware" replaces the previously used term "EMFware." See related article.)

Making the learning of Maxwell's equations painless
Why Study Electromagnetics?
Spreading Maxwell's Message
On Maxwell and Maxwell's Equations
The MEF approach of teaching FEM
Details on Books
Open-Access Materials

Dedication, in gratitude to Everitt, Jordan, and VanValkenburg

“Gratitude and appreciation are the key virtues for a better life. They are the spell that is cast to dissolve hatred, hurt and sadness, the medicine which heals subjective states of mind, restoring self-respect, confidence and security.” – the late Gurudeva Sivaya Subramuniyaswami of the Kauai Aadheenam, Kauai, Hawaii

“I am grateful for being here, for being able to think, for being able to see, for being able to taste, for appreciating love – for knowing that it exists in a world so rife with vulgarity, with brutality and violence, and yet love exists.  I am grateful to know that it exists.” – Celebrated author and poet Maya Angelou looking back on 80 years of blessings, in American Way Magazine of the American Airlines, November 15, 2008.

This site is dedicated to three noble and legendary personalities in engineering education, with whom I have had the good fortune of being associated, in my academic career at the University of Illinois at Urbana-Champaign – William L. Everitt, the “grandfather” of my department; Edward C. Jordan, the “father” of my department; and Mac E. VanValkenburg, the “uncle” of my department.  Please read speech at the Jordan Professor Investiture Ceremony on April 14, 2004. 

William L. Everitt     Edward C. Jordan     Mac E. VanValkenburg

Making the learning of Maxwell’s equations painless


“… I am talking about the areas of science and learning that have been at the heart of what we know and what we do, that which has supported and guided us and which is fundamental to our thinking.  It is electromagnetism (EM) in all its many forms that has been so basic, that haunts us and guides us…” – Nick Holonyak, Jr., John Bardeen Endowed Chair Professor of Electrical and Computer Engineering and of Physics, UIUC, and the inventor of the semiconductor visible LED, laser, quantum-well laser, and the transistor laser, in a foreword in the special Indian Edition of “Elements of Engineering Electromagnetics, Sixth Edition,” Pearson Education India, 2006.  See picture of presentation of copy of book to Professor Holonyak and read the foreword by him in the book.

The 80th birthday of my distinguished colleague and good friend, Nick Holonyak, Jr., was celebrated on the UIUC campus on October 24, 2008, with a symposium held in his honor during the day, followed by a dinner in the evening.  When I was standing at the registration table to pick up my badge, another attendee happened to walk in.  When he saw me, he said spontaneously, “Professor Rao, you are the guy that made Maxwell’s equations painless for me!”  Further he said that students considered two things, semiconductors and Maxwell’s equations, difficult and painful to learn.   He was a student in my undergraduate electromagnetics class in 1983 and was attending the symposium as the President of a company founded by him.

Incidents such as this either by direct personal encounters or telephone conversations or through e-mails from former students and users of my books have been numerous over the years.  I am grateful for the opportunities I have been provided with in my life to do “duty to others,” through my educational activities.  Throughout my life, I have been involved in education, as a student, professor, researcher, teacher, and administrator.  The sheer enjoyment of my work led me to coining the word “grattitude,” combining the words, “gratitude” and “attitude,” and meaning an “attitude of gratitude.”  Please read the section entitled, “Gratitude and Grattitude” in the front matter of my book, “Fundamentals of Electromagnetics for Electrical and Computer Engineering,” Pearson Prentice Hall, 2009, and its Indian version, “Fundamentals of Electromagnetics for Engineering,” Pearson Education, India, 2009, published in May 2008 and August 2008, respectively. See also article in The Integrator, Fall 2008, Newsletter of the University of Washington.

At one time, someone asked Professor Jordan the question, “What do your faculty do?”  He meant what areas of research and all that.  Professor Jordan responded: “Our faculty do what they like to do best.”  That was how Edward C. Jordan, whose name I carry in my title, built the distinguished Department of Electrical and Computer Engineering, as its head for a period of 25 years from 1954 to 1979.  He enjoyed best bringing to the department people from among the best in their respective fields.  I am very fortunate to have joined the department in 1965, during his tenure as head, and doing what I liked to do best for a period of 42 years, prior to retirement in 2007 (see article and letter from Board of Trustees), and continue to do it following retirement.  And what I liked to do best and continue to like to do best is writing textbooks on electromagnetics, for which I am grateful to Professor Jordan and an antenna, which existed at one time near Champaign, Illinois. See story with the antenna and the books.   So, when a former student walks up to me and says, “Professor Rao, you are the guy that made Maxwell’s equations painless for me,” it is natural for me to feel gratified.  Here, I am providing materials, that can be downloaded freely and used with my books (see below) or otherwise, for the purpose of teaching and learning electromagnetics. I shall consider it a worthwhile effort if I have made the teaching and learning of Maxwell’s equations less painful, if not painless.

Why Study Electromagnetics?


“The electromagnetic theory, as we know it, is surely one of the supreme accomplishments of the human intellect, reason enough to study it. But its usefulness in science and engineering makes it an indispensable tool in virtually any area of technology or physical research.” – George W. Swenson, Jr., Professor Emeritus of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, in the special Indian Edition of “Elements of Engineering Electromagnetics, Sixth Edition,” Pearson Education India, 2006.

The above quote and the one from Nick Holonyak, Jr. underscore the fact that electromagnetics is all around us.  In simple terms, every time we turn a switch on for electrical power or for an electronic equipment, every time we press a key on our computer key board or on our cell phone, or every time we perform a similar action involving an everyday electrical device, electromagnetics comes into play.  It is the foundation for the technologies of electrical and computer engineering, spanning the entire electromagnetic spectrum, from dc to light.  As such, in the context of engineering education, it is fundamental to the study of electrical and computer engineering.

Still, very often, the question is asked why one should study EM as a required course for all branches of electrical and computer engineering. In 1963, the American Institute of Electrical Engineers (AIEE) and the Institute of Radio Engineers (IRE) were merged into the Institute of Electrical and Electronics Engineers (IEEE). The IEEE is a global nonprofit organization with over 375,000 members. It is “the world's leading professional association for the advancement of technology.” The IEEE logo or badge is a merger of the badges of the two parent organizations. It contains a vertical arrow surrounded by a circular arrow, within a kite-shaped border. No letters clutter the badge because a badge without letters can be read in any language. The AIEE badge had the kite shape which was meant to symbolize the kite from Benjamin Franklin’s famous kite experiment to study electricity. The IRE badge had the two arrows that symbolize the right hand rule of electromagnetism.  Alternatively, the vertical arrow can be thought of as representing one of the two fields, electric or magnetic, and the circular arrow surrounding it representing the second field, produced by it, so that together they represent an electromagnetic field.

Whether this logo of IEEE was intended to be a recognition of the fact that electromagnetics is fundamental to all of electrical and computer engineering or not, it is a fact that all electrical phenomena are governed by the laws of electromagnetics, and hence, the study of electromagnetics is essential to all branches of electrical and computer engineering, and indirectly impacts many other branches.  Read more in Preface to the Indian Edition in “Fundamentals of Electromagnetics for Engineering,” Pearson Education India, 2009, and in the section on “Introduction: Why Study Electromagnetics?” in the Indian Edition of Elements of Engineering Electromagnetics, Sixth Edition, Pearson Education, 2006.

Spreading Maxwell's Message

One day in September 2009, I was sitting in the Jordan faculty lounge in the Everitt Laboratory along with a distinguished faculty colleague of mine, sipping coffee. Another distinguished faculty colleague, James Coleman, Intel Alumni Endowed Chair Professor in Electrical and Computer Engineering, walked in to collect his mail, and noticing me, he said that he was at Maxwell’s family home in Scotland in August for the IEEE’s unveiling ceremony of the plaque honoring Maxwell, and he thought about me not once, but several times during the ceremony. After a couple of days, I sent him a thank you note with a request for a couple of pictures, and asking him to tell me in a couple of sentences or so what made him think of me at the ceremonies. He very graciously obliged, and sent me the following reply, which I am posting with his consent:

“Why did I think of you at the ceremony? I have often thought that physicists do a better job than electrical engineers at honoring their professional predecessors. You are a most notable exception. I have always admired the warm words you have for such luminaries as Maxwell, Everitt, Jordan and others. When I think of luminaries of our department, especially in E&M, it is those men but also Y.T. Lo, Paul Mayes, etc. – wonderful men whom I was fortunate enough to have as teachers long ago. I consider you to be one of the luminaries. And even though I never was honored to take a class from you, I have learned much from your book, your words, and your example. So when I think of Maxwell, it is only logical that you should also come to mind. I’m delighted to hear – and not surprised – that you are still spreading Maxwell’s message to those young people in India.”

I felt greatly honored by Jim’s response. He also sent me a link to a web site maintained by the Glenair trust, established to conserve and preserve for the benefit of the public Glenair House, the family home of James Clerk Maxwell. I was able to get to the plaque, unveiled on August 13, 2009, through the link. See picture of the plaque. As I read the stunningly precise and most beautifully descriptive last sentence on the citation, “Maxwell’s equations today underpin all modern information and communication technologies,” I felt immensely grateful for the fortune of having been greatly influenced and guided by Maxwell’s equations in my professional career, and of being able to continue to spread the message of this Mahatma (Great Soul) of Electromagnetics.

On Maxwell and Maxwell’s Equations

Equations 1     James Clerk Maxwell     Equations 2

James Clerk Maxwell, born on 13th June 1831, at 14 India Street, Edinburgh, Scotland, was destined to become one of the 19th century's greatest scientific figures.  He died at the early age of 48, on 5th November 1879, but only after developing his unified theory of electricity, magnetism, and light through his crucial modification to one of the laws of magnetism, and predicting the phenomenon of electromagnetic wave propagation, thereby ushering in the field of electromagnetism, or electromagnetics, defined by the four equations, known as “Maxwell’s Equations.”  He put an end to speculation as to the nature of light with his discovery that it is a form of wave motion, by which electromagnetic waves travel through a medium at a speed, which is determined by the electric and magnetic properties of that medium. This is still the basis for explanations of all the phenomena of light and accompanying optical properties. He did not live until his theory was tested, but the follow-up developments by others have given the world the electromagnetic technologies the impact of which has become so common place in our daily lives in numerous ways.

In 2004, PHYSICS WORLD magazine conducted a poll asking its readers to send their short lists of the greatest equations ever and also asked them to explain why their nominations belonged on the list and why, if at all, the topic matters. They received about 120 responses -- including single candidates as well as lists -- proposing about 50 different equations. They ranged from obvious classics to "overlooked" candidates, personal favorites and equations invented by the respondents themselves. The result was published under CRITICAL POINT in the October 6, 2004 issue.

The Result and Explanation: Maxwell’s equations of electromagnetism and the Euler equation top a poll to find the greatest equations of all time.

The article said: “Although Maxwell’s equations are relatively simple, they daringly recognize our perception of nature, unifying electricity and magnetism and linking geometry, topology and physics.  They are essential to understanding the surrounding world.  And as the first field equations, they not only showed scientists a new way of approaching physics but also took them on the first step towards the unification of the fundamental forces of nature.”

In the words of Albert Einstein: "The work of James Clerk Maxwell changed the world forever."  More tributes by Einstein and other notables are reproduced below from the web site of the Glenair Trust:

"One scientific epoch ended and another began with James Clerk Maxwell." - Albert Einstein
"The special theory of relativity owes its origins to Maxwell's equations of the electromagnetic field." - Albert Einstein
"There can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade." - Richard Feynman
"He achieved greatness unequalled." - Max Planck
"One of the most penetrating intellects of all time." - R.A. Millikan
"The discovery of electrical waves has not merely scientific interest though that alone inspired it. ... it has had a profound influence on civilization..." - Sir J. J. Thomson
"Maxwell's importance in the history of scientific thought is comparable to Einstein's (whom he inspired) and to Newton's (whose influence he curtailed)." - Ivan Tolstoy
"...when he crossed the bridge from Astronomy to Physics he left behind him forever the prospect of becoming a great astronomer - but only to become the greatest mathematical physicist the world has seen since Newton." - Sir James Jeans
"Maxwell's equations have had a greater impact on human history than any ten presidents." - Carl Sagan

The MEF (Maxwell’s Equations First) approach of teaching FEM (Fundamentals of Electromagnetics)

Historically, the development of major technologies based on Maxwell’s equations occurred in the sequence of electrically and magnetically based technologies (electromechanics and electrical power) in the nineteenth century; electronics hardware and software in the twentieth century; and photonics technologies, entering into the twenty-first century. The teaching of electromagnetics evolved following this sequence, that is, beginning with a course on electrostatics, magnetostatics, energy and forces, and in some cases quasistatic fields, followed by Maxwell’s equations for time-varying fields and an introduction to electromagnetic waves.  This course was then followed by one or more courses on transmission lines, electromagnetic waves, waveguides and antennas.  In fact, when I joined the University of Illinois at Urbana-Champaign in 1965, a three- semester sequence of courses in electromagnetics (Electric and Magnetic Fields, Transmission Lines, and Electromagnetic Fields) for a total of nine credit-hours, were required in the Electrical Engineering (EE) curriculum of a total of 145 credit-hours.  The treatment of material in this manner in the first course was largely due to development and application of electrical machinery.  The course on transmission lines was to extend the low frequency (lumped) circuit theory, into the high frequency regime by beginning with the introduction of transmission line as a distributed circuit.  The third course was then based on full electromagnetic wave solution of Maxwell’s equations.

In 1973, the Computer Engineering curriculum (CompE) was introduced, and the total number of credit-hours was decreased to 124 credit-hours (now 128 credit-hours) for both curricula.  Only one course in electromagnetics, the first of the above three course sequence, was made required in both curricula, with the remaining courses moved to elective status.  Soon, this was found to be unsuitable, because the course was based on the traditional approach (also called the inductive approach) of beginning with the particular laws of static fields, followed by the complete set of Maxwell’s equations for time-varying fields.  Since much time was taken up for the coverage of static fields before getting to the time-varying fields, the time was cut short for the material centered on electromagnetic waves. This principal drawback of the traditional approach of teaching electromagnetics was unnecessarily aggravating for both curricula, because students were already exposed to the traditional treatment in a prerequisite physics course on electricity and magnetism.  It was further aggravating for the Computer Engineering curriculum, because generally that was the only EM course they would take, whereas the EE students would take one or more of the remaining courses as electives.

To resolve this problem, without having separate courses for the two curricula (EE and CompE), a new completely revised course was introduced beginning in 1974 based on an approach, which I call the MEF (Maxwell’s Equations First) approach (also called the deductive approach), beginning with the complete set of Maxwell’s equations for time-varying fields and developing their applications, as well as considering special cases of static and quasistatic fields.  With reference to the figure below, which depicts the frequency range of the electromagnetic spectrum extending from dc to light, the traditional approach, which I shall now call the MEL (Maxwell’s Equations Later) approach, takes up considerable amount of time and effort on laws applicable at one point (zero frequency) along the entire horizontal axis and then extends the treatment to the laws applicable for the entire horizontal axis, whereas the MEF approach introduces at the outset the laws pertaining to the entire horizontal axis and develops the concepts, and then treats the zero frequency case as a special case.  For a more detailed discussion, read Preface to the Indian Edition in “Fundamentals of Electromagnetics for Engineering,” Pearson Education, India, 2009.  For more than 30 years from 1974, the MEF approach served the needs of the Illinois curricula well by imparting to the students the elements of engineering electromagnetics that (a) constitute the foundation for preparing the EE majors to take follow-on courses, and (b) represent the essentials for the CompE majors taking this course only.

Deductive versus Inductive Approach graphic

The MEL approach served the purpose of teaching and learning EM well when the dominant technologies were at the low end of the frequency spectrum, because for low frequencies, an approximation, known as the “quasistatic approximation,” can be made. In the quasistatic approximation, the time-varying field in a given physical structure is approximated to have the same spatial variations as the static field in the structure, obtained by setting the source frequency equal to zero. This gives the same result as the actual dynamic field solution, approximated for low frequencies. Thus, although the actual situation in the structure is one of electromagnetic wave nature, it is approximated by a dynamic but not wavelike nature. As the frequency becomes higher and higher, this approximation violates the actual situation more and more, and it becomes increasingly necessary to consider the wave solution.

Thus, as the technologies based on Maxwell’s equations progressed from nineteenth century low-frequency applications to those of the present high-speed era, it has become increasingly necessary to deviate from the MEL approach, employed in my very first book in 1972 (see below) and base the teaching and learning of EM on the MEF approach, employed from my second book (1977) onwards. I have learned from personal experience that, while the MEL approach is enlightening from the cultural aspect of the historical development of electromagnetics, for engineering students the MEF approach of teaching and learning EM is more rewarding than the MEL approach, not only because of its importance for the sake of modern applications, but also because the latter is redundant in view of the prior exposure of engineering students to the historical treatment of EM in a physics course. Furthermore, I firmly believe that the cultural and intellectual aspect of the historical development of electromagnetics can be better understood and appreciated when one has grasped the broader impact of the complete set of Maxwell’s equations through the MEF approach.

Details on Books

(See picture showing all books)

Book employing the MEL approach

“Basic Electromagnetics with Applications,” by Nannapaneni Narayana Rao, U.S., and Indian Editions, Prentice-Hall, 1972



Books (most recent editions) employing the MEF approach

U.S., and International Editions:

"Fundamentals of Electromagnetics for Electrical and Computer Engineering," by Nannapaneni Narayana Rao, Pearson Prentice Hall, 2009 (Published in May 2008)
"Elements of Engineering Electromagnetics, Sixth Edition," by Nannapaneni Narayana Rao, Pearson Prentice Hall, 2004

Indian Editions:

"Fundamentals of Electromagnetics for Engineering," by Nannapaneni Narayana Rao, Low-Priced Indian Edition, Pearson Education, 2009 (Published in August 2008)

Complete Front Matter of Book
About the Author
Gratitude and “Grattitude”
Preface to the Indian Edition

"Elements of Engineering Electromagnetics, Sixth Edition," by Nannapaneni Narayana Rao, Low-Priced Indian Edition, Pearson Education, 2006

Complete Front Matter of Book
Message from A. P. J. Abdul Kalam
Foreword by Richard H. Herman
Foreword by Linda P. B. Katehi
Foreword by Nick Holonyak, Jr.
About the Author
A Tribute to Edward C. Jordan
About the Illinois ECE Series
Introduction: Why Study Electromagnetics?

Open-Access Materials


Note: The material presented here is incomplete and will be updated continuously for additions, changes, and corrections. Please communicate errors to  Thank you.

2009 India Programs on Fundamentals of Electromagnetics for training faculty: The material here is from two two-week intensive courses for training faculty in electrical-, electronics-, communication, and computer- related engineering departments in India, offered from June 3 to June 11, 2009, at Hyderabad, and from June 22 to July 3, 2009, at Mysore, attended by faculty from colleges from various parts of India. Go to 2009 India Programs.

2008 Amrita course on Fundamentals of Electromagnetics for training faculty: The material here is from a two-week, 8-day, intensive course for training faculty in electrical-, electronics-, communication, and computer- related engineering departments, offered at Amrita in Ettimadai, from August 11 to August 21, 2008, attended by faculty from the Amrita campuses at Ettimadai, Bangalore, and Amritapuri, and from other colleges.  Go to 2008 Amrita course.

2007 Tutorial Workshops at Amrita and at MIT, Chromepet:  The material here is from tutorial presentations made at a workshop on Electromagnetic Compatibility at Amrita in Ettimadai on January 8-12, 2007, and also at a workshop on “Fundamentals of Electromagnetics Revisited” at the Madras Institute of Technology in Chromepet, Madras, on January 26-27, 2007.  See picture at MIT, Chromepet.  Download the document (PDF, 1.1 MB), “Fundamentals of Electromagnetics for Electrical and Computer Engineering in 108 slides: A Tutorial.”

2006 Amrita course on Electromagnetics to students: The material here is from the inaugural offering of a course for students under the Indo-US Interuniversity Collaborative Initiative in Higher Education and Research, at Amrita Vishwa Vidyapeetham in Ettimadai, Coimbatore, Tamil Nadu, India, from July 10 to August 11, 2006, and broadcast over the Edusat Satellite to remote institutions.  See article in INGENUITY, a publication of the ECE Department, UIUC, following the course offering.  Go to 2006 Amrita Course.

2005 Video Lectures Recorded in Hyderabad: These are a set of five lectures on revisiting the fundamentals of electromagnetics, entitled, “The Science of Electromagnetics, in Essence,” captured on video camera in a studio in Hyderabad, India, in August 2005. Please note that these videos can be viewed only with Internet Explorer as the browser. Also, you may find a few editing errors still remaining, including an occasional slide in the video that does not match with the audio. Go to 2005 Hyderabad video lectures.

1983 UIUC Course Materials from EE 350, "Lines, Fields, and Waves:"  These are materials, consisting of class notes from an on-campus offering of EE 350 in Summer 1983, and PowerPoint presentations created with slide materials from Fall 1983 off-campus videotape offering of the same course.  Work is in progress for including the video recordings in this item and it is expected that all 43 recordings, each of 50-minutes duration, will be available for open-access by December 2009.  Go to Transmission Lines and Waveguides.

1974 Introduction of the MEF approach in UIUC Course EE 229, "Introduction to Electromgagnetic Fields:"  The material here is the class notes from EE 229 in Fall 1974, which constitutes the beginning of the introduction of the MEF (Maxwell's  Equations First) approach of teaching FEM (Fundamentals of Electromagnetics), necessitated by changes in the EE and CompE curricula, including a reduction in the number of required courses in EM from three to one, as explained above.  Download 1974 EE 229 Class Notes (219 pages; PDF 5.28 MB).

1972 Solution Manual for "Basic Electromagnetics with Applications." Despite the discussion above advocating the use of the deductive approach for the present-day engineering curricula, for those fascinated by electromagnetics the inductive approach provides the opportunity to tackle some challenging problems. "Basic Electromagnetics with Applications" contains numerous such problems. With the kind permission of my publisher, I am here making available the solution manual for the book, abridged from more than 900 handwritten pages of complete solutions. Download solution manual (117 pages; PDF 3.08 MB).