Classical Electromagnetism

Classical Electromagnetism

Richard Fitzpatrick
Published in: The University of Texas at Austin
Release Year: 1997
ISBN: 978-1530325917
Pages: 227
Edition: First Edition
File Size: 1 MB
File Type: pdf
Language: English

Description of Classical Electromagnetism

The main topic of this course is Maxwell’s equations. These are a set of eight first-order partial differential equations which constitute a complete description of electric and magnetic phenomena. To be more exact, Maxwell’s equations constitute a complete description of the behavior of electric and magnetic fields. You are all, no doubt, quite familiar with the concepts of electric and magnetic fields, but I wonder how many of you can answer the following question. “Do electric and magnetic fields have a real physical existence or are they just theoretical constructs which we use to calculate the electric and magnetic forces exerted by charged particles on one another?” In trying to formulate an answer to this question we shall, hopefully, come to a better understanding of the nature of electric and magnetic fields and the reasons why it is necessary to use these concepts in order to fully describe electric and magnetic phenomena.
At any given point in space, an electric or magnetic field possesses two properties, a magnitude, and a direction. In general, these properties vary from point to point. It is conventional to represent such a field in terms of its components measured with respect to some conveniently chosen set of Cartesian axes (i.e., x, y, and z axes). Of course, the orientation of these axes is arbitrary. In other words, different observers may well choose different coordinate axes to describe the same field. Consequently, electric and magnetic fields may have different components according to different observers. We can see that any description of electric and magnetic fields is going to depend on two different things. 
Firstly, the nature of the fields themselves and, secondly, our arbitrary choice of the coordinate axes with respect to which we measure these fields. Likewise, Maxwell’s equations, the equations which describe the behavior of electric and magnetic fields, depending on two different things. Firstly, the fundamental laws of physics which govern the behavior of electric and magnetic fields and, secondly, our arbitrary choice of coordinate axes. It would be nice if we could easily distinguish those elements of Maxwell’s equations which depend on physics from those which only depend on coordinates. In fact, we can achieve this using what mathematicians call vector field theory. This enables us to write Maxwell’s equations in a manner that is completely independent of our choice of coordinate axes. As an added bonus, Maxwell’s equations look a lot simpler when written in a coordinate-free manner. 
In fact, instead of eight first-order partial differential equations, we only require four such equations using vector field theory. It should be clear, by now, that we are going to be using a lot of vector field theory in this course. In order to help you with this, I have decided to devote the first few lectures of this course to a
review of the basic results of vector field theory. I know that most of you have already taken a course on this topic. However, that course was taught by somebody from the mathematics department. Mathematicians have their own agenda when it comes to discussing vectors. They like to think of vector operations as a sort of algebra which takes place in an abstract “vector space.” This is all very well, but it is not always particularly useful. So, when I come to review this topic I shall emphasize those aspects of vectors which make them of particular interest to physicists; namely, the fact that we can use them to write the laws of physics in a coordinate-free fashion.
Traditionally, an upper-division college-level course on electromagnetic theory is organized as follows. First, there is a lengthy discussion of electrostatics (i.e., electric fields generated by stationary charge distributions) and all of its applications. Next, there is a discussion of magnetostatics (i.e., magnetic fields generated by steady current distributions) and all of its applications. At this point, there is usually some mention of the interaction of steady electric and magnetic fields with the matter. Next, there is an investigation of induction (i.e., electric and magnetic fields generated by time-varying magnetic and electric fields, respectively) and its many applications. Only at this rather late stage in the course is it possible to write down the full set of Maxwell’s equations. The course ends with a discussion of electromagnetic waves.
The organization of my course is somewhat different from that described above. There are two reasons for this. Firstly, I do not think that the traditional course emphasizes Maxwell’s equations sufficiently. After all, they are only written down in their full glory for more than three-quarters of the way through the course. I find this a problem because, as I have already mentioned, I think that Maxwell’s equations should be the principal topic of an upper-division course on electromagnetic theory. Secondly, in the traditional course it is very easy for the lecturer to fall into the trap of dwelling too long on the relatively uninteresting subject matter at the beginning of the course (i.e., electrostatics and magnetostatics) at the expense of the really interesting material towards the end of the course (i.e., induction, Maxwell’s equations, and electromagnetic waves). I vividly remember that this is exactly what happened when I took this course as an undergraduate. I was very disappointed! I had been looking forward to hearing all about Maxwell’s equations and electromagnetic waves, and we were only able to cover these topics in a hurried and rather cursory fashion because the lecturer ran out of time at the end of the course.
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