Studies on the electromagnetic interaction.
Here are collected the documents of an ongoing project (which ‘properly’ began in ~2007) to obtain a better understanding of electromagnetic theory. The impetus that drives this effort is a vision of the electrical engineer as one who, by the powers of his/her intellect, is able to harness, manipulate, and exploit the electromagnetic fields and forces, and hopefully for good.
This vision may serve as a reminder that electromagnetism, one of the 4 fundamental forces of Nature as we understand her, underlies a great deal of the work that we do as electrical engineers. It is also a childish re-envisioning of the electrical engineer as a Wizard of Nature or a Master of Force and Field, which I believe, is not an entirely fruitless notion to indulge upon.
What does it mean to have a good understanding of electromagnetic theory, or any theory for that matter? I think it simply means, to possess a clear sense of:
1) What questions should the theory answer?
2) How exactly does it answer those questions?
A good understanding of theory is a pre-requisite for enlightened practice of the art, which is the duty of an engineer. But to go beyond engineering and into the business of expanding the boundaries of human knowledge, which is the chief concern of the scientist, we find the need to append a third:
3) Of the questions a theory is supposed to answer, what are those to which it gives no satisfactory answer, or no answer at all, and why?
In pursuing this project one may follow the standard process of picking up a textbook (or two) and working through the pages and exercises, and/or enrolment in a suitable course at university. Indeed, this was how I began wth it. However, these measures did little to satisfy my ends, as questions often arose that could not be answered, or were not directly addressed in standard treatments of the subject. These initial frustrations of mine have shaped the general approach that has since been taken for this project, and hence they are worth the discussion that follows:
Most presentations of electromagnetic theory proceed in a fashion that begins with the following sequence, more or less: electrostatics, current electricity, magnetostatics, electromagnetic induction.
Electrostatics and magnetostatics start out reasonably, with Coulomb’s law of electrostatics, and the Biot-Savart law, which is regarded in some sense as the former’s counterpart in magnetostatics. An elementary treatment of these subjects will include these laws along with the Lorentz force law. Altogether, they directly address the issue of calculating the forces and fields that arise from stationary charges and steady currents, which is something one would expect to get out of an electromagnetic theory after exposure to classical mechanics and Newtownian gravitation.
The standard treatment up to the pre-university level ends at this point, and so far the concepts are comfortable, except for a slight unease I felt about magnetism, where the theory would not quite correspond to electrostatics. It seemed as if magnetism was an alternate but more tortuous path through the theory, that began and ended with the electric charge, with many meanderings amongst the electric currents; could we not write everything in terms of electric charges instead?
At a more advanced level (i.e. early undergraduate courses), Gauss’ law of electrostatics and its magnetostatics counterpart, Ampere’s circuital law, enter the picture, and things quickly take a turn for the worse. To the unsuspecting student (as I was), these were queer mathematical statements that appeared out of the blue. What sort of motivation could possibly lead someone to these things? If they could bring further insight into electromagnetism, this benefit was not obvious, and apart from facilitating the solution of certain physical problems (which is frequently touted by professors as their ‘main selling point’), they did not appear to be very useful.
Along the whole way from pre-university to undergraduate levels, other ‘accessories’ are introduced, such as the concepts of inductance, capacitance, and most notably, the potential. At the elementary levels, the potential comes across as a distraction, an exotic cousin of the potential energy, and just one of the many ways to fool around with the terms of Coulomb’s law (and inverse square laws in general). It may have been one and the same in the more familiar context of the electric circuit, where often it is referred to as the voltage, but in my own mind the connection was weak and not properly appreciated. As it turned out, this was the proverbial tip of the rest of the iceberg, which was discovered to be very large indeed upon more serious initial encounters with a university text on electromagnetism. It was only through this project that I finally began to appreciate this huge enterprise that is the theory of potentials.
Lastly, concerning electromagnetic induction, perhaps the most easily forgotten fact is that the resulting electric field is non-conservative. Early on, some people may find it slightly disturbing that Faraday’s law seems to imply that the strength of a magnetic field may well depend on the relative velocity between the observer and the electric charge. Unfortunately, under the standard treatment of things, one has to wait for too long to seek ‘resolution’ of this issue.
All the above may be considered the first part of a standard treatment on electromagnetism. What follows after this is so shocking that I believe it marks the beginning of the second part of the standard treatment: Maxwell’s equations and its corollaries. In this part, it is revealed to the student that the entire classical theory of electromagnetism is to be found contained within the fabled Maxwell’s equations (plus the Lorentz force law, which is not often mentioned in the same breath). It is as if everything that came before had been melted down and recast in an entirely different mould.
The shock is doubled when, peering through the smoke, one identifies Gauss’ law of electrostatics and Ampere’s circuital law (the one modified by Maxwell of course) as two of the four ‘Maxwell’s equations’; how could this happen, that the weird creatures have taken a place of priority over the Coulomb and Biot-Savart laws?! Some sense of familiarity is retained by the persistence of Faraday’s law in this ‘new order’, but its appearance in differential form is rather uncanny. Finally, there is the sometimes-called Gauss’ law of magnetism, which does not appear in the first part but has for some reason managed a seat at the table; of it little is spoken, except of the doom that looms over its verity should the so-far-elusive magnetic monopoles be found.
These were my first impressions with Maxwell’s equations, and they left me very puzzled. I wondered about the disappearance of Coulomb’s law and the Biot-Savart law, and how to get them back from Maxwell’s equations. I wondered why the Gauss’ and Ampere’s laws have greater significance in summarising the ideas of classical electromagnetism than the force laws. I was intrigued by the idea that Maxwell’s equations, in such a form, plus the Lorentz force law (let us not forget), make a complete theory of electromagnetism.
Thus the naive student begins a study of electromagnetism by collecting a set of phenomenological laws (i.e. Coulomb, Biot-Savart, Faraday, Lorentz) that enable the calculation of forces and fields, plus a few other ‘distractions’ thrown in, only to be suddenly thrust with a reformulation of electromagnetism that looks so different, and be told that from those everything else may be derived. What a shock it must be, to find the tables turned so suddenly!
The frustrations and confusions desribed above have played a great part in translating the general desire for ‘good understanding’ of electromagnetic theory, into more precise points of investigation for this project. These points serve as a general guide to possible areas of inquiry for the project, and they are as follows:
1. To understand the state of electromagnetic theory before the work of Maxwell, and to understand what was perceived to be lacking from it, particularly as Maxwell perceived it himself.
Most of the phenomenological laws, and in fact, much of the material covered in what we have called above the ‘first part’ of the standard treatment of electromagnetism, was discovered before Maxwell. It is important to understand the purpose they serve in electromagnetic theory, then and now. This is particularly true for the topic of potentials.
2. To understand Maxwell’s theory of electromagnetism from Maxwell’s point of view, as well as from our contemporary perspective.
Maxwell’s equations were a response to the theoretical deficiencies of his time, and through an understanding of his work, it may be possible to better understand what their purpose was and how they served it. These historical investigations, along with a study of the modern form of Maxwell’s equations and its corollaries (e.g. Jefimenko’s equations), should together accomplish the ultimate objective of understanding how Maxwell’s equations accomplish their purpose in modern electromagnetic theory.
3. To understand the developments that occurred after Maxwell, and finally to understand how the various parts of modern electromagnetism come together to make a complete and coherent theory as we know it today.
Major developments are undoubtedly, the experimental investigations that led to further elucidation of the nature of electricity and magnetism (e.g. discovery of the electron), and the theoretical work that led to the present day electromagnetic theory (e.g. the contributions of Oliver Heaviside).
A reminder that Maxwell’s equations were formulated before the nature of electricity and magnetism was known as precisely as in modern times may come as a surprise to some (I certainly was). In the light of this fact, it will be useful to take a closer look at the part played by Maxwell’s equations, in relation to the other parts of modern electromagnetic theory, particularly the Lorentz force equation and the theory of relativity.
4. From Maxwell’s equations and the Lorentz force law, to make the mathematical and conceptual connections back to the phenomenological laws that are first encountered in a course of study in electromagnetism.
To address the frustrations that started it all, and come full circle.
Off I go into the maze
for hidden keys inside that place,
that shall unlock the chains that bind
my thoughts encaged in weary mind.
And though the labyrinth’s corridors
have many times been mapped before,
still I must lay a string on floor,
with every step, through every door,
that I may never lose the way,
but leave to see the light of day,
and look upon a maze no more,
for t’is a pretty garden after all.
- Project Bibliography (2007) — A very old list of references compiled at the start of this project, during the initial period of frenzied reading. Document is being converted to BibTeX.
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