# Electromagnetism & Classical Electrodynamics

Syllabus:

Magnetism

Oscillations

Wave motion and Sound

Test questions

## What is electrodynamics, and how does it fit into the general scheme of physics?

In the diagram below  the four great realms of mechanics are sketched out:

1. Classical Mechanics ( Newton, Lagrange,
2. Special Relativity (Einstein)
3. Quantum Mechanics (Bohr, Heisenberg, Schrodinger, et al.)
4. Quantum Field Theory (Dirac, , Schwinger, et al.)

Newtonian mechanics was found to be inadequate in the early years of the 20th century- it’s all right in “everyday life,” but for objects moving at high speeds (near the speed of light) it is incorrect, and must be replaced by special relativity (introduced by Einstein in 1905); for objects that are extremely small (near the size of atoms) it fails for different reasons, and is superseded by quantum mechanics (developed by Bohr, Schrodinger, Heisenberg, and many others, in the twenties, mostly). For objects that are both very fast and very small (as is common in modem particle physics), a mechanics that combines relativity and quantum principles is in order: this relativistic quantum mechanics is known as quantum field theory–it was worked out in the thirties and forties, but even today it cannot claim to be a completely satisfactory system. We shall work exclusively in the domain of classical mechanics, although electrodynamics extends with unique simplicity to the other three realms. (In fact, the theory is in most respects automatically consistent with special relativity, for which it was, historically, the main stimulus.)

## Four Kinds of Forces

Mechanics tells us how a system will behave when subjected to a given force. There are
just four basic forces known (presently) to physics: (in the order of decreasing
strength):

1. Strong
2. Electromagnetic
3. Weak
4. Gravitational

The brevity of this list may surprise you. Where is friction? Where is the “normal” force that keeps you from falling through the floor? Where are the chemical forces that bind molecules together? Where is the force of impact between two colliding billiard balls? The answer is that all these forces are electromagnetic. Indeed, it is scarcely an exaggeration to say that we live in an electromagnetic world–for virtually every force we experience in everyday life, with the exception of gravity, is electromagnetic in origin.

The strong forces, which hold protons and neutrons together in the atomic nucleus, have extremely short range, so we do not “feel” them, in spite of the fact that they are a hundred times more powerful than electrical forces. The weak forces, which account for certain kinds of radioactive decay, are not only of short range; they are far weaker than
electromagnetic ones to begin with. As for gravity, it is so pitifully feeble (compared to all
of the others) that it is only by virtue of huge mass concentrations (like the earth and the sun) that we ever notice it at all. The electrical repulsion between two electrons is $10^{42}$ times as large as their gravitational attraction, and if atoms were held together by gravitational (instead of electrical) forces, a single hydrogen atom would be much larger than the known universe.

Not only are electromagnetic forces overwhelmingly the dominant ones in everyday
life, they are also, at present, the only ones that are completely understood. There is, of
course, a classical theory of gravity (Newton’s law of universal gravitation) and a relativistic one (Einstein’s general relativity), but no entirely satisfactory quantum mechanical theory of gravity has been constructed (though many people are working on it). At the present time there is a very successful (if cumbersome) theory for the weak interactions, and a strikingly attractive candidate (called chromodynamics) for the strong interactions. All these theories draw their inspiration from electrodynamics; none can claim conclusive experimental verification at this stage. So electrodynamics, a beautifully complete and successful theory, has become a kind of paradigm for physicists: an ideal model that other theories strive to emulate.

The laws of classical electrodynamics were discovered in bits and pieces by Franklin,
Coulomb, Ampere, Faraday, and others, but the person who completed the job, and packaged it all in the compact and consistent form it has today, was James Clerk Maxwell. The theory is now a little over a hundred years old.

## The Unification of Physical Theories

In the beginning, electricity and magnetism were entirely separate subjects. The one dealt with glass rods and cat’s fur, pith balls, batteries, currents, electrolysis, and lightning; the other with bar magnets, iron filings, compass needles, and the North Pole. But in 1820 Oersted noticed that an electric current could deflect a magnetic compass needle. Soon afterward, Ampere correctly postulated that all magnetic phenomena are due to electric charges in motion. Then, in 1831, Faraday discovered that a moving magnet generates an electric current. By the time Maxwell and Lorentz put the finishing touches on the theory, electricity and magnetism were inextricably intertwined. They could no longer be regarded as separate subjects, but rather as two aspects of a single subject: electromagnetism.

Faraday had speculated that light, too, is electrical in nature. Maxwell’s theory provided
spectacular justification for this hypothesis, and soon optics–the study of lenses, mirrors, prisms, interference, and diffraction–was incorporated into electromagnetism. Hertz, who presented the decisive experimental confirmation for Maxwell’s theory in 1888, put it this way: “The connection between light and electricity is now established… In every flame, in every luminous particle, we see an electrical process… Thus, the domain of electricity extends over the whole of nature. It even affects ourselves intimately: we perceive that we possess … an electrical organ–the eye.” By 1900, then, three great branches of physics, electricity, magnetism, and optics, had merged into a single unified theory. (And it was soon apparent that visible light represents only a tiny “window” in the vast spectrum of electromagnetic radiation, from radio though microwaves, infrared and ultraviolet, to x- rays and gamma rays.)

Einstein dreamed of a further unification, which would combine gravity and electrody-
namics, in much the same way as electricity and magnetism had been combined a century earlier. His unified field theory was not particularly successful, but in recent years the same impulse has spawned a hierarchy of increasingly ambitious (and speculative) unification schemes, beginning in the 1960s with the electroweak theory of Glashow, Weinberg, and Salam (which joins the weak and electromagnetic forces), and culminating in the 1980s with the superstring theory (which, according to its proponents, incorporates all four forces in a single “theory of everything”). At each step in this hierarchy the mathematical difficulties mount, and the gap between inspired conjecture and experimental test widens; nevertheless, it is clear that the unification of forces initiated by electrodynamics has become a major theme in the progress of physics.

## The Field Formulation of Electrodynamics

The fundamental problem a theory of electromagnetism hopes to solve is this: I hold up
a bunch of electric charges here (and maybe shake them around)–what happens to some
other charge, over there? The classical solution takes the form of a field theory: We say
that the space around an electric charge is permeated by electric and magnetic fields (the electromagnetic “odor,” as it were, of the charge). A second charge, in the presence of these fields, experiences a force; the fields, then, transmit the influence from one charge to the other–they mediate the interaction.

When a charge undergoes acceleration, a portion of the field “detaches” itself, in a
sense, and travels off at the speed of light, carrying with it energy, momentum, and angular momentum. We call this electromagnetic radiation. Its existence invites (if not compels) us to regard the fields as independent dynamical entities in their own right, every bit as “real” as atoms or baseballs. Our interest accordingly shifts from the study of forces between charges to the theory of the fields themselves. But it takes a charge to produce an electromagnetic field, and it takes another charge to detect one, so we had best begin by reviewing the essential properties of electric charge.

## Electric Charge

1. Charge comes in two varieties, which we call “plus” and “minus,” because their effects
tend to cancel (if you have +q and -q at the same point, electrically it is the same as having no charge there at all). This may seem too obvious to warrant comment, but I encourage you to contemplate other possibilities: what if there were 8 or 10 different species of charge? (In chromodynamics there are, in fact, three quantities analogous to electric charge, each of which may be positive or negative.) Or what if the two kinds did not tend to cancel? The extraordinary fact is that plus and minus charges occur in exactly equal amounts, to fantastic precision, in bulk matter, so that their effects are almost completely neutralized. Were it not for this, we would be subjected to enormous forces: a potato would explode violently if the cancellation were imperfect by as little as one part in $10^{10}$.

2. Charge is conserved: it cannot be created or destroyed–what there is now has always
been. (A plus charge can “annihilate” an equal minus charge, but a plus charge cannot simply disappear by itself–something must account for that electric charge.) So the total charge of the universe is fixed for all time. This is called global conservation of charge. Actually, I can say something much stronger: Global conservation would allow for a charge to disappear in New York and instantly reappear in San Francisco (that wouldn’t affect the total), and yet we know this doesn’t happen. If the charge was in New York and it went to San Francisco, then it must have passed along some continuous path from one to the other. This is called local conservation of charge. The precise mathematical law expressing local conservation of charge–it’s called the continuity equation.

3. Charge is quantized. Although nothing in classical electrodynamics requires that it be
so, the fact is that electric charge comes only in discrete lumps–integer multiples of the
basic unit of charge. If we call the charge on the proton +e, then the electron carries charge -e, the neutron charge zero, the pi mesons +e, 0, and -e, the carbon nucleus +6e, and so on (never 7.392e, or even 1/2e)*. This fundamental unit of charge is extremely small, so for practical purposes it is usually appropriate to ignore quantization altogether. Water, too, “really” consists of discrete lumps (molecules); yet, if we are dealing with reasonably large quantities of it we can treat it as a continuous fluid. This is in fact much closer to Maxwell’s own view; he knew nothing of electrons and protons–he must have pictured charge as a kind of “jelly” that could be divided up into portions of any size and smeared out at will.

* Actually, protons and neutrons are composed of three quarks, which carry fractional charges (±2/3 e and ±1/3 e). However, free quarks do not appear to exist in nature, and in any event this does not alter the fact that charge is quantized; it merely reduces the size of the basic unit.