Classical Electromagnetism

Chapter 1: Introduction

By Mark Lawrence

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This book is highly enhanced course notes from the standard 1st year graduate course in Classical Electromagnetism. This is not at all an introductory course - it's presumed that before you take this course, you had a semester of E&M at the sophomore level, and another semester at the junior or senior level. It's also presumed you're familiar with trigonometry, full and partial differential equations and vector mathematics. Properly speaking, this isn't really a course in E&M, or even physics for that matter: it's an applied math course in how to expand functions over various interesting complete sets. The complete sets we'll be using are all solutions to the Laplace equation.

The introduction is completely non-mathematical, and suitable for any interested person over about the age of 14. The second chapter should be suitable for anyone who has completed sophomore or junior physics. The rest of the book is basically non-stop advanced mathematics.


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Why study Electromagnetism?

There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the business of experimental Philosophy to find them out. - Isaac Newton

We recognize four fundamental forces in physics today - the color force, which binds quarks to each other; electromagnetism, which is responsible for essentially all of the phenomenon in our day to day lives; the weak force, which regulates radioactivity; and gravity. We have a theory for each of these four forces. Of these four theories, by far the most powerful and accurate is our theory of electromagnetism.

We recognize only a few stable fundamental particles in physics today: two different quarks, which we arbitrarily label "up" and "down," the electron, and the neutrino. In addition, we say that particles have fields. Each particle has mass, so it also includes a gravitational field. We may consider the mass as a gravitational charge. The quarks and the electron are electrically charged, so they include an electromagnetic field. Quarks have another kind of non-electromagnetic charge which we whimsically label "color," and so also include a "color" field. This "color" has nothing at all to do with the colors you see with your eyes. All four particles apparently also have a "weak" field. These various fields can hold energy and momentum in packets which are much like particles. These field particles are called the graviton, the photon, the weak boson, and the gluon. So, it seems we live in a universe where most everything is made up of only eight different particles. We know of four more types of quarks, two more types of electron and two more types of neutrino, but these particles seem to exist only for incredibly short portions of a second, and only in our highest energy particle accelerators.

Three of these eight basic particles, the two quarks and the electron, are most frequently found bound into atoms. The gluons hold the quarks together in triples that make up protons and neutrons, and also bind the protons and neutrons together into an atomic nucleus. Photons bind the electrons to the nucleus, and also bind atoms to each other so that they may form molecules. Gravitons hold large collections of atoms together, like planets and stars. The neutrino and weak boson are essential for the process in the center of the sun that combines hydrogen into helium and releases energy. In the 300 or so years since Newton gave us his directive, we have made remarkable and surprising progress.

For technical reasons, the color force is almost completely contained inside the atomic nucleus. The weak force is an exceedingly short ranged force, and its effects almost never leave the nucleus. Thus our world of rocks and clouds and oceans is nearly completely governed by the laws of electromagnetism and gravity. Surprisingly, although gravity seemed originally to be a very simple force, we have learned since Einstein's General Relativity that it is a very subtly complicated force. So subtle that a complete quantum field theory of gravity continues to elude us to this day.

Of the four fundamental forces, it turns out that electromagnetism is the simplest to understand, and therefore by far the best understood. In fact, electromagnetic theory is so well understood and so successful that our theories of the other three forces are based on electromagnetism. This is one reason why we study electromagnetism: our theories of other forces and effects will all be similar, although somewhat more complicated.

Our subjective world is filled with physical phenomenon: the sky is bright and warm during the day, due to the light and heat from the Sun. At night we see thousands of other stars. All the matter in our world is made of atoms, which frequently bind together into chemical compounds. Atoms seem to completely fill their space: two atoms cannot be in the same place at the same time, giving the sensation that matter is hard. Our physical sensations are dominated by sight, which means we can see light; hearing, which means we can hear vibrations of atoms transmitted through the air; taste and smell, which directly sense the presence of particular molecules; and touch, which tells us if something is hot or cold, hard or soft, wet or dry.

The light we see is simply photons. Atoms bounce off each other and can transmit sound waves due to their electromagnetic interactions. Molecules hold together and activate taste and smell sensors due to the details of the complex electromagnetic interactions of their electrons with the electrons in our tongue and nose. Heat is increased motion of the atoms. Hard is a group of atoms that are tightly bound to each other by electromagnetism, soft is a group that is less tightly bound. Wet is a group of atoms that are electromagnetically bound into a collection, but not directly to their nearest neighbors, allowing them to easily slide around each other. Except for gravity, our entire subjective physical world is made up of various electromagnetic effects. Light, the structure and hardness of atoms, chemical bonds, and heat and cold are all now understood as governed by the laws of electromagnetism. So here is another reason: electromagnetic effects are central to our existence.

A brief history of Electromagnetism

Basic observations of electricity and magnetism have been around for more than 2000 years. The ancient Greeks wrote about lightning, magnetic rocks, and static electricity. The mathematical theory of electromagnetism was developed over a 90 year period from about 1785 to about 1875. In 1785, Coulomb established that the electric force, like gravity, was a 1/r2 force. This meant that all of Newton's mechanics could be brought to bear on the problems of electricity immediately. In 1820, Ampere found that an electric current generated a magnetic field, giving the first hint that these two forces were deeply related. In the 1820s, Gauss, perhaps the most gifted mathematician ever, published Gauss' law, the law of divergence. In 1831, Faraday, an unschooled inventor, showed that just as Ampere's electric currents could produce a magnetic field, so a moving magnetic field could produce an electric current. This lead directly to generators and alternators and electric motors. Faraday also introduced the idea of lines of force. In 1873, Maxwell brought all the work of Coulomb, Ampere, Gauss, and Faraday together into one set of equations, Maxwell's equations. Maxwell was able to use his equations to calculate the speed of light, which strongly suggested that light was also an electromagnetic phenomenon.  In 1875 Lorentz added his force law to Maxwell's work, completing the theory. In 1886, Heinrich Hertz demonstrated that a purely electrical apparatus could create light waves, which pretty much nailed down the theory of electromagnetism.

Charles Augustin Coulomb, 1736 to 1806

André-Marie Ampère, 1775 to 1836

Carl Friedrich Gauss, 1777 to 1855

Michael Faraday, 1791 to 1867

James Clerk Maxwell, 1831 to 1879 

Hendrik Antoon Lorentz, 1853 to 1928

Electrostatics in a nutshell

Let's consider an electron and its electric field. We'll draw the electric field as a set of field lines. The field lines will go in the direction of the field. The strength of the field will be indicated by how close the field lines are to each other. Here's an electron just sitting around:

Figure 1.1: An electron and its field

We'll be looking at two dimensional drawings of this electron and its field, but we must keep in mind that the electron's field is a three dimensional object.

Above, we see 16 field lines. As we get further from the electron, the field lines spread out from each other, but the number of field lines does not change. If you are a distance r from the electron, the strength of the field goes down as 1/r2. However, if you think of a sphere of radius r centered on the electron, the area of this sphere is 4/3 π r2. As the sphere gets larger the electron's field through any small area on the sphere gets weaker as 1/r2, but the total amount of area on the sphere gets larger as r2. So the total amount of electric field on the sphere is a constant, the total number of field lines leaving the sphere is constant.

If we imagine moving the sphere so that the electron is no longer inside, we see something quite different: every field line that enters the sphere exits somewhere else.

Figure 1.2: The number of field lines
does not depend on the circle's radius

Figure 1.3: If the circle doesn't contain the
electron, each field line both enters and exits.

This is the most basic law of electrostatics: if you draw a sphere and count the net number of field lines leaving the sphere, you know how many electrons are in the sphere. The net number of field lines leaving the sphere is, of course, the number leaving the sphere minus the number entering the sphere. Mathematically we count net field lines with an operator called the divergence. In words, we can now state two of the fundamental laws of electromagnetism. First, the divergence of the electric field on a closed surface tells us the number of electrons inside the closed surface. We have just seen that we can know how many electrons are in a basketball by counting the electric field lines leaving the basketball. So we have a choice, we can count electrons or we can count field lines. Second, the divergence of the magnetic field on a closed surface is always zero. This is because although you can isolate an electron, you cannot isolate the north pole of a magnet - there will always be a south pole next to it, no matter how small the magnet. If you break a magnet in half you don't get one north pole and one south pole, you get two magnets. No one understands why this is true. We simply (tentatively) accept that God did not make magnetic monopoles. However, just as Ponce de Leon spent his life searching for the Fountain of Youth, there are many scientists who have spent their careers searching for a Magnetic Monopole and the certain Nobel Prize that would accompany it.

A key insight from this law is that all electric field lines come from charged particles. There are no electric field lines that simply appear out of nowhere, and there are no electric field lines that simply disappear into empty space. If there were, our counting of field lines could get screwed up. Perhaps more subtly, there are no electric field lines that loop upon themselves. You can imagine a circular electric field line. This circular field line would have no beginning or end, and would not require a charge as an anchor. Although I haven't justified this yet, there are no such electric field lines. All electric field lines are attached at one end to a charged particle. The other end of the field line is either attached to an oppositely charged particle, or it goes on to infinity.

Since field lines must always start on a charge, and field lines cannot be broken, charges cannot teleport. Of course we would all like to have a Star Trek transporter, but our best understanding right now is that no such device is possible. Imagine you had two spheres a mile apart, and an electron teleported from one sphere to the other. The field lines in one sphere would suddenly end, and field lines would appear seemingly from nowhere coming out of the second sphere. This is a violation of our law. Now, we can shrink the two spheres to the size of an atom, or perhaps smaller, and put them very close to each other. If the electron teleports even such a very small distance, still there will be breaks in the field lines as they disappear from one sphere and appear from the other. Our law tells us that if you want to get an electron from here to there, it has to take some path from here to there, it can't just "quantum leap." Sorry.

Another key insight we learn from this is that 1 + 1 = 2. If there are two electrons inside the basketball, there will be twice as many field lines leaving the basketball. If there are 150 electrons inside, there will be 150 times as many field lines leaving. Now, this may seem so simple and obvious as to be bordering on stupid, but it didn't necessarily have to be that way. We can imagine that if you bring two electrons close to each other, perhaps the electrons could effect each other. Perhaps when very close, electrons might enhance or suppress the fields of each other. But, they don't. We have two important ways of saying this. One is to say that the electric field is linear. If A and B are two electrons, then the electric field from ( A plus B ) is identical to the electric field from ( A ) plus the electric field from ( B ). The other way we say this is that the electric field obeys the law of superposition, which is a big word meaning that you just simply add all the individual fields together to find the resulting total field. Did it have to be this way? As Fred Brooks famously said, "No matter how many women you assign to the task, it still takes nine months to make a baby." Things do not always simply add, but electric fields do.

This leads us to an important result in physics, something called a null result. In hydrogen, the electrons have a charge of minus one. The protons have a charge of plus one. So, the hydrogen atoms are net uncharged. Why is this? A proton is a very different beast from an electron. Electrons are fundamental particles as best as we can tell, but protons are made up of three quarks, two with charges of plus two-thirds, and one with a charge of minus one-third. Why do three quarks add up to exactly the opposite charge of one electron? Do they really? There is a very simple experiment we can do to test this. You get a large metal container, for example a welding tank. You fill it with a lot of hydrogen gas under very high pressure. Then you carefully ground the metal tank so that you are certain the net charge on and in the tank is zero. Now, you electrically isolate the tank from its surroundings and slowly bleed off the hydrogen gas. If the protons had, for example, a billionth more charge than the electrons did, then each time a hydrogen atom left the tank, the tank would lose a billionth of an electron charge. You can easily fill a tank with a ton of hydrogen - this is about 1030 atoms. If the charge on each atom was unbalanced by as much as one part in 1030, you would see the tank pick up an electric charge as it emptied. You can repeat the experiment with nitrogen, which has seven protons and seven electrons. No matter what gas we use, no matter how much, we never see the tank pick up a charge. So, while we're a bit mystified about how the quarks and electrons know to balance each other so perfectly, we're pretty confident that they do.

That's about it for electrostatics. We've learned that charge comes in discrete packets, which we call electrons or quarks. The electrons and quarks balance each other out to a perfection better than we can currently measure. Each electrically charged particle has an electric field. The electric fields from lots of particles simply add up. As you get further away from a charge, the electric field in a small area drops off precisely as the distance squared from the charge. All electric field lines end on a charged particle. There are no electric field lines that simply appear and/or disappear, and no electric field lines that loop upon themselves with no beginning or end. As simple as it is to state and understand these results, it will take several weeks of advanced mathematics to appreciate them in what detail is currently understood.

Electrodynamics in a nutshell

Einstein wrote down the laws of physics as seen from a moving platform. These laws are called Special Relativity. The most important point of Special Relativity is that you can have your physics lab floating in space, or sitting on the earth, or moving on a train, or flying in a spaceship at some horrendous speed. In any of these cases, the laws of physics will be the same. In any of these labs, you can start up your iPod, and it will work and play music for you. Remember, your iPod contains several million transistors, a battery, knobs and switches, a couple motors and some mechanical stuff, and finally the ones and zeros that represent the music. Frankly, I consider it a near miracle that all this stuff works at all. If physics changed just a little bit, it's hard to imagine the music would still play.

However, people in different labs moving at different speeds will see very different things when they look into other labs. We're all familiar with doppler shifts - a car horn seems to sound a higher note as it speeds towards us, then a lower note as it passes and speeds away. Similarly, when the car is speeding towards us, the color of the light from the headlights is shifted a tiny bit towards blue, and when the car is speeding away we see the headlight's color as shifted a tiny bit towards red. For automobiles this effect is so tiny that we aren't actually aware of it with our naked eyes, but for things moving very fast this effect can be quite pronounced.

Also, at very high speeds things appear to get shorter. If a man with a 10 foot long pole runs at you at 160,000 miles per second, about 87% of the speed of light, his pole appears to you to be only 5 feet long. This is not something that comes up in the Olympics, but it is something that we have to worry about with the laws of physics. We have to worry about this effect in electromagnetism.

Imagine you have a friend who is shooting pool. Meanwhile, you're flying by above him at 160,000 miles per second. Let's see what this looks like. First, as you're approaching, you notice the cue ball is blue, not white. The stripped balls are all blue with strange stripe colors, too. The table is a strange shape - instead of 4.5' x 9', it's a square, 4.5' x 4.5'. Also, the pool balls are not round, they look compressed in the direction of your flight to only half thickness. So, you and your friend don't agree on the shape of the table, the color of the felt, the shape of the balls, the color of the balls. . . but you do agree on the number of pool balls that are on the table. The things you agree upon are called Lorentz scalars. These are numbers that are the same no matter how fast you are moving.

Electric charge is a Lorentz scalar. No matter who is counting, they will always agree on how many electrons are present. However, the electromagnetic field around the electrons is not a Lorentz scalar - as you move faster, the shape of the field changes. The electromagnetic field appears to compress, exactly as the pool balls did. Relativity assures us that only the relative speed matters, not who is moving. So Figure 1.5 is also a picture of an electron moving towards us at 87% the speed of light.

Figure 1.4: An electron at rest and its field

Figure 1.5: The same electron as seen by
someone going 87% the speed of light

Now, let's imagine that there's an electron sitting near us at rest, and that it has been at rest for a long time. Suddenly it accelerates to 87% the speed of light. We know what the electric field looks like for an electron at rest, and we know what the field looks like for an electron going 87% the speed of light. We also know that field lines are continuous - they never break, they never start or stop. So, we can draw the field lines for an electron at rest, then in the center of that field we can put the field lines for an electron zipping along. All we need to do now is connect up the lines. This is shown below in Figure 1.6. As you see, the field lines get very pronounced kinks in them from the acceleration. The field lines are straight so long as the electron is moving at a constant velocity, but any acceleration of the electron will cause kinks in the field lines like these. These kinks travel out along the field lines at the speed of light. Suppose you are holding one end of a rope, the other end of the rope is tied to a tree. You flick your wrist quickly, as if to crack a whip. There will be a disturbance in the rope, a wave which will travel along the rope until it hits the tree. These kinks are just like that, except there's no tree. The kinks in the electric field lines have no rest mass, so they must always move at the speed of light. Imagine there's an electron far away from us at rest, then it accelerates to a high speed. This acceleration will cause kinks in the field lines. If we're far away from all this, we'll see the field of an electron at rest, then suddenly some of these kinks will hit us, sweep by us at the speed of light, then the field will settle down into the correct field for an electron moving at a constant velocity.

Figure 1.6: An electron at rest suddenly
accelerates to 87% the speed of light

All the fifty years of conscious brooding have brought me no closer to the answer to the question, 'What are light quanta?' Of course today every rascal thinks he knows the answer, but he is deluding himself. - Albert Einstein

These kinks have a name - we call them "photons." Normally, the field lines point straight at the electron, but we can see that inside one of the kinks the field is not pointing at the electron. The field at this point is a vector combination of a line pointing at the electron and a line pointing sideways to the electron. To be more precise, we call the component of the kink that's sideways the "photon." We even have a special rule: the electric field of a photon must be precisely sideways to its direction of travel. These photons are constrained to move along the field lines - there's nowhere else they can go. They are not allowed to disconnect or break away from the field lines, as this would leave a gap in the field line, and that's against the laws. Remember, field lines are only allowed to start or stop on charges, or at infinity. If you're a physicist, infinity is more than just the great beyond, it's also the big rug, a big rug that you can sweep pretty much anything under.

If we look at all the same pictures, but instead of an electron with electric field lines think of it as a mass with gravitational field lines, then we have an equally valid picture of a graviton. Although our casual language uses the word "charge" for electricity, we're using it in a more general fashion. A charge is something that produces a field. There are many types of field, and many types of charge. We can think of mass or energy as the gravitational charge. Quarks have gravitational fields due to their mass; they have electric fields due to their electric charge; they have weak fields due to their weak charge; and they have color fields due to their color charge. Quarks live in very crowded neighborhoods.

Are photons real? This is a question which is difficult to answer - our sense of reality does not extend very well into these realms where things are moving at very high speeds. Imagine there is an electron in the Sun's atmosphere, and due to heat it's wriggling and changing velocity a lot. This electron is going to have field lines with a lot of kinks in them. The photons coming off this wriggling electron will most likely be of the correct frequency to be seen as light by your eyes or felt as heat by your hands. Something you can see with your eyes or feel with your hands seems awfully real. However, now lets imagine we have two marble pedestals with an electron sitting on each. One of these pedestals is sitting on the ground, and the other is sitting in a spaceship which is also sitting on the ground. Next, we'll quickly accelerate the spaceship to 87% the speed of light. If you are an observer standing on the ground, you see the electron next to you as sitting still - it never changes speed, so there are no kinks, no photons. The electron in the spaceship, however, is changing speeds, so the spaceship electron's field has kinks. If the spaceship accelerates quickly enough, those kinks could be absorbed by your hands or eyes and interpreted as light or heat.

Meanwhile, an observer standing in the spaceship sees the electron next to him as always going at the same speed he is. He sees no kinks in his electron's field. However, the electron that's standing on the pedestal on the ground looks to the spaceship pilot like it's accelerating, and therefore there are kinks in the field of that electron. If the spaceship accelerates quickly enough, the kinks in the field lines from the electron on the ground may hit the pilot's eyes or hands and be absorbed and interpreted as heat or light. So, two different observers that are accelerating relative to each other disagree on whether any particular electron is producing photons. This disagreement may be verified by eyes, hands, or photographic film. Who is right? Both observers may have photographs to verify their claim that it was the other electron that was radiating. Which picture is "real," and which is "fake?" The source of this confusion is that while the number of electrons is a Lorentz scalar, the number of photons is not.

Does it really work this way? Classical Electromagnetism theory tells us that all accelerating charges radiate photons. Let's test this. Consider again the electron sitting on the marble pedestal. It's on the surface of the Earth, which means it's accelerating constantly at 1g. Why isn't this electron radiating? Actually, if you had very sensitive meters and were in free fall near the Earth, you would measure that it was. But in the Earth laboratory, the observer and the electron are accelerating at precisely the same rate, so the Earth bound observer sees no radiation.

Now let's imagine a wire carrying a current. Before the current starts, the electrons in the wire are all near their respective atoms, and all the charges cancel out. For each electron there's a proton with the opposite charge, so there are no net field lines coming out of the wire. But, now we'll start up a current, perhaps by hooking up a battery and a light bulb. Once we start the current flowing, the atoms stay in their places, but the electrons start moving away from the battery and towards the light bulb. When the electrons start moving, there will be a Lorentz contraction - when something is moving, it shrinks a bit in the direction it's moving. Normally we are completely unaware of this tiny effect - at 60 miles per hour, your car shrinks by about a 10 millionth of an inch. However, there are an enormous number of charges in this wire, something like 1022 protons per inch and 1022 electrons per inch. Normally, these protons and electrons cancel out each other's electric fields just about perfectly, and we feel no electric field from the wire. However, now the electrons are moving, and that means a clump of electrons in the wire is a little shorter than it used to be. The number of electrons doesn't change, that's a Lorentz scalar, but the number of electrons per inch increases by about one part in a billion. Since there is such a huge number of conduction electrons, that means suddenly there are seemingly trillions of unpaired electrons in this wire, and the wire emits an electric field.

Figure 1.7: A wire carrying a current
1,000,000,000,000,000,000,000 protons per inch
1,000,000,000,100,000,000,000 electrons per inch

This is a particular kind of electric field - the strength and direction of the field depends on the velocity of the electrons in the wire, and your velocity as you perhaps move past the wire. We have a special name for this peculiar electric field - we call it a magnetic field. This is what magnetism is, it's a relativistic effect of the electric field. When charges move, their field lines get compressed and the charge density changes, and the resulting squished electric field is now commonly called a normal electric field plus a magnetic field. The conduction electrons in the wire are moving quite slowly, about one to five miles per hour. You could walk and keep up with them. If you do walk at the same pace as the electrons, then the electrons look normal to you and the protons seem to be clumped together a little more than normal. This is because if you're moving at the same speed as the electrons, then they are at rest compared to you and look normal. However, now the atoms (protons) look to you like they're moving backwards at a few miles per hour, and the protons look clumped up. The field from the clumped up protons will be just like the field from the clumped up electrons, except in the opposite direction as the protons are oppositely charged. This strange electric field can reverse direction if you are moving.

The reason we can feel magnetism is that the electrons and protons cancel each other out to such high precision that even tiny little changes like one part in a billion are apparent. Moving mass has a similar property - the gravitational field around a train is subtly different if the train is moving, there's a magneto-gravity effect. However, because the base gravity of the Earth and the train are never cancelled out - there's no such thing as anti-gravity or opposite gravity, as there is opposite electric charge - the very tiny magneto-gravitic effects are never noticed. In fact, magneto-gravitic effects have never been measured.