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1. Preliminaries: Looking Over Jackson's Preface and Introduction

    Michael Fowler, UVa

What Does this Course Cover?

These notes cover a standard two-semester graduate E&M course.  Jackson was used as the text (3^rd edition) but the notes are complete in themselves.

To understand and appreciate E&M at this level, fluency in its native language$—$three-dimensional vector calculus$—$is essential.  Therefore, after discussing Jackson’s Preface and Introduction, and a short section on electromagnetic units, we present a “boot camp”: a detailed review of the necessary definitions and theorems in vector calculus we need. (This boot camp is not exactly an original approach: it’s not in Jackson, but has appeared in E&M texts from Abraham and Becker (1932) to Zangwill (2013).)

After that, we more or less follow topics in the standard order:  electrostatics, magnetostatics, Maxwell’s equations, radiation, waveguides, scattering, special relativity and relativistic particle dynamics. That covers the first twelve chapters in Jackson.  We only cover selected topics in the final four chapters, thirteen to sixteen. 

We have added a few topics not discussed in Jackson: conformal mapping methods for two dimensional electrostatics, some uses of confocal coordinate systems, waves in anisotropic media, etc. We believe that the effort needed to become familiar with spheroidal coordinates pays off bigtime when dealing with systems having these symmetries, and these coordinates are far more natural. Compare, for example, the treatment here of field through a hole in a conducting plane with Jackson’s.

As for other texts, an excellent backup at a slightly lower level is Griffiths.  Another good book is Zangwill.  You should look over it. A classic treatment by a great physicist is Landau’s, but that is split between two of his books with Lifshitz, and no unworked problems are given (but reading the terse solutions provided can be very instructive!) Feynman’s notes are fun to read, and have some neat insights.

Jackson's Preface …

… is worth looking through. First, the bedeviling question of units: many physicists use the traditional Gaussian system, which is more elegant, but unfortunately not the standardized system (SI) used by all engineers. We’ll discuss these in detail in the next lecture.  Jackson’s solution is to have the first half of the book in SI, the second half, more relevant especially to particle physicists, in Gaussian. This has its downsides$—$for example, some equations appear in both halves of the book, in different units, so you can't refer back without careful thought. Watch out.

Jackson’s Introduction and Survey

Don’t take this too seriously, and in particular don’t worry if you can’t follow some of it. Glance through my notes below:

On page two, Jackson writes down Maxwell’s equations in differential form, then a little later (page 16, 17) in integral form.  No doubt the idea is you're already familiar with them from undergraduate studies, but if at this point in your studies you’re not completely comfortable with them, that’s ok, we’re going to spend a large part of the first semester working towards them, and, in particular, developing and practicing the necessary mathematical tools.  Anyway, here they are, for the record:

$\begin{array}{l}\overrightarrow{\nabla }\cdot \overrightarrow{D}=\rho \\ \overrightarrow{\nabla }\times \overrightarrow{H}-\frac{\partial \overrightarrow{D}}{\partial t}=\overrightarrow{J}\\ \overrightarrow{\nabla }\times \overrightarrow{E}+\frac{\partial \overrightarrow{B}}{\partial t}=0\\ \overrightarrow{\nabla }\cdot \overrightarrow{B}=0\end{array}$

where for external sources in vacuum $\overrightarrow{D}={\epsilon }_0\overrightarrow{E}$ and $\overrightarrow{B}={\mu }_0\overrightarrow{H}.$ The subtle differences between $\overrightarrow{D}$ and $\overrightarrow{E}$, and between $\overrightarrow{B}$ and $\overrightarrow{H}$, will be explored in detail later.   If you're feeling rusty on these differences, don’t worry.

The first two equations lead to charge conservation $\overrightarrow{\nabla }\cdot \overrightarrow{J}=-\partial \rho /\partial t.$ (Prove it!)

The force on a point charge $q$ in an electromagnetic field is (in SI units)

$\overrightarrow{F}=q\left(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}\right),$

written by Lorentz in 1892, often called the Lorentz force.  (But it was written years earlier by Heaviside, a telegraph engineer.)

$\overrightarrow{E}$ is the electric field, defined as force/unit charge on a small test charge (small enough not to disturb the system).

$\overrightarrow{B}$ is the magnetic field, defined as force per unit current, same conditions.

Notation alert!  Jackson calls $\overrightarrow{B}$ the "magnetic induction".  This is ridiculous (Griffiths, p 282, calls it "absurd".)  $\overrightarrow{B}$ is the fundamental field, not $\overrightarrow{H}$, which we'll refer to just as H, without giving it a name.   (Unfortunately Jackson calls H the magnetic field.)

Here $\rho \left(\overrightarrow{x},t\right),\overrightarrow{J}\left(\overrightarrow{x},t\right)$ are charge and current densities.

Maxwell’s equations can equally be written in integral form, the first two over closed surfaces, the other two equations relate integrals around a closed curve in three-dimensional space (the left-hand side) to integrals over surfaces spanning those curves:

$\begin{array}{l}{\displaystyle \underset{S}{\oint }\overrightarrow{D}\cdot \overrightarrow{n}da={\displaystyle \underset{V}{\int }\rho d^{3}x}}\\ {\displaystyle \underset{S}{\oint }\overrightarrow{B}\cdot \overrightarrow{n}da=0}\\ {\displaystyle \underset{C}{\oint }\overrightarrow{H}\cdot d\overrightarrow{l}}={\displaystyle \underset{S^{\prime }}{\int }\left[\overrightarrow{J}+\frac{\partial \overrightarrow{D}}{\partial t}\right]}\cdot {\overrightarrow{n}}^{\prime }da\\ {\displaystyle \underset{C}{\oint }\overrightarrow{E}\cdot d\overrightarrow{l}}=-{\displaystyle \underset{S^{\prime }}{\int }\frac{\partial \overrightarrow{B}}{\partial t}\cdot {\overrightarrow{n}}^{\prime }da}\end{array}$

and these are a bit easier to interpret than the differential version.  (We’re going to go over all this in detail, of course.)

If you want a gentle review of Maxwell’s equations and how they lead to prediction of the speed of light, see my notes here.

How does classical E&M fit into the big picture of modern physics?

This is a reasonable question to ask at this point: if we’re going to all this trouble to master this subject, when can we rely on it and when not?  Let’s review some of its basic assumptions:

Inverse square law:

It’s fantastically successful over a huge range of sizes$—$from atoms to galaxies.

In particular, the inverse-square law of the Coulomb electrostatic attraction/repulsion correctly predicts Rutherford scattering of alpha particles (each a few Mev kinetic energy) by nuclei, which (you can figure this out) means it’s still good at ${10}^{-13}$ meters, and the inverse square law is certainly accurate for atoms: in the hydrogen atom, the experimentally observed degeneracies of energy levels could only happen for an exactly inverse-square force. (OK, there are very tiny corrections for quantum electrodynamic effects, the “polarization of the vacuum”, briefly discussed below.)

Accelerating charges radiate:

Some classical E&M predictions come unstuck for atoms.  In particular, the Bohr orbital model, with the classical field strength, gives correct energy levels for hydrogen but the same classical E&M theory predicts that accelerating charges always radiate$—$a prediction amply borne out for currents in antennae.  So the orbiting Bohr electron, accelerating since it’s in circular motion, should be losing energy by radiation. It isn’t.  So what’s wrong?

Photons:

In fact, the picture of continuous classical fields is inadequate: the fields are quantized: in particular, radiation at frequency $\omega$ emitted by an accelerating electron can only be in chunks of energy $\hslash \omega$.  Physicists at the end of the nineteenth century were forced to this conclusion to explain black body radiation. The electromagnetic field has particle-like properties: the field is a collection of photons.  And there’s more: the electron in the hydrogen atom, traditionally regarded as a point charged particle turns out to have wavelike properties: in particular, the only definite energy orbit it can occupy in the hydrogen atom correspond to standing circular waves.  Bohr’s model succeeded because he imposed an equivalent requirement, without understanding its origin.

Correspondence Principle:

So classical E&M doesn’t work for low electron orbits in the hydrogen atom, but it clearly works well on a macroscopic scale$—$at what scale does it become OK to use it? In fact, it’s already pretty good for highly excited hydrogen atoms: say the thirtieth excited level.  Such atoms are called Rydberg atoms (my colleague Tom Gallagher has written a book about them). As the electron cascades through energy levels, $n=30,29,28\dots$ it emits photons corresponding to the energy differences $\hslash {\omega }_n=E_n-E_{n-1}$, but for larger and larger atomic orbits that frequency $\underset{n\to \infty }{\lim }{\omega }_n=\omega ,$ the circular frequency of the electron in orbit$—$exactly the frequency the classical theory predicts! In fact, it was this Correspondence Principle (as he called it) that Bohr used to fix the size of the quantum, and hence of the atom.

The photon as a particle:

How does the photon fit into the family of elementary particles? The proton and electron were familiar more than a hundred years ago, the neutron, and the picture of the nucleus, emerged in the 1930’s.  At the same time, Yukawa and others began constructing a quantum theory of elementary particles. They knew the nucleus was held together by short-range forces, Yukawa’s model suggested the fermions attracted each other by exchanging bosons of mass somewhere between electrons and protons, called mesons (Greek for in the middle). The potential energy corresponding to the force has the form $e^{-\mu r}/r$, where $\mu$ is the mass of the boson exchanged.  This is called a Yukawa potential. The appropriate mass, from the estimate of the force range in nuclei, is around 200 electron masses. (It was first thought that the $\mu$ meson might be the force carrier, in fact the $\pi$ mesons are the appropriate carriers in this simple model, which is now superseded by quantum chromodynamics, with quarks and gluons.)

Evidently, for the photon to fit into this quasi potential theory, with the known Coulomb inverse-square force, we need $\mu =0$, the photon must be massless. Of course, it could have a really small mass, but measurements of the accuracy of the inverse-square force show that any photon mass is less than ${10}^{-20}$ electron masses.  So we’re not going to worry about that any further.

Maxwell’s equations are linear. How accurate is that?

Certainly the linearity is incorrect for strong fields in materials: lasers depend on the nonlinearity. But even in vacuum there are limitations: light can be scattered by light, but it’s a tiny effect at available intensities. Nevertheless, it’s there, and in fact can be precisely predicted from the well-established theory called quantum electrodynamics, QED.  Light-light scattering (which is observed) depends on photons splitting into virtual electron-positron pairs.

If the electron is a point particle, does the inverse square law hold all the way in?

No: because, as we shall see, an electric field has an energy density $\propto {\overrightarrow{E}}^2$, and if you find the total energy in the electric field of a point particle, by doing the integral down to $r=0,$ you get infinity and therefore infinite local mass, clearly nonsense.  Nobody believes particles are mathematical points anyway$—$in fact distances below the Planck length, ${10}^{-35}$ meters, are meaningless (to measure that length you’d need a photon with energy great enough to completely distort the space, this from general relativity). However, things go bad long before getting down to the Planck length: at the electron Compton wavelength, $\lambda =h/mc\sim {10}^{-12}$ meters, the field can polarize the vacuum.  How can you polarize nothing?  What happens is that with a sufficient electric field intensity you can create electron-positron pairs, separated to gain potential energy. These are virtual pairs, meaning they don’t last long but are continuously recreated. Nevertheless, they lower the field strength sufficiently to give a finite mass. (This handwaving presentation nowhere near does justice to the incredibly precise theory, of course: the important point is that it is well understood how to extend classical electrodynamics into this quantum regime$—$in fact QED is often termed the most successful physics theory ever).

Back to earth:

So what, exactly, are $\overrightarrow{D},\overrightarrow{H}$?   In vacuum, just multiples $\overrightarrow{D}={\epsilon }_0\overrightarrow{E}$, $\overrightarrow{B}={\mu }_0\overrightarrow{H}.$

Dielectric materials have polarizable molecules.  When the externally generated electric field is switched on and penetrates the material, the internal positive and negative charges are slightly displaced, in opposite directions of course.  Imagine a cube of material with the field perpendicular to a pair of faces: the effect of the displacement is equivalent to a uniform (macroscopically speaking) cube of positive material, and a slightly displaced cube of negative material. The net result is a layer of negative charge on one face of the cube, a layer of positive charge on the opposite face. The electric field from these layers is the polarization.

Again, we’ll be discussing this in detail later$—$this is just to give you some idea.

The macroscopic field is called the electric displacement and written

$\overrightarrow{D}={\epsilon }_0\overrightarrow{E}+\overrightarrow{P}.$

Here $\overrightarrow{P}$ is the polarization with units defined appropriately (see later).

From the discussion above, it’s clear that polarization is equivalent to a charge displacement, but $\overrightarrow{D}$ here also includes a vacuum term. This has historical roots:  the thinking long ago was that the vacuum was also really a substance, the aether, and that somehow it too contained charges that were displaced by an electric field.  (Actually not completely crazy$—$in quantum electrodynamics, as we’ve outlined above, an electric field in a vacuum causes distortion of virtual electron-positron pairs which are constantly appearing and disappearing, but it’s a much more dynamic picture.)

The true local electric field in a solid varies dramatically from atom to atom, and certainly varies a lot within an atom. The electric displacement is the smooth electric field measured with macroscopic instruments that average over distances far larger than atomic sizes.

Similarly,

$\overrightarrow{H}=\frac{1}{{\mu }_0}\overrightarrow{B}+\overrightarrow{M}.$

Here $\overrightarrow{M}$ is the magnetization, and, confusingly, as we've discussed and will again, Jackson calls $\overrightarrow{H}$ the magnetic field, $\overrightarrow{B}$ the magnetic induction.  We won't: $\overrightarrow{B}$ is the magnetic field.   

Some materials are not isotropic, molecular polarization may be easier in one direction than another:

$\begin{array}{l}{D}_{\alpha }={\epsilon }_{\alpha \beta }{E}_{\beta }\\ H_{\alpha }={{\mu }^{\prime }}_{\alpha \beta }{B}_{\beta }\end{array}$

using dummy suffix notation (meaning repeated suffixes are automatically summed over).  Such materials are crucial in optics.

More jargon:      ${\epsilon }_{\alpha \beta }$ is the electric permittivity or dielectric tensor.

                              ${{\mu }^{\prime }}_{\alpha \beta }$ is the inverse magnetic permeability tensor.

Real world responses: these tensors are frequency dependent: for example, water molecules have electric dipole moments, $\epsilon /{\epsilon }_0\approx 80$ at room temperature$—$strong molecular alignment in a static field. But at optical frequencies, a much lower response $\epsilon /{\epsilon }_0\approx 1.8$, the field oscillation is too fast for the molecule to turn, only the electrons within the molecule can move fast enough to respond at optical frequencies.

Broadly speaking, there are three types of magnetic materials: diamagnetic, paramagnetic, ferromagnetic:

Diamagnetic atoms are a miniature version of Lenz’ law: if you move a magnet towards a loop of wire, a current is set up opposing motion (Maxwell’s Fourth Equation). In the atom, the orbit distorts in a way to reduce the total magnetic field.

Actually everything’s diamagnetic$—$but if the atoms already have magnetic moments, these will tend to line up with an imposed field and enhance it. This paramagnetism is stronger than the diamagnetic effect.

Ferromagnetic materials have atoms with magnetic moments, but also interactions between atoms that tend to align them. These forces are not themselves magnetic, but much stronger, and quantum mechanical in origin.

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