Preface

Most of mathematics is concerned at some level with setting up and solving various types of equations. Algebraic geometry is the mathematical discipline which handles solution sets of systems of polynomial equations. These are called algebraic sets.

By making use of a correspondence which relates algebraic sets to ideals in polynomial rings, problems concerning the geometry of algebraic sets can be translated into algebra. As a consequence, algebraic geometers have developed a multitude of often highly abstract techniques for the qualitative and quantitative study of algebraic sets, without, in the first instance, considering the equations. Modern computer algebra algorithms, on the other hand, allow us to manipulate the equations and, thus, to study explicit examples. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, is now also added to the toolbox of the algebraic geometer.

In these notes, we discuss some of the basic operations in geometry and describe their counterparts in algebra. We explain how the operations can be carried out using computation, and give a number of explicit examples, worked out with the computer algebra system SINGULAR. In this way, our book may serve as a first introduction to SINGULAR, guiding the reader to performing his own experiments.

In this chapter, we will explain how to solve a Sudoku puzzle using ideas from algebraic geometry and computer algebra. In fact, we will represent the solutions of a Sudoku as the points in the vanishing locus of a polynomial ideal I in 81 variables, and we will show that the unique solution of a well-posed Sudoku can be read off from the reduced Gröbner basis of I. We should point out, however, that attacking a Sudoku can be regarded as a graph coloring problem, with one color for each of the numbers 1, . . . ,9, and that graph theory provides much more efficient methods for solving Sudoko than do Gröbner bases.

A completed Sudoku is a particular example of what is called a Latin square. A Latin square of order n is an n Ⅹ n square grid whose entries are taken from a set of n different symbols, with each symbol appearing exactly once in each row and each column. For a Sudoku, usually n = 9, and the symbols are the numbers from 1 to 9. In addition to being a Latin square, a completed Sudoku is subject to the condition that each number from 1 to 9 appears exactly once in each of the nine distinguished 3 Ⅹ 3 blocks.

Overview

The existence of this book is owed (both figuratively and literally) to the fact that the building blocks of matter possess a quality called charge. Two important aspects of charge are conservation and quantization. The electric force between two charges is given by Coulomb’s law. Like the gravitational force, the electric force falls off like 1/r 2. It is conservative, so we can talk about the potential energy of a system of charges (the work done in assembling them). A very useful concept is the electric field, which is defined as the force per unit charge. Every point in space has a unique electric field associated with it. We can define the flux of the electric field through a given surface. This leads us to Gauss’s law, which is an alternative way of stating Coulomb’s law. In cases involving sufficient symmetry, it is much quicker to calculate the electric field via Gauss’s law than via Coulomb’s law and direct integration. Finally, we discuss the energy density in the electric field, which provides another way of calculating the potential energy of a system.

Electric charge

Electricity appeared to its early investigators as an extraordinary phenomenon. To draw from bodies the “subtle fire,” as it was sometimes called, to bring an object into a highly electrified state, to produce a steady flow of current, called for skillful contrivance. Except for the spectacle of lightning, the ordinary manifestations of nature, from the freezing of water to the growth of a tree, seemed to have no relation to the curious behavior of electrified objects. We know now that electrical forces largely determine the physical and chemical properties of matter over the whole range from atom to living cell. For this understanding we have to thank the scientists of the nineteenth century, Ampère, Faraday, Maxwell, and many others, who discovered the nature of electromagnetism, as well as the physicists and chemists of the twentieth century who unraveled the atomic structure of matter.

Preface to the first edition of Volume 2

The subject of this volume of the Berkeley Physics Course is electricity and magnetism. The sequence of topics, in rough outline, is not unusual: electrostatics; steady currents; magnetic field; electromagnetic induction; electric and magnetic polarization in matter. However, our approach is different from the traditional one. The difference is most conspicuous in Chaps. 5 and 6 where, building on the work of Vol. I, we treat the electric and magnetic fields of moving charges as manifestations of relativity and the invariance of electric charge. This approach focuses attention on some fundamental questions, such as: charge conservation, charge invariance, the meaning of field. The only formal apparatus of special relativity that is really necessary is the Lorentz transformation of coordinates and the velocity-addition formula. It is essential, though, that the student bring to this part of the course some of the ideas and attitudes Vol. I sought to develop—among them a readiness to look at things from different frames of reference, an appreciation of invariance, and a respect for symmetry arguments. We make much use also, in Vol. II, of arguments based on superposition.

Our approach to electric and magnetic phenomena in matter is primarily “microscopic,” with emphasis on the nature of atomic and molecular dipoles, both electric and magnetic. Electric conduction, also, is described microscopically in the terms of a Drude-Lorentz model. Naturally some questions have to be left open until the student takes up quantum physics in Vol. IV. But we freely talk in a matter-of-fact way about molecules and atoms as electrical structures with size, shape, and stiffness, about electron orbits, and spin. We try to treat carefully a question that is sometimes avoided and sometimes beclouded in introductory texts, the meaning of the macroscopic fields E and B inside a material.

Overview

The first half of this chapter deals mainly with the potential associated with an electric field. The second half covers a number of mathematical topics that will be critical in our treatment of electromagnetism. The potential difference between two points is defined to be the negative line integral of the electric field. Equivalently, the electric field equals the negative gradient of the potential. Just as the electric field is the force per unit charge, the potential is the potential energy per unit charge. We give a number of examples involving the calculation of the potential due to a given charge distribution. One important example is the dipole, which consists of two equal and opposite charges. We will have much more to say about the applications of dipoles in Chapter 10.

Turning to mathematics, we introduce the divergence, which gives a measure of the flux of a vector field out of a small volume. We prove Gauss’s theorem (or the divergence theorem) and then use it to write Gauss’s law in differential form. The result is the first of the four equations known as Maxwell’s equations (the subject of Chapter 9). We explicitly calculate the divergence in Cartesian coordinates. The divergence of the gradient is known as the Laplacian operator. Functions whose Laplacian equals zero have many important properties, one of which leads to Earnshaw’s theorem, which states that it is impossible to construct a stable electrostatic equilibrium in empty space. We introduce the curl, which gives a measure of the line integral of a vector field around a small closed curve. We prove Stokes’ theorem and explicitly calculate the curl in Cartesian coordinates. The conservative nature of a static electric field implies that its curl is zero. See Appendix F for a discussion of the various vector operators in different coordinate systems.

Overview

In this chapter we study how electric fields affect, and are affected by, matter. We concern ourselves with insulators, or dielectrics, characterized by a dielectric constant. The study of electric fields in matter is largely the study of dipoles. We discussed these earlier in Chapter 2, but we will derive their properties in more generality here, showing in detail how the multipole expansion comes about. The net dipole moment induced in matter by an electric field can come about in two ways. In some cases the electric field polarizes the molecules; the atomic polarizability quantifies this effect. In other cases a molecule has an inherent dipole moment, and the external field serves to align these moments. In any case, a material can be described by a polarization density P. The electric susceptibility gives (up to a factor of ϵ0) the ratio of P to the electric field. The effect of the polarization density is to create a surface charge density on a dielectric material. This explains why the capacitance of a capacitor is increased when it is filled with a dielectric; the surface charge on the dielectric partially cancels the free charge on the capacitor plates.

We study the special case of a uniformly polarized sphere, which interestingly has a uniform electric field in its interior. We then extend this result to the case of a dielectric sphere placed in a uniform electric field. By considering separately the free charge and bound charge, we are led to the electric displacement vector D, whose divergence involves only the free charge (unlike the electric field, whose divergence involves all the charge, by Gauss’s law). We look at the effects of temperature on the polarization density, how the polarization responds to rapidly changing fields, and how the bound-charge current affects the “curl B” Maxwell equation. Finally, we consider an electromagnetic wave in a dielectric. We find that only a slight modification to the vacuum case is needed.

Overview

Magnetic fields in matter are a bit more involved than electric fields in matter. Our main goal in this chapter is to understand the three types of magnetic materials: diamagnetic materials, which are weakly repelled by a solenoid; paramagnetic materials, which are somewhat strongly attracted; and ferromagnetic materials, which are very strongly attracted. As was the case in Chapter 10, we will need to understand dipoles. The far field of a magnetic dipole has the same form as that of an electric dipole, with the magnetic dipole moment replacing the electric dipole moment. However, the near fields are fundamentally different due to the absence of magnetic charge. We will find that diamagnetism is due to the fact that an applied magnetic field causes the magnetic dipole moment arising from the orbital motion of electrons in atoms to pick up a contribution pointing opposite to the applied field. In contrast, in the case of paramagnetism, the spin dipole moment is the relevant one, and it picks up a contribution pointing in the same direction as the applied field. Ferromagnetism is similar to paramagnetism, although a certain quantum phenomenon makes the overall effect much larger; a ferromagnetic dipole moment can exist in the absence of an external magnetic field. Magnetized materials can be described by the magnetization M, the curl of which gives the bound currents (which arise from both orbital motion and spin). By considering separately the free and bound currents, we are led to the field H (also called the “magnetic field”) whose curl involves only the free current (unlike the magnetic field B, whose curl involves all the current, by Ampère’s law).

The metal lead is a moderately good conductor at room temperature. Its resistivity, like that of other pure metals, varies approximately in proportion to the absolute temperature. As a lead wire is cooled to 15 K its resistance falls to about 1 ∕ 20 of its value at room temperature, and the resistance continues to decrease as the temperature is lowered further. But as the temperature 7.22 K is passed, there occurs without forewarning a startling change: the electrical resistance of the lead wire vanishes! So small does it become that a current flowing in a closed ring of lead wire colder than 7.22 K – a current that would ordinarily die out in much less than a microsecond – will flow for years without measurably decreasing. This phenomenon has been directly demonstrated. Other experiments indicate that such a current could persist for billions of years. One can hardly quibble with the flat statement that the resistivity is zero. Evidently something quite different from ordinary electrical conduction occurs in lead below 7.22 K. We call it superconductivity.

Superconductivity was discovered in 1911 by the great Dutch low-temperature experimenter Kamerlingh Onnes. He observed it first in mercury, for which the critical temperature is 4.16 K. Since then hundreds of elements, alloys, and compounds have been found to become superconductors. Their individual critical temperatures range from roughly a millikelvin up to the highest yet discovered, 138 K. Curiously, among the elements that do not become superconducting are some of the best normal conductors such as silver, copper, and the alkali metals.

Overview

In the first two chapters, we were concerned with the electric field and potential due to charges whose positions were fixed and known. We will now study the field and potential due to charges on conductors, where the charges are free to move around. This is a more difficult task, because on one hand we need to know the field to determine the positions of the charges, but on the other hand we need to know the positions of the charges to determine the field. Fortunately, there are some facts and theorems that make this tractable, and indeed in some cases trivial. The most important fact is that in an electrostatic setup, the electric field inside the material of a conductor is zero. Equivalently, all points in a given conductor have the same potential. This leads to the somewhat surprising effect called electrical shielding; the electric field inside an empty conducting shell is zero, independent of whatever arbitrary charge distribution exists outside. We prove the very helpful uniqueness theorem, which states that, given the values of the potential ϕ on the surfaces of a set of conductors, the solution for ϕ throughout space is unique. This theorem often makes things so easy that you may wonder if you’re actually cheating. A byproduct of the theorem is the topic of image charges, which allow us to construct the electric field near conductors in certain cases. We define the capacitance coefficient(s) of a set of conductors; these tell us how much charge resides on a conductor at a given potential. Capacitors are a fundamental circuit element, as we will see in Chapter 8. Finally, we discuss the energy stored in a capacitor.

Overview

In this chapter we discuss charge in motion, or electric current. The current density is defined as the current per cross-sectional area. It is related to the charge density by the continuity equation. In most cases, the current density is proportional to the electric field; the constant of proportionality is called the conductivity, with the inverse of the conductivity being the resistivity. Ohm’s law gives an equivalent way of expressing this proportionality. We show in detail how the conductivity arises on a molecular level, by considering the drift velocity of the charge carriers when an electric field is applied. We then look at how this applies to metals and semiconductors. In a circuit, an electromotive force (emf) drives the current. A battery produces an emf by means of chemical reactions. The current in a circuit can be found either by reducing the circuit via the series and parallel rules for resistors, or by using Kirchhoff’s rules. The power dissipated in a resistor depends on the resistance and the current passing through it. Any circuit can be reduced to a Thévenin equivalent circuit involving one resistor and one emf source. We end the chapter by investigating how the current changes in an RC circuit.