This revision of “Electricity and Magnetism,” Volume 2 of the Berkeley Physics Course, has been made with three broad aims in mind. First, I have tried to make the text clearer at many points. In years of use teachers and students have found innumerable places where a simplification or reorganization of an explanation could make it easier to follow. Doubtless some opportunities for such improvements have still been missed; not too many, I hope.

A second aim was to make the book practically independent of its companion volumes in the Berkeley Physics Course. As originally conceived it was bracketed between Volume 1, which provided the needed special relativity, and Volume 3, “Waves and Oscillations,” to which was allocated the topic of electromagnetic waves. As it has turned out, Volume 2 has been rather widely used alone. In recognition of that I have made certain changes and additions. A concise review of the relations of special relativity is included as Appendix A. Some previous introduction to relativity is still assumed. The review provides a handy reference and summary for the ideas and formulas we need to understand the fields of moving charges and their transformation from one frame to another. The development of Maxwell's equations for the vacuum has been transferred from the heavily loaded Chapter 7 (on induction) to a new Chapter 9, where it leads naturally into an elementary treatment of plane electromagnetic waves, both running and standing.

CONDUCTORS AND INSULATORS

The earliest experimenters with electricity observed that substances differed in their power to hold the “Electrick Vertue.” Some materials could be easily electrified by friction and maintained in an electrified state; others, it seemed, could not be electrified that way, or did not hold the Vertue if they acquired it. Experimenters of the early eighteenth century compiled lists in which substances were classified as “electricks” or “nonelectricks.” Around 1730, the important experiments of Stephen Gray in England showed that the Electrick Vertue could be conducted from one body to another by horizontal string, over distances of several hundred feet, provided that the string was itself supported from above by silk threads. Once this distinction between conduction and nonconduction had been grasped, the electricians of the day found that even a nonelectrick could be highly electrified if it were supported on glass or suspended by silk threads. A spectacular conclusion of one of the popular electric exhibitions of the time was likely to be the electrification of a boy suspended by many silk threads from the rafters; his hair stood on end and sparks could be drawn from the tip of his nose.

After the work of Gray and his contemporaries the elaborate lists of electricks and non-electricks were seen to be, on the whole, a division of materials into electrical insulators and electrical conductors. This distinction is still one of the most striking and extreme contrasts that nature exhibits.

ELECTRIC CURRENT AND CURRENT DENSITY

An electric current is charge in motion. The carriers of the charge can be physical particles like electrons or protons, which may or may not be attached to larger objects, atoms or molecules. Here we are not concerned with the nature of the charge carriers but only with the net transport of electric charge their motion causes. The electric current in a wire is the amount of charge passing a fixed mark on the wire in unit time. In CGS units current will be expressed in esu/sec. The SI unit is coulombs/sec, or amperes (amps). A current of 1 ampere is the same as a current of 2.998 × 109 esu/sec, which is equivalent to 6.24 × 1018 elementary electronic charges per second.

It is the net charge transport that counts, with due regard to sign. Negative charge moving east is equivalent to positive charge moving west. Water flowing through a hose could be said to involve the transport of an immense amount of charge—about 3 × 1023 electrons per gram of water! But since an equal number of protons move along with the electrons (every water molecule contains 10 of each), the electric current is zero. On the other hand, if you were to charge negatively a nylon thread and pull it steadily through a nonconducting tube, that would constitute an electric current, in the direction opposite the motion of the thread.

HOW VARIOUS SUBSTANCES RESPOND TO A MAGNETIC FIELD

Imagine doing some experiments with a very intense magnetic field. To be definite, suppose we have built a solenoid of 10-cm inside diameter, 40 cm long, like the one shown in Fig. 11.1. Its outer diameter is 40 cm, most of the space being filled with copper windings. This coil will provide a steady field of 30,000 gauss, or 3.0 teslas, at its center if supplied with 400 kilowatts of electric power—and something like 30 gallons of water per minute, to carry off the heat. We mention these practical details to show that our device, though nothing extraordinary, is a pretty respectable laboratory magnet. The field strength at the center is nearly 105 times the earth's field, and probably 5 or 10 times stronger than the field near any iron bar magnet or horseshoe magnet you may have experimented with. The field will be fairly uniform near the center of the solenoid, falling, on the axis at either end, to roughly half its central value. It will be rather less uniform than the field of the solenoid in Fig. 6.18, since our coil is equivalent to a “nested” superposition of solenoids with length-diameter ratio varying from 4:1 to 1:1. In fact, if we analyze our coil in that way and use the formula (Eq. 44 of Chapter 6) which we derived for the field on the axis of a solenoid with a single-layer winding, it is not hard to calculate the axial field exactly. A graph of the field strength on the axis, with the central field taken as 30 kilogauss, is included in Fig. 11.1.

1. The power which electricity of tension possesses of causing an opposite electrical state in its vicinity has been expressed by the general term Induction; which, as it has been received into scientific language, may also, with propriety, be used in the same general sense to express the power which electrical currents may possess of inducing any particular state upon matter in their immediate neighbourhood, otherwise indifferent. It is with this meaning that I purpose using it in the present paper.

2. Certain effects of the induction of electrical currents have already been recognised and described: as those of magnetization; Ampere's experiments of bringing a copper disc near to a flat spiral; his repetition with electromagnets of Arago's extraordinary experiments, and perhaps a few others. Still it appeared unlikely that these could be all the effects which induction by currents could produce; especially as, upon dispensing with iron, almost the whole of them disappear, whilst yet an infinity of bodies, exhibiting definite phenomena of induction with electricity of tension, still remain to be acted upon by the induction of electricity in motion.

3. Further: Whether Ampére's beautiful theory were adopted, or any other, or whatever reservation were mentally made, still it appeared very extraordinary, that as every electric current was accompanied by a corresponding intensity of magnetic action at right angles to the current, good conductors of electricity, when placed within the sphere of this action, should not have any current induced through them, or some sensible effect produced equivalent in force to such a current.

The vector concepts and techniques described in the previous chapters are important for two reasons: they allow you to solve a wide range of problems in physics and engineering, and they provide a foundation on which you can build an understanding of tensors (the “facts of the universe”). To achieve that understanding, you'll have to move beyond the simple definition of vectors as objects with magnitude and direction. Instead, you'll have to think of vectors as objects with components that transform between coordinate systems in specific and predictable ways. It's also important for you to realize that vectors can have more than one kind of component, and that those different types of component are defined by their behavior under coordinate transformations.

So this chapter is largely about the different types of vector component, and those components will be a lot easier to understand if you have a solid foundation in the mathematics of coordinate-system transformation.

Coordinate-system transformations

In taking the step from vectors to tensors, a good place to begin is to consider this question: “What happens to a vector when you change the coordinate system in which you're representing that vector?” The short answer is that nothing at all happens to the vector itself, but the vector's components may be different in the new coordinate system. The purpose of this section is to help you understand how those components change.

The real value of understanding vectors and how to manipulate them becomes clear when you realize that your knowledge allows you to solve a variety of problems that would be much more difficult without vectors. In this chapter, you'll find detailed explanations of four such problems: a mass sliding down an inclined plane, an object moving along a curved path, a charged particle in an electric field, and a charged particle in a magnetic field. To solve these problems, you'll need many of the vector concepts and operations described in Chapters 1 and 2.

Mass on an inclined plane

Consider the delivery woman pushing a heavy box up the ramp to her delivery truck, as illustrated in Figure 3.1. In this situation, there are a number of forces acting on the box, so if you want to determine how the box will move, you need to know how to work with vectors. Specifically, to solve problems such as this, you can use vector addition to find the total force acting on the box, and then you can use Newton's Second Law to relate that total force to the acceleration of the box.

To understand how this works, imagine that the delivery woman slips off the side of the ramp, leaving the box free to slide down the ramp under the influence of gravity.