A particle with charge q has been moving in a straight line at constant speed v0 for a long time. It runs into something, let us imagine, and in a short period of constant deceleration, of duration τ, the particle is brought to rest. The graph of velocity versus time in Fig. B.l describes its motion. What must the electric field of this particle look like after that? Figure B.2 shows how to derive it.

We shall assume that v0 is small compared with c. Let t = 0 be the instant the deceleration began, and let x = 0 be the position of the particle at that instant. By the time the particle has completely stopped it will have moved a little farther on, to x = 1/2;v0τ. That distance, although we tried to indicate it on our diagram, is small compared with the other distances that will be involved.

We now examine the electric field at a time t = T ≫ τ. Observers farther away from the origin than R = cT cannot have learned that the particle was decelerated. Throughout that region, region I in Fig. B.2, the field must be that of a charge which has been moving and is still moving at the constant speed v0. That field, as we discovered in Section 5.7, appears to emanate from the present position of the charge, which for an observer anywhere in region I is the point x = v0T on the x axis.

DEFINITION OF THE MAGNETIC FIELD

A charge which is moving parallel to a current of other charges experiences a force perpendicular to its own velocity. We can see it happening in the deflection of the electron beam in Fig. 5.3. We discovered in Section 5.9 that this is consistent with—indeed, is required by—Coulomb's law with charge invariance and special relativity. And we found that a force perpendicular to the charged particle's velocity also arises in motion at right angles to the current-carrying wire. For a given current the magnitude of the force, which we calculated for the particular case in Fig. 5.20a, is proportional to the product of the particle's charge q and its speed v in our frame. Just as we defined the electric field E as the vector force on unit charge at rest, so we can define another field B by the velocity-dependent part of the force that acts on a charge in motion. The defining relation was introduced at the beginning of Chapter 5. Let us state it again more carefully.

At some instant t a particle of charge q passes the point (x, y, z) in our frame, moving with velocity v. At that moment the force on the particle (its rate of change of momentum) is F. The electric field at that time and place is known to be E.

This chapter provides examples of how to apply the tensor concepts contained in Chapters 4 and 5, just as Chapter 3 provided examples of how to apply the vector concepts presented in Chapters 1 and 2. As in Chapter 3, the intent for this chapter is to include more detail about a small number of selected applications than can be included in the chapters in which tensor concepts are first presented.

The examples in this chapter come from the fields of Mechanics, Electromagnetics, and General Relativity. Of course, there's no way to comprehensively cover any significant portion of those fields in one chapter; these examples were chosen only to serve as representatives of the types of tensor application you're likely to encounter in those fields.

The inertia tensor

A very useful way to think of mass is this: mass is the characteristic of matter that resists acceleration. This means that it takes a force to change the velocity of any object with mass. You may find it helpful to think of moment of inertia as the rotational analog of mass. That is, moment of inertia is the characteristic of matter that resists angular acceleration, so it takes a torque to change the angular velocity of an object.

Many students find that rotational motion is easier to understand by keeping the relationships between translational and rotational quantities in mind.

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.