Overview

In earlier chapters we encountered resistors, capacitors, and inductors. We will now study circuits containing all three of these elements. If such a circuit contains no emf source, the current takes the form of a decaying oscillation (in the case of small damping). The rate of decay is described by the Q factor. If we add on a sinusoidally oscillating emf source, then the current will reach a steady state with the same frequency of oscillation as the emf source. However, in general there will be a phase difference between the current and the emf. This phase, along with the amplitude of the current, can be determined by three methods. The first method is to guess a sinusoidal solution to the differential equation representing the Kirchhoff loop equation. The second is to guess a complex exponential solution and then take the real part to obtain the actual current. The third is to use complex voltages, currents, and impedances. These complex impedances can be combined via the same series and parallel rules that work for resistors. As we will see, the third method is essentially the same as the second method, but with better bookkeeping; this makes it far more tractable in the case of complicated circuits. Finally, we derive an expression for the power dissipated in a circuit, which reduces to the familiar V 2/R result if the circuit is purely resistive.

Overview

Having shown in Chapter 5 that the magnetic force must exist, we will now study the various properties of the magnetic field and show how it can be calculated for an arbitrary (steady) current distribution. The Lorentz force gives the total force on a charged particle as F = q E + q v × B. The results from the previous chapter give us the form of the magnetic field due to a long straight wire. This form leads to Ampère’s law, which relates the line integral of the magnetic field to the current enclosed by the integration loop. It turns out that Ampère’s law holds for a wire of any shape. When supplemented with a term involving changing electric fields, this law becomes one of Maxwell’s equations (as we will see in Chapter 9). The sources of magnetic fields are currents, in contrast with the sources of electric fields, which are charges; there are no isolated magnetic charges, or monopoles. This statement is another of Maxwell’s equations.

As in the electric case, the magnetic field can be obtained from a potential, but it is now a vector potential; its curl gives the magnetic field. The Biot–Savart law allows us to calculate (in principle) the magnetic field due to any steady current distribution. One distribution that comes up often is that of a solenoid (a coil of wire), whose field is (essentially) constant inside and zero outside. This field is consistent with an Ampère’s-law calculation of the discontinuity of B across a sheet of current. By considering various special cases, we derive the Lorentz transformations of the electric and magnetic fields. The electric (or magnetic) field in one frame depends on both the electric and magnetic fields in another frame. The Hall effect arises from the q v × B part of the Lorentz force. This effect allows us, for the first time, to determine the sign of the charge carriers in a current.

Overview

The goal of this chapter is to show that when relativity is combined with our theory of electricity, a necessary conclusion is that a new force, the magnetic force, must exist. In nonstatic situations, charge is defined via a surface integral. With this definition, charge is invariant, that is, independent of reference frame. Using this invariance, we determine how the electric field transforms between two frames. We then calculate the electric field due to a charge moving with constant velocity; it does not equal the spherically symmetric Coulomb field. Interesting field patterns arise in cases where a charge starts or stops.

The main result of this chapter, derived in Section 5.9, is the expression for the force that a moving charge (or a group of moving charges) exerts on another moving charge. On our journey to this result, we will consider setups with increasing complexity. More precisely, in calculating the force on a charge q due to another charge Q, there are four basic cases to consider, depending on the charges’ motions. (1) If both charges are stationary in a given frame, then we know from Chapter 1 that Coulomb’s law gives the force. (2) If the source Q is moving and q is at rest, then we can use the transformation rule for the electric field mentioned above. (3) If the source Q is at rest and q is moving, then we can use the transformation rule for the force, presented in Appendix G, to show that the Coulomb field gives the force, as you would expect. (4) Finally, the case we are most concerned with: if both charges are moving, then we will show in Section 5.9 that a detailed consideration of relativistic effects implies that there exists an additional force that must be added to the electrical force; this is the magnetic force. In short, the magnetic force is a consequence of Coulomb’s law, charge invariance, and relativity.

In this appendix we discuss the differences between the SI and Gaussian systems of units. First, we will look at the units in each system, and then we will talk about the clear and not so clear ways in which they differ.

SI units

Consider the SI system, which is the one we use in this book. The four main SI units that we deal with are the meter (m), kilogram (kg), second (s), and coulomb (C). The coulomb actually isn’t a fundamental SI unit; it is defined in terms of the ampere (A),which is a measure of current (charge per time). The coulomb is a derived unit, defined to be 1 ampere-second.

The reason why the ampere, and not the coulomb, is the fundamental unit involving charge is one of historical practicality. It is relatively easy to measure current via a galvanometer (see Section 7.1). More crudely, a current can be determined by measuring the magnetic force that two pieces of a current-carrying wire in a circuit exert on each other (see Fig. 6.4). Once we determine the current that flows onto an object during a given time, we can then determine the charge on the object. On the other hand, although it is possible to measure charge directly via the force that two equally charged objects exert on each other (imagine two balls hanging from strings, repelling each other, as in Exercise 1.36), the setup is a bit cumbersome. Furthermore, it tells us only what the product of the charges is, in the event that they aren’t equal.

Preface to the third edition of Volume 2

For 50 years, physics students have enjoyed learning about electricity and magnetism through the first two editions of this book. The purpose of the present edition is to bring certain things up to date and to add new material, in the hopes that the trend will continue. The main changes from the second edition are (1) the conversion from Gaussian units to SI units, and (2) the addition of many solved problems and examples.

The first of these changes is due to the fact that the vast majority of courses on electricity and magnetism are now taught in SI units. The second edition fell out of print at one point, and it was hard to watch such a wonderful book fade away because it wasn’t compatible with the way the subject is presently taught. Of course, there are differing opinions as to which system of units is “better” for an introductory course. But this issue is moot, given the reality of these courses.

For students interested in working with Gaussian units, or for instructors who want their students to gain exposure to both systems, I have created a number of appendices that should be helpful. Appendix A discusses the differences between the SI and Gaussian systems. Appendix C derives the conversion factors between the corresponding units in the two systems. Appendix D explains how to convert formulas from SI to Gaussian; it then lists, side by side, the SI and Gaussian expressions for every important result in the book. A little time spent looking at this appendix will make it clear how to convert formulas from one system to the other.

Overview

In this chapter we study the effects of magnetic fields that change with time. Our main result will be that a changing magnetic field causes an electric field. We begin by using the Lorentz force to calculate the emf around a loop moving through a magnetic field. We then make the observation that this emf can be written in terms of the rate of change of the magnetic flux through the loop. The sign of the induced emf is determined by Lenz’s law. If we shift frames so that the loop is now stationary and the source of the magnetic field is moving, we obtain the same result for the emf in terms of the rate of change of flux, as expected. Faraday’s law of induction states that this result holds independent of the cause of the flux change. For example, it applies to the case in which we turn a dial to decrease the magnetic field while keeping all objects stationary. The differential form of Faraday’s law is one of Maxwell’s equations. Mutual inductance is the effect by which a changing current in one loop causes an emf in another loop. This effect is symmetrical between the two loops, as we will prove. Self-inductance is the effect by which a changing current in a loop causes an emf in itself. The most commonly used object with self-inductance is a solenoid, which we call an inductor, symbolized by L. The current in an RL circuit changes in a specific way, as we will discover. The energy stored in an inductor equals LI 2/2, which parallels the CV 2/2 energy stored in a capacitor. Similarly, the energy density in a magnetic field equals B 2/2μ 0, which parallels the ϵ0E 2/2 energy density in an electric field.

Preface to the second edition of Volume 2

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 I, 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. The propagation of a wave in a dielectric medium can then be treated in Chapter 10 on Electric Fields in Matter.

Foundations of relativity

We assume that the reader has already been introduced to special relativity. Here we shall review the principal ideas and formulas that are used in the text beginning in Chapter 5. Most essential is the concept of an inertial frame of reference for space-time events and the transformation of the coordinates of an event from one inertial frame to another.

A frame of reference is a coordinate system laid out with measuring rods and provided with clocks. Clocks are everywhere. When something happens at a certain place, the time of its occurrence is read from a clock that was at, and stays at, that place. That is, time is measured by a local clock that is stationary in the frame. The clocks belonging to the frame are all synchronized. One way to accomplish this (not the only way) was described by Einstein in his great paper of 1905. Light signals are used. From a point A, at time t A, a short pulse of light is sent out toward a remote point B. It arrives at B at the time t B, as read on a clock at B, and is immediately reflected back toward A, where it arrives at t′ A. If t B = (t A + t′ A) ∕ 2, the clocks at A and B are synchronized. If not, one of them requires adjustment. In this way, all clocks in the frame can be synchronized. Note that the job of observers in this procedure is merely to record local clock readings for subsequent comparison.