Introduction To Electrodynamics

I NT R ODUCT I ONt o E L E CT R ODYNAMI CS T hi r dE di t i on Da v i dJ .Gr i f f i t hs Introduction to Electrodynamics David 1. Griffiths Reed College Prentice Hall Prentice Hall Upper Saddle River, New Jersey 07458 Library of Congress (::ataloging-in-Publication Data Griffiths, David 1. (David Jeffrey) Introduction to electrodynamics / David J. Griffiths - 3rd ed. p. em. Includes biqliographical references and index. ISBN 0-13-805326-X I. Electrodynamics. I. Title. OC680.G74 1999 537.6--- R (outside). Express your answers in tenns of the total charge q on the sphere. [Hint: Use the law of cosines to write!z. in tenns of Rand (). Be sure to take the positive square root: R2 + z2 - 2Rz = (R - z) if R > Z, but it's (z - R) if R <> J Problem 2.8 Use your result in Prob. 2.7 to find the field inside and outside a sphere of radius R, which carries a unifonn volume charge density p. Express your answers in terms of the total charge of the sphere, q. Draw a graph of IE I as a function of the distance from the center. ,P I I IZ I Figure 2. 10 Figure 2.11 2.2. DIVERGENCE AND CURL OF ELECTROSTATIC FIELDS 65 2.2 Divergence and Curl of Electrostatic Fields 2.2.1 Field Lines, Flux, and Gauss's Law In principle, we are done with the subject of electrostatics. Equation 2.8 tells us how to compute the field of a charge distribution, and Eq. 2.3 tells us what the force on a charge Q placed in this field will be. Unfortunately, as you may have discovered in working Prob. 2.7, the integrals involved in computing E can be formidable, even for reasonably simple charge distributions. Much of the rest of electrostatics is devoted to assembling a bag of tools and tricks for avoiding these integrals. It all begins with the divergence and curl of E. I shall calculate the divergence of E directly from Eq. 2.8, in Sect. 2.2.2, but first I want to show you a more qualitative, and perhaps more illuminating, intuitive approach. Let's begin with the simplest possible case: a single point charge q, situated at the origin: 1 q A E(r) = - - - r . 4nEo r 2 (2.10) To get a "feel" for this field, I might sketch a few representative vectors, as in Fig. 2.12a. Because the field falls off like 1/ r 2 , the vectors get shorter as you go farther away from the origin; they always point radially outward. But there is a nicer way to represent this field, and that's to connect up the arrows, to form field lines (Fig. 2.12b). You might think that I have thereby thrown away information about the strength of the field, which was contained in the length of the arrows. But actually I have not. The magnitude of the field is indicated by the density of the field lines: it's strong near the center where the field lines are close together, and weak farther out, where they are relatively far apart. In truth, the field-line diagram is deceptive, when I draw it on a two-dimensional surface, for the density of lines passing through a circle of radius r is the total number divided by the circumference (nj2nr), which goes like (ljr), not Ojr 2 ). But if you imagine the model in three dimensions (a pincushion with needles sticking out in all directions), then the density of lines is the total number divided by the area of th