Problems and Solutions on Electromagnetism Major American Universities Ph. D. Qualifying Questions and Solutions Problems and Solutions on Electromagnetism Compiled by: The Physics Coaching Class University of Science and Technology of China Refereed by: Zhao Shu-ping, You Jun-han, Zhu Jun-jie Edited by: Lim Yung-kuo World Scientific Singapore New Jersey London Hong Kong Publrrhed by World Scientific Publi~hingC O Pte Ltd P 0 Box 128, Parrer Road. Singapore 9 12805 USA ofJe Suite IB, 1060 Main Street, Rner Edge, NJ 07661 lily office 57 Sheiton Street, Covent Garden, London WC2H YHE British Library Cataloguing-in-~biirationD ata A catalogue record for ?hi<b ook I\ available from the Bnrtdt Library Fmt published tYY3 Reprinted I9Y4, 1996. l99X, 2000,2(X)2 For p~ototopyino~f rnatertdl in tht5 volume, plca\c pay a copying fee through the ~opyri~ht Clearante Center, Irtc , 222 Rocewood Ilrive, I)anvcr\, MA 01923, USA fn thir case ~ ~ i ~ ~ ito~ s i o n photctcopy 15 not requtred froin the pubfisher iSBN 981-01-0625-9 ISBN 981-02-0626-7 (pbk) Pnnted rn Sing~pore PREFACE This series of physics problems and solutions, which consists of seven - volumes Mechanice, Electromagnetism, Optics, Atomic, Nuclear and Particle Physics, Thermodynamics and Statistical Physics, Quantum M e chanice, Solid State Physics and Relativity, contains a selection of 2550 problems from the graduate school entrance and qualifying examination papers of seven U.S. universities - California University Berkeley Cam- pus, Columbia University, Chicago University, Massachusetts Institute of Technology, New York State University Buffalo Campus, Princeton Uni- veraity, Wisconsin University - as well as the CUSPEA and C. C. Ting’s papers for selection of Chinese students for further studies in U.S.A. and their solutions which represent the effort of more than 70 Chinese physicists plus some 20 more who checked the solutions. The series is remarkable for its comprehensive coverage. In each area the problems span a wide spectrum of topics while many problems overlap several areas, The problems themselves are remarkable for their versatil- ity in applying the physical laws and principles, their uptodate realistic situations, and their scanty demand on mathematical skills. Many of the problems involve order of magnitude calculations which one often requires in an experimental situation for estimating a quantity from a simple model. In short, the exerciees blend together the objectives of enhancement of one’s understanding of the physical principles and ability of practical application. The solutions as presented generally just provide a guidance to solving the problem, rather than step by step manipulation, and leave much to the students to work out for themselves, of whom much is demanded of the basic knowledge in physics. Thus the series would provide an invaluable complement to the textbooks. The present volume for Electromagnetism consists of five parts: elec- trostatics, magnetostatic and quasi-stationary electromagnetic fields, cir- cuit analysis, electromagnetic waves, relativity and particle-field interac- tions, and contains 440 problems. 34 Chinese physicists were involved in the task of preparing and checking the solutions. In editing, no attempt has been made to unify the physical terms, unita and symbols. Rather, they are left to the setters’ and solvers’ own preference so as to reflect the realistic situation of the usage today. Great V vi Preface pains has been taken to trace the logical steps from the first principles to the final solutions, frequently even to the extent of rewriting the entire solution. In addition, a subject index has been included to facilitate the location of topics. These editorial efforts hopefully will enhance the value of the volume to the students and teachers alike. Yung-Kuo Lim Editor INTRODUCTION Solving problems in school work is the exercise of the mind and ex- amination questions are usually picked from the problems in school work. Working out problems is an essential and important aspect of the study of Physics. Major American Universiiy Ph. D. Qualifying Quesiions and Soluiione is a series of books which consists of seven volumes. The subjects of each volume and the respective referees (in parentheses) are as follows: 1. Mechanics (Qiang Yan-qi, Gu En-pu, Cheng Jia-fu, Li Ze-hua, Yang De-tian) 2. Electromagnetism (Zhao Shu-ping, You Jun-han, Zhu Jun-jie) 3. Optics (Bai Gui-ru, Guo Guang-can) 4. Atomic, Nuclear and Particle Physics (Jin Huai-cheng, Yang Bao- zhong, Fan Yang-mei) 6. Thermodynamics and Statistical Physics (Zheng Jiu-ren) 6. Quantum Mechanics (Zhang Yong-de, Zhu Dong-pei, Fan Hong-yi) 7. Solid State Physics, Relativity and Miscellaneous Topics (Zhang Jia-lu, Zhou You-yuan, Zhang Shi-ling) This series covers almost all aspects of University Physics and contains 2550 problems, most of which are solved in detail. The problems have been carefully chosen from 3100 problems, of which some came from the China-U.S. Physics Examination and Application Pro- gram, some were selected from the Ph.D. Qualifying Examination on Ex- perimental High Energy Physics sponsored by Chao Chong Ting. The rest came from the graduate school entrance examination questions of seven world-renowned American universities: Columbia University, University of California at Berkeley, Massachusetts Institute of Technology, University of Wisconsin, University of Chicago, Princeton University and State Univer- sity of New York, Buffalo. In general, examination problems in physics in American universities do not involve too much mathematics; however, they are to a large ex- tent characterized by the following three aspects: some problems involving various frontier subjects and overlapping domains of science are selected by professors directly from their own research work and show a “modern style”. Some problems involve broad fields and require a quick mind to Vii Viii Introduction analyse, while the methods needed for solving the other problems are sim- ple and practical but requires a full “touch of physics”. Indeed, we ven- ture to opine that the problems, as a whole, embody to some extent the characteristics of American science and culture, as well as the philosophy underlying American education. Therefore, we considered it worthwhile to collect and solve these prob- lems and introduce them to students and teachers, even though the effort involved was extremely strenuous. As many as a hundred teachers and graduate students took part in this time-consuming task. A total of 440 problems makes up this volume of five parts: electrostat- ics (loti), magnetostatic and quasi-stationary electromagnetic fields (119), circuit analysis (go), electromagnetic waves (67), and relativity, particle field interactions (56). In scope and depth, most of the problems conform to the undergrad- uate physics syllabi for electromagnetism, circuit analysis and electrody- namics in most universities. However, many of them are rather profound, sophisticated and broad-based. In particular, problems from American uni- versities often fuse fundamental principles with the latest research activities. Thus the problems may help the reader not only to enhance understanding in the basic principles, but also to cultivate the ability of solving practical problems in a realistic environment. International units are used whenever possible, but in order to conform to some of the problems, Gaussian units are also used. This in fact would give the student broader training and wider experience. This volume is the result of collective efforts of 34 physicists involved in working out and checking of the solutions, among them Zheng D-chen, Eu You-qiu, Ning Bo, Zhu Xue-liang, and Zhao Shu-ping.