A guide to physics problems

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Foreword
For many graduate students of physics the written qualifying examina-
tion is the last and one of the most important of the hundreds of grueling
examinations that they have had to take in their career. I remember vividly
my own experience in 1947 at the University of Chicago. After the quali-
fying examination, I knew I was finally free from all future examinations,
and that generated a wonderful feeling of liberation and relief.
Be that as it may, the written qualifying examination does serve a useful
purpose, both for the faculty and for the students themselves. That is why
so many universities give these exams year after year in all parts of the
world.
Sidney Cahn and Boris Nadgorny have energetically collected and pre-
sented solutions to about 140 problems from the exams at many universities
in the United States and one university in Russia, the Moscow Institute
of Physics and Technology. Some of the problems are quite easy, others
quite tough; some are routine, others ingenious. Sampling them I am re-
minded of the tripos questions of Cambridge University that I had spent so
many hours on when I was an undergraduate student in China during the
years 1938–1942, studying such books as Whittaker’s Analytical Dynamics,
Whittaker and Watson’s Modern Analysis, Hardy’s Pure Mathematics, and
Jeans’ Electricity and Magnetism.
It is perhaps interesting to the readers of this volume to note that the
famous Stokes’ theorem, so important to modern differential geometry and
to physics, first appeared in public as problem No. 8 of the Smith Prize
Examination of 1854. Stokes was the examiner and Maxwell was one of the
takers of the examination. That Maxwell was impressed with this theorem,
and made extensive use of it in 1856 in t h e first of his epoch-making series
v
vi
FOREWORD
of papers that led to Maxwell’s equations, is obvious from his papers and
from his A Treatise on Electricity and Magnetism (1873). Maybe a hundred
years from now somebody will remember one of the problems of the present
collection?
C.N. Yang
Stony Brook
Preface
The written qualifying examination, a little publicized requirement of
graduate physics programs in most universities, brings some excitement to
the generally dull life of the graduate student. While undergoing this ordeal
ourselves, we were reminded of the initiation ceremonies into certain strict
monastic orders, designed to cause the novices enough pain to make them
consider their vocation seriously. However, as the memory of the ghastly
experience grows dim, our attitudes are gradually changing, and we now
may agree that these exams help assure a minimal level of general physics
knowledge necessary for performing successful research. Still, the affair
is rather stressful, sometimes more a test of character than of knowledge
(see Figure P.1). Perhaps it is the veteran’s memory of this searing, yet
formative experience that preserves the Institution of the Qualifying Exam.
Some schools do not have written exams, for instance: Brown, Cal-
Tech, Cornell, Harvard, UT Austin, Univ. of Toronto, Yale. However, the
majority do administer them and do so in a more or less standard form,
though, the level of difficulty of the problems, their style, etc., may differ
substantially from school to school. Our main purpose in publishing this
book — apart from the obvious one to become rich and famous — is to
assemble, as far as possible, a universal set of problems that the graduate
student should be able to solve in order to feel comfortable and confident at
the exam. Some books containing exam problems from particular univer-
sities (Chicago, Berkeley, Princeton
)
have been published; however, this is
the first book to contain problems from different American schools, and for
comparison, problems from Moscow Phys-Tech, one of the leading Russian
universities.
vii
viii
PREFACE
The other goal of the book is much more complicated and only partly
realized: to allow comparison of problems from different schools in terms of
breadth of material, style, difficulty, etc. This would have required analysis
of a greater number of problems than we were able to include, and the
use of approximately the same number of problems from each department
(we had only a few problems from some universities and hundreds from
others). We were much more concerned to present problems that would
cover as much material as possible. We should note in this regard that
the exams with the most difficult problems to solve are not necessarily the
most difficult to pass — that depends on the number of problems that have
to be solved, the amount of time given for each problem, and the way in
which the problems are graded. We have not attempted to present such
information, but we wish to point out that it is an important consideration
in the selection of a graduate school and well worth investigating.
Quite often the written exam consists of two parts: the first part, cover-
ing “fundamental” physics, usually includes classical mechanics, electrody-
namics, quantum mechanics, statistical physics and thermodynamics, and
PREFACE
ix
sometimes special relativity and optics; the second part, containing “mod-
ern” physics, includes nuclear, atomic, elementary particle, and solid state
physics, and sometimes general relativity and astrophysics. The scope and
difficulty of the second part vary too much from school to school to allow
generalization, and we will only deal with the first part. The problems will
appear in two volumes: Part 1 — Mechanics, Relativity, and Electrody-
namics, and Part 2 — Quantum Mechanics and Statistical Physics.
While reviewing the material submitted to us, we were not surprised to
find that often the same problems, maybe in slightly different formulations,
were part of the exams at several schools. For these problems, we have
noted the name of the school whose particular version we solved next to
the name we assigned to the problem, followed by the name or names of
schools whose exams contained variants of the problem. If only part of
the problem was used at a different school, we have indicated which one.
We have also tried to establish a balance between standard problems that
are popular with many physics departments and more original problems,
some of which we believe have never been published. Many of the standard
problems used in the exams have been published previously. In most cases,
though, it is difficult to determine when the problem was first presented;
almost as difficult as it is to track down the origin of a fairy tale. However,
when we could refer to a standard textbook where the problem may be
found, we have done so. Although it may be boring to solve a lot of the
standard problems, it is worthwhile – usually they comprise more than half
of all the problems given in the exams. We have to acknowledge grudgingly
that all errors in the formulation of the problems and solutions are the
sole responsibility of the authors. We have tried to provide solutions that
are as detailed as possible and not skip calculations even if they are not
difficult. We cannot claim that we have the best possible solutions and
inevitably there must be some errors, so we would welcome any comments
or alternative solutions from the reader.
We were encouraged by the response from most of the schools that we
approached, which furnished us with problems for inclusion in this book.
We would like to take this opportunity to thank the Physics Departments at
Boston University
(
Boston
)
, University of Colorado at Boulder
(
Colorado
)
,
Columbia University [Applied Physics]
(
Columbia
)
, University of Mary-
land
(
Maryland
)
, Massachusetts Institute of Technology
(
MIT
),
Univer-
sity of Michigan
(
Michigan
)
, Michigan State University
(
Michigan State
)
,
Michigan Technological University
(
Michigan Tech
)
, Princeton University
(
Princeton
)
, Rutgers University
(
Rutgers
)
, Stanford University
(
Stanford
)
,
State University of New York at Stony Brook
(
Stony Brook
)
, University of
Wisconsin (Wisconsin-Madison
)
. The problems from Moscow Institute of
Physics and Technology
(
Moscow Phys-Tech
)
came from different sources
x
PREFACE
— none from graduate qualifying exams, rather from undergraduate exams,
oral exams, and magazines (Kvant). A few were published before, in a book
containing a lot of interesting problems from Moscow Phys-Tech, but most
were compiled by the authors. We wish to thank Emmanuel I. Rashba, one
of the authors of that book, for his advice. We realize that there are many
schools which are not represented here, and we welcome any submissions
for Part 2 of this project.
It is our pleasure to thank many members of the Department of Physics
at Stony Brook for their encouragement during the writing of this book,
especially Andrew Jackson, Peter Kahn and Gene Sprouse, as well as Kirk
McDonald of Princeton. We are indebted to Chen Ning Yang, who agreed
to write the foreword for this book. We are grateful to: Dmitrii Averin,
Fabian Essler, Gerald Gwinner, Sergey Panitkin, Babak Razzaghe-Ashrafi,
Sergey Shokhor and Henry Silsbee for numerous discussions of problems
and many useful suggestions, and especially to Bas Peeters, who read most
of the manuscript; and to Michael Bershadsky, Claudio Corianò, and Sergey
Tolpygo for contributing some of the problems. One of the authors (B.N.)
wishes to thank the students at Oxford University and Oxford’s Student
Union for their invaluable help without which this book might not have been
written. Finally, we would like to thank Vladimir Gitt and Yair Minsky for
drawing the humorous pictures, and Susan Knapp for typing part of the
manuscript.
Sidney B. Cahn
Boris E. Nadgorny
Stony Brook
Textbooks Used in the
Preparation of This Volume
Chapter 1
An exhaustive bibliography may be found in Goldstein.
1)
2)
3)
4)
5)
6)
Landau, L.D., and Lifshitz, E.M., Mechanics, Volume 1 of Course of
Theoretical Physics, 3rd ed., Elmsford, New York: Pergamon Press,
1976
Goldstein, H., Classical Mechanics, 2nd ed., Reading, MA: Addison-
Wesley, 1981
Barger, V.D., and Olsson, M.G., Classical Mechanics, A Modern Per-
spective, New York: McGraw-Hill, 1973
Routh, E., Dynamics of a System of Rigid Bodies, New York: Dover,
1960
Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed.,
New York: Springer-Verlag, 1978
Landau, L.D., and Lifshitz, E.M., Fluid Mechanics, Volume 6 of
Course of Theoretical Physics, 2nd ed., Elmsford, New York: Perga-
mon Press, 1987
Chapter 2
1)
Taylor, E.F., and Wheeler, J.A., Spacetime Physics, San Francisco,
California: W.H. Freeman and Company, 1966
xi
xii
TEXTBOOKS
2)
3)
4)
Landau, L.D., and Lifshitz, E.M., Classical Theory of Fields, Volume
2 of Course of Theoretical Physics, 4th ed., Elmsford, New York:
Pergamon Press, 1975
Halzen, F., and Martin, A., Quarks and Leptons, New York: John
Wiley & Sons, Inc., 1984
Jackson, J.D., Classical Electrodynamics, New York: John Wiley &
Sons, Inc., 1975
Chapter 3
An exhaustive bibliography may be found in Jackson.
1)
2)
3)
4)
5)
6)
Jackson, J.D., Classical Electrodynamics, New York: John Wiley &
Sons, Inc., 1975
Landau, L.D., and Lifshitz, E.M., Classical Theory of Fields, Volume
2 of Course of Theoretical Physics, 4th ed., Elmsford, New York:
Pergamon Press, 1975
Landau, L.D., Lifshitz, E.M., and
L.P., Electrodynamics
of Continuous Media, Volume 8 of Course of Theoretical Physics, 2nd
ed., Elmsford, New York: Pergamon Press, 1984
Panofsky, W., and Philips, M., Classical Electricity and Magnetism,
2nd ed., Reading, MA: Addison-Wesley, 1962
Marion, J.B., and Heald, M.A., Classical Electromagnetic Radiation,
2nd ed., New York: Academic Press, 1980
Smythe, W.R., Static and Dynamic Electricity, 3rd ed., New York:
Hemisphere Publishing Corp., 1989
Note:
CGS
units
are
uniformly
used
in
Chapter
3 for the
purpose
of
con-
sistency, even if the original problem was given in other units.
PART I: PROBLEMS
l.
Mechanics
1.1.
1.2.
1.3.
1.4.
Falling Chain (MIT, Stanford)
Cat and Mouse Tug of War (Moscow Phys-Tech, MIT)
Cube Bouncing off Wall (Moscow Phys-Tech)
Cue-Struck Billiard Ball (Rutgers, Moscow Phys-Tech, Wisconsin-
Madison (a))
Stability on Rotating Rollers (Princeton)
1.5.
1.6.
1.7.
1.8.
1.9.
Swan and Crawfish (Moscow Phys-Tech)
Mud from Tire (Stony Brook)
Car down Ramp up Loop (Stony Brook)
Pulling Strings (MIT)
Thru-Earth Train (Stony Brook, Boston (a), Wisconsin-
1.10.
Madison (a))
1.11.
1.12.
1.13.
1.14.
1.15.
1.16.
1.17.
1.18.
1.19.
1.20.
String Oscillations (Moscow Phys-Tech)
Hovering Helicopter (Moscow Phys-Tech)
Astronaut Tether (Moscow Phys-Tech, Michigan)
Spiral Orbit (MIT)
Central Force with Origin on Circle (MIT, Michigan State)
Central Force Orbit (Princeton)
Dumbbell Satellite (Maryland, MIT, Michigan State)
Yukawa Force Orbit (Stony Brook)
Particle Colliding with Reflecting Walls (Stanford)
Earth–Comet Encounter (Princeton)
xiii
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3
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