Classical Electromagnetism - Fitzpatrick.pdf - pomieszane ENG - bartus24

PHY 352K

CLASSICAL ELECTROMAGNETISM

AN UPPER-DIVISION UNDERGRADUATE LEVEL LECTURE COURSE GIVEN BY

RICHARD FITZPATRICK

ASSISTANT PROFESSOR OF PHYSICS

THE UNIVERSITY OF TEXAS AT AUSTIN

FALL 1997

EMAIL: RFITZP@FARSIDE.PH.UTEXAS.EDU, TEL.: 512-471-9439

HOMEPAGE: HTTP://FARSIDE.PH.UTEXAS.EDU/EM1/EM.HTML

1 INTRODUCTION

1.1 MAJOR SOURCES

THE TEXTBOOKS WHICH I HAVE CONSULTED MOST FREQUENTLY WHILST DEVELOPING COURSE

MATERIAL ARE:

INTRODUCTION TO ELECTRODYNAMICS: D.J. GRI±THS, 2ND EDITION (PRENTICE HALL,

ENGLEWOOD CLI®S NJ, 1989).

ELECTROMAGNETISM: I.S. GRANT AND W.R. PHILLIPS (JOHN WILEY & SONS, CHICH-

ESTER, 1975).

CLASSICAL ELECTROMAGNETIC RADIATION: M.A. HEALD AND J.B. MARION, 3RD EDI-

TION (SAUNDERS COLLEGE PUBLISHING, FORT WORTH TX, 1995).

THE FEYNMAN LECTURES ON PHYSICS: R.P. FEYNMAN, R.B. LEIGHTON, AND M.

SANDS, VOL. II (ADDISON-WESLEY, READING MA, 1964).

1.2 OUTLINE OF COURSE

THE MAIN TOPIC OF THIS COURSE IS MAXWELL'S EQUATIONS. THESE ARE A SET OF EIGHT

¯RST ORDER PARTIAL DI®ERENTIAL EQUATIONS WHICH CONSTITUTE A COMPLETE DESCRIPTION

OF ELECTRIC AND MAGNETIC PHENOMENA. TO BE MORE EXACT, MAXWELL'S EQUATIONS CON-

STITUTE A COMPLETE DESCRIPTION OF THE BEHAVIOUR OF ELECTRIC AND MAGNETIC ¯ELDS.

YOU ARE ALL, NO DOUBT, QUITE FAMILIAR WITH THE CONCEPTS OF ELECTRIC AND MAGNETIC

¯ELDS, BUT I WONDER HOW MANY OF YOU CAN ANSWER THE FOLLOWING QUESTION. \DO

ELECTRIC AND MAGNETIC ¯ELDS HAVE A REAL PHYSICAL EXISTENCE OR ARE THEY JUST THE-

ORETICAL CONSTRUCTS WHICH WE USE TO CALCULATE THE ELECTRIC AND MAGNETIC FORCES

EXERTED BY CHARGED PARTICLES ON ONE ANOTHER?" IN TRYING TO FORMULATE AN ANSWER

TO THIS QUESTION WE SHALL, HOPEFULLY, COME TO A BETTER UNDERSTANDING OF THE NATURE

OF ELECTRIC AND MAGNETIC ¯ELDS AND THE REASONS WHY IT IS NECESSARY TO USE THESE

CONCEPTS IN ORDER TO FULLY DESCRIBE ELECTRIC AND MAGNETIC PHENOMENA.

AT ANY GIVEN POINT IN SPACE AN ELECTRIC OR MAGNETIC ¯ELD POSSESSES TWO PROPER-

TIES, A MAGNITUDE AND A DIRECTION. IN GENERAL, THESE PROPERTIES VARY FROM POINT TO

POINT. IT IS CONVENTIONAL TO REPRESENT SUCH A ¯ELD IN TERMS OF ITS COMPONENTS MEA-

SURED WITH RESPECT TO SOME CONVENIENTLY CHOSEN SET OF CARTESIAN AXES (I.E., X, Y,

AND Z AXES). OF COURSE, THE ORIENTATION OF THESE AXES IS ARBITRARY. IN OTHER WORDS,

DI®ERENT OBSERVERS MAY WELL CHOOSE DI®ERENT COORDINATE AXES TO DESCRIBE THE SAME

¯ELD. CONSEQUENTLY, ELECTRIC AND MAGNETIC ¯ELDS MAY HAVE DI®ERENT COMPONENTS

ACCORDING TO DI®ERENT OBSERVERS. WE CAN SEE THAT ANY DESCRIPTION OF ELECTRIC AND

MAGNETIC ¯ELDS IS GOING TO DEPEND ON TWO DI®ERENT THINGS. FIRSTLY, THE NATURE OF

THE ¯ELDS THEMSELVES AND, SECONDLY, OUR ARBITRARY CHOICE OF THE COORDINATE AXES

WITHRESPECTTOWHICHWEMEASURETHESE¯ELDS. LIKEWISE, MAXWELL'SEQUATIONS, THE

EQUATIONSWHICHDESCRIBETHEBEHAVIOUROFELECTRICANDMAGNETIC¯ELDS, DEPENDON

TWO DI®ERENT THINGS. FIRSTLY, THE FUNDAMENTAL LAWS OF PHYSICS WHICH GOVERN THE

BEHAVIOUR OF ELECTRIC AND MAGNETIC ¯ELDS AND, SECONDLY, OUR ARBITRARY CHOICE OF

COORDINATE AXES. IT WOULD BE NICE IF WE COULD EASILY DISTINGUISH THOSE ELEMENTS OF

MAXWELL'S EQUATIONS WHICH DEPEND ON PHYSICS FROM THOSE WHICH ONLY DEPEND ON

COORDINATES. IN FACT, WE CAN ACHIEVE THIS USING WHAT MATHEMATICIANS CALL VECTOR

¯ELD THEORY. THIS ENABLES US TO WRITE MAXWELL'S EQUATIONS IN A MANNER WHICH

IS COMPLETELY INDEPENDENT OF OUR CHOICE OF COORDINATE AXES. AS AN ADDED BONUS,

MAXWELL'S EQUATIONS LOOK A LOT SIMPLER WHEN WRITTEN IN A COORDINATE FREE MANNER.

IN FACT, INSTEAD OF EIGHT ¯RST ORDER PARTIAL DI®ERENTIAL EQUATIONS, WE ONLY REQUIRE

FOUR SUCH EQUATIONS USING VECTOR ¯ELD THEORY. IT SHOULD BE CLEAR, BY NOW, THAT WE

ARE GOING TO BE USING A LOT OF VECTOR ¯ELD THEORY IN THIS COURSE. IN ORDER TO HELP

YOU WITH THIS, I HAVE DECIDED TO DEVOTE THE ¯RST FEW LECTURES OF THIS COURSE TO A

REVIEW OF THE BASIC RESULTS OF VECTOR ¯ELD THEORY. I KNOW THAT MOST OF YOU HAVE

ALREADY TAKEN A COURSE ON THIS TOPIC. HOWEVER, THAT COURSE WAS TAUGHT BY SOME-

BODY FROM THE MATHEMATICS DEPARTMENT. MATHEMATICIANS HAVE THEIR OWN AGENDA

WHEN IT COMES TO DISCUSSING VECTORS. THEY LIKE TO THINK OF VECTOR OPERATIONS AS A

SORT OF ALGEBRA WHICH TAKES PLACE IN AN ABSTRACT \VECTOR SPACE." THIS IS ALL VERY

WELL, BUT IT IS NOT ALWAYS PARTICULARLY USEFUL. SO, WHEN I COME TO REVIEW THIS TOPIC

I SHALL EMPHASIZE THOSE ASPECTS OF VECTORS WHICH MAKE THEM OF PARTICULAR INTEREST

TO PHYSICISTS; NAMELY, THE FACT THAT WE CAN USE THEM TO WRITE THE LAWS OF PHYSICS

IN A COORDINATE FREE FASHION.

TRADITIONALLY, AN UPPER DIVISION COLLEGE LEVEL COURSE ON ELECTROMAGNETIC THEORY

IS ORGANIZED AS FOLLOWS. FIRST, THERE IS A LENGTHY DISCUSSION OF ELECTROSTATICS (I.E.,

ELECTRIC ¯ELDS GENERATED BY STATIONARY CHARGE DISTRIBUTIONS) AND ALL OF ITS APPLICA-

TIONS. NEXT,THEREISADISCUSSIONOFMAGNETOSTATICS(I.E.,MAGNETIC¯ELDSGENERATED

BY STEADY CURRENT DISTRIBUTIONS) AND ALL OF ITS APPLICATIONS. AT THIS POINT, THERE IS

USUALLYSOMEMENTIONOFTHEINTERACTIONOFSTEADYELECTRICANDMAGNETIC¯ELDSWITH

MATTER. NEXT, THERE IS AN INVESTIGATION OF INDUCTION (I.E., ELECTRIC AND MAGNETIC

¯ELDS GENERATED BY TIME VARYING MAGNETIC AND ELECTRIC ¯ELDS, RESPECTIVELY) AND ITS

MANY APPLICATIONS. ONLY AT THIS RATHER LATE STAGE IN THE COURSE IS IT POSSIBLE TO

WRITE DOWN THE FULL SET OF MAXWELL'S EQUATIONS. THE COURSE ENDS WITH A DISCUSSION

OF ELECTROMAGNETIC WAVES.

THE ORGANIZATION OF MY COURSE IS SOMEWHAT DI®ERENT TO THAT DESCRIBED ABOVE.

THERE ARE TWO REASONS FOR THIS. FIRSTLY, I DO NOT THINK THAT THE TRADITIONAL COURSE

EMPHASIZESMAXWELL'SEQUATIONSSU±CIENTLY. AFTERALL,THEYAREONLYWRITTENDOWN

IN THEIR FULL GLORY MORE THAN THREE QUARTERS OF THE WAY THROUGH THE COURSE. I ¯ND

THISAPROBLEMBECAUSE,ASIHAVEALREADYMENTIONED,ITHINKTHATMAXWELL'SEQUA-

TIONS SHOULD BE THE PRINCIPAL TOPIC OF AN UPPER DIVISION COURSE ON ELECTROMAGNETIC

THEORY. SECONDLY, IN THE TRADITIONAL COURSE IT IS VERY EASY FOR THE LECTURER TO FALL

INTOTHETRAPOFDWELLINGTOOLONGONTHERELATIVELYUNINTERESTINGSUBJECTMATTERAT

THEBEGINNINGOFTHECOURSE(I.E.,ELECTROSTATICSANDMAGNETOSTATICS)ATTHEEXPENSE

OF THE REALLY INTERESTING MATERIAL TOWARDS THE END OF THE COURSE (I.E., INDUCTION,

MAXWELL'S EQUATIONS, AND ELECTROMAGNETIC WAVES). I VIVIDLY REMEMBER THAT THIS

IS EXACTLY WHAT HAPPENED WHEN I TOOK THIS COURSE AS AN UNDERGRADUATE. I WAS

VERY DISAPPOINTED! I HAD BEEN LOOKING FORWARD TO HEARING ALL ABOUT MAXWELL'S

EQUATIONS AND ELECTROMAGNETIC WAVES, AND WE WERE ONLY ABLE TO COVER THESE TOPICS

IN A HURRIED AND RATHER CURSORY FASHION BECAUSE THE LECTURER RAN OUT OF TIME AT

THE END OF THE COURSE.

MY COURSE IS ORGANIZED AS FOLLOWS. THE ¯RST SECTION IS DEVOTED TO MAXWELL'S

EQUATIONS. I SHALL DESCRIBE HOW MAXWELL'S EQUATIONS CAN BE DERIVED FROM THE

FAMILIAR LAWS OF PHYSICS WHICH GOVERN ELECTRIC AND MAGNETIC PHENOMENA, SUCH

AS COULOMB'S LAW AND FARADAY'S LAW. NEXT, I SHALL SHOW THAT MAXWELL'S EQUA-

TIONS POSSESS PROPAGATING WAVE LIKE SOLUTIONS, CALLED ELECTROMAGNETIC WAVES, AND,

FURTHERMORE, THAT LIGHT, RADIO WAVES, AND X-RAYS, ARE ALL DI®ERENT TYPES OF ELEC-

TROMAGNETIC WAVE. FINALLY, I SHALL DEMONSTRATE THAT IT IS POSSIBLE TO WRITE DOWN

A FORMAL SOLUTION TO MAXWELL'S EQUATIONS, GIVEN A SENSIBLE CHOICE OF BOUNDARY

CONDITIONS. THE SECOND SECTION OF MY COURSE IS DEVOTED TO THE APPLICATIONS OF

MAXWELL'S EQUATIONS. WE SHALL INVESTIGATE ELECTROSTATIC ¯ELDS GENERATED BY STA-

TIONARY CHARGE DISTRIBUTIONS, CONDUCTORS, RESISTORS, CAPACITORS, INDUCTORS, THE EN-

ERGY AND MOMENTUM CARRIED BY ELECTROMAGNETIC ¯ELDS, AND THE GENERATION AND

TRANSMISSION OF ELECTROMAGNETIC RADIATION. THIS ARRANGEMENT OF MATERIAL GIVES

THE PROPER EMPHASIS TO MAXWELL'S EQUATIONS. IT ALSO REACHES THE RIGHT BALANCE

BETWEEN THE INTERESTING AND THE MORE MUNDANE ASPECTS OF ELECTROMAGNETIC THE-

ORY. FINALLY, IT ENSURES THAT EVEN IF I DO RUN OUT OF TIME TOWARDS THE END OF THE

COURSE I SHALL STILL HAVE COVERED MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES

IN ADEQUATE DETAIL.

ONETOPICWHICHIAMNOTGOINGTOMENTIONATALLINMYCOURSEISTHEINTERACTION

OF ELECTROMAGNETIC ¯ELDS WITH MATTER. IT IS IMPOSSIBLE TO DO JUSTICE TO THIS TOPIC

AT THE COLLEGE LEVEL, WHICH IS WHY I ALWAYS PREFER TO LEAVE IT TO GRADUATE SCHOOL.

2 VECTOR ASSAULT COURSE

2.1 VECTOR ALGEBRA

INAPPLIEDMATHEMATICSPHYSICALQUANTITIESAREREPRESENTEDBYTWODISTINCTCLASSES

OF OBJECTS. SOME QUANTITIES, DENOTED SCALARS, ARE REPRESENTED BY REAL NUMBERS.

PQ. NOTE

OTHERS, DENOTED VECTORS, AREREPRESENTEDBYDIRECTEDLINEELEMENTS: E.G.

THAT LINE ELEMENTS (AND THEREFORE VECTORS) ARE MOVABLE AND DO NOT CARRY INTRINSIC

POSITION INFORMATION. IN FACT, VECTORS JUST POSSESS A MAGNITUDE AND A DIRECTION,

WHEREAS SCALARS POSSESS A MAGNITUDE BUT NO DIRECTION. BY CONVENTION, VECTOR

QUANTITIESAREDENOTEDBYBOLD-FACEDCHARACTERS(E.G. A)INTYPESETDOCUMENTSAND

BY UNDERLINED CHARACTERS (E.G. A) IN LONG-HAND. VECTORS CAN BE ADDED TOGETHER

BUT THE SAME UNITS MUST BE USED, LIKE IN SCALAR ADDITION. VECTOR ADDITION CAN BE

PR=

PQ +

QR. SUPPOSETHAT A ´

PQ´

SR,

REPRESENTEDUSINGAPARALLELOGRAM:

QR´ !

PS, AND C ´ !

PR. IT IS CLEAR FROM THE DIAGRAM THAT VECTOR ADDITION IS

B ´ !

Classical Electromagnetism - Fitzpatrick.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: