FUNDAMENTALS OF ELECTROMAGNETICS: QUASISTATICS AND WAVES
INTRODUCTION 1 PART I THE PHYSICS OF ELECTROMAGNETISM 6 CHAPTER 1 ELECTRIC FORCES AND FIELDS 6 CHAPTER 2 GAUSS S LAW 31 CHAPTER 3 MAGNETIC FORCES AND FIELDS 55 CHAPTER 4 AMP ERE S LAW 81 CHAPTER 5 ELECTROSTATIC POTENTIAL 108 CHAPTER 6 ELECTROMAGNETIC INDUCTION 132 CHAPTER 7 MAXWELL S TRIUMPH 159 PART II THE MATHEMATICS OF ELECTROMAGNETISM 185 CHAPTER 8 MATHEMATICAL TOOLKIT 185 SOLUTIONS TO EXERCISES 245 CONTENTS INTRODUCTION FOUNDERS OF ELECTROMAGNETISM MICHAEL FARADAY CAME FROM A POOR FAMILY AT TIMES HIS WEEKLY DIET CONSISTED OF NO MORE THAN A LOAF OF BREAD HE WAS APPRENTICED TO A BOOKBINDER AND BECAME AWARE OF SCIENCE BY READING BOOKS AND ATTENDING PUBLIC LECTURES AT THE ROYAL INSTITUTION HE SECURED HIS RST JOB IN SCIENCE BY TAKING LECTURE NOTES BINDING THEM AND PRESENTING THEM TO THE LECTURER HUMPHRY DAVY WHO TOOK HIM ON AS AN ASSISTANT BUT FARADAY WAS ENDLESSLY INQUISITIVE AND SOON EXPLORED ON HIS OWN INITIATIVE BY THE EARLY 1830S HE HAD DISCOVERED ELECTROMAGNETIC INDUCTION BUILT THE RST TRANSFORMER ESTABLISHED THE PRINCIPLE OF THE ELECTRIC MOTOR AND PRODUCED THE RST CONTINUOUS ELECTRICAL GENERATOR IN ORDER TO ACCOUNT FOR HIS OBSERVATIONS FARADAY INTRODUCED THE CONCEPT OF A ELD FIGURE 1 MICHAEL FARADAY 1791 1867 JAMES CLERK MAXWELL PRODUCED A UNI ED THEORY OF THE ELECTROMAGNETIC ELD AND USED IT TO SHOW THAT LIGHT IS A TYPE OF ELECTROMAGNETIC WAVE THIS PREDICTION DATES FROM THE EARLY 1860S WHEN MAXWELL WAS AT KING S COLLEGE LONDON SHORTLY AFTERWARDS MAXWELL DECIDED TO RETIRE TO HIS FAMILY ESTATE IN GALLOWAY IN ORDER TO CONCENTRATE ON RESEARCH UNHINDERED BY OTHER DUTIES HE WAS LURED OUT OF RETIREMENT IN 1871 WHEN HE BECAME THE RST PROFESSOR OF EXPERIMENTAL PHYSICS IN THE CAVENDISH LABORATORY CAMBRIDGE GIVEN MAXWELL S PRESENT STATUS AS ONE OF THE GREATEST OF ALL PHYSICISTS IT IS ASTONISHING TO LEARN THAT HE WAS THE THIRD CHOICE FOR THIS JOB INCIDENTALLY CLERK MAXWELL WITHOUT A HYPHEN IS A SURNAME MAXWELL S FATHER JOHN CLERK SIMPLY APPENDED MAXWELL TO HIS OWN NAME IN ORDER TO SMOOTH A LEGAL TRANSACTION FIGURE 2 JAMES CLERK MAXWELL 1831 1879 1 INTRODUCTION WHY STUDY ELECTROMAGNETISM ELECTROMAGNETISM WAS SLOW TO GET GOING ABOUT A CENTURY AFTER NEWTON PUBLISHED THE LAW OF UNIVERSAL GRAVITATION COULOMB DISCOVERED THE LAW OF ELECTROSTATIC FORCE BUT IT TOOK ANOTHER CENTURY BEFORE ALL THE LAWS OF ELECTROMAGNETISM WERE ESTABLISHED IN RETROSPECT THIS SLOW DEVELOPMENT OF ELECTROMAGNETISM IS NOT SURPRISING NEWTON PROVIDED A MECHANICAL WORLD VIEW WHICH WORKED MARVELLOUSLY WELL IN THE CONTEXT OF GRAVITY BUT WHICH STRUGGLED TO EXPLAIN THE MORE SUBTLE EFFECTS OF ELECTRICITY AND MAGNETISM IT TURNED OUT THAT A KEY IDEA WAS MISSING THAT OF A ELD THROUGH LONG EXPERIENCE IN THE LABORATORY MICHAEL FARADAY CONVINCED HIMSELF THAT MAGNETIC ELDS FORM PART OF THE FABRIC OF THE WORLD EVERY BIT AS REAL AS PARTICLES HE EXPLAINED THE RESULTS OF HIS EXPERIMENTS IN TERMS OF ELDS DRAWING DIAGRAMS TO HELP HIM VISUALIZE THEM BUT THE MATHEMATICS NEEDED TO DESCRIBE ELDS WAS A SIGNI CANT HURDLE WHICH FARADAY WAS ILL EQUIPPED TO SURMOUNT THE FULL POWER OF FARADAY S ELD CONCEPT WAS REVEALED BY JAMES CLERK MAXWELL WHO DELIBERATELY CHOSE TO READ FARADAY S EXPERIMENTAL RESEARCHES IN ELECTRICITY BEFORE APPLYING ANY MATHEMATICS TO THE SUBJECT IN 1864 MAXWELL DISTILLED ALL THE KNOWN PROPERTIES OF ELECTRIC AND MAGNETIC ELDS INTO A SET OF FOUR EQUATIONS KNOWN AS MAXWELL S EQUATIONS THESE EQUATIONS LED TO ONE OF THE GREATEST DISCOVERIES IN SCIENCE THE REALIZATION THAT LIGHT IS AN ELECTROMAGNETIC WAVE THE EARTH IS BATHED IN LIGHT FROM THE SUN THE PROPAGATION OF THIS LIGHT THE BLUENESS OF THE SKY THE SPARKLING OF OCEAN SPRAY THE COLOURS OF RAINBOWS AND BUTTER Y WINGS CAN ALL BE EXPLAINED BY MAXWELL S EQUATIONS IT IS NOT EVERY DAY THAT A WHOLE BRANCH OF PHYSICS IS MERGED WITH ANOTHER YET THIS IS WHAT MAXWELL ACHIEVED OPTICS BECAME A BRANCH OF ELECTROMAGNETISM BEFORE LONG IT BECAME CLEAR THAT ELECTROMAGNETIC WAVES ALSO EXIST BEYOND THE NARROW BAND DETECTED BY OUR EYES RADIO WAVES AND X RAYS ARE OTHER TYPES OF ELECTROMAGNETIC WAVE IMPORTANT IN ENGINEERING MEDICINE AND ASTRONOMY NO WONDER THAT MAXWELL S PUBLICATION OF THE LAWS OF ELECTROMAGNETISM HAS BEEN DESCRIBED AS THE MOST SIGNI CANT EVENT OF THE NINETEENTH CENTURY THESE LAWS ARE THE MAIN SUBJECT OF THIS BOOK THERE ARE MANY GOOD REASONS FOR STUDYING ELECTROMAGNETISM AND THEREFORE MAXWELL S EQUATIONS IN DEPTH YOUR MOTIVATION MIGHT COME FROM A DESIRE TO UNDERSTAND FUNDAMENTAL PHYSICS APPLICATIONS IN SCIENCE APPLICATIONS IN TECHNOLOGY AND MEDICINE OR YOU MIGHT BE INTERESTED IN SEEING HOW MATHEMATICS IS USED TO EXPLAIN PHYSICAL EFFECTS LET ME SAY A FEW WORDS ABOUT EACH OF THESE MOTIVES 1 FUNDAMENTAL PHYSICS ELECTROMAGNETISM IS ONE OF FOUR FUNDAMENTAL FORCES OF NATURE THE OTHERS BEING GRAVITY THE STRONG NUCLEAR FORCE AND THE WEAK NUCLEAR FORCE OF THESE FORCES ELECTROMAGNETISM IS THE ONE THAT PHYSICISTS UNDERSTAND BEST NUCLEAR FORCES ARE COMPLICATED AND GRAVITY IS HARD TO RECONCILE WITH QUANTUM MECHANICS BUT PHYSICISTS ARE VERY CON DENT THAT THEY KNOW HOW ELECTROMAGNETISM WORKS PART OF THE IMPORTANCE OF ELECTROMAGNETISM STEMS FROM THE FACT THAT IT IS A THEORY OF ELDS ELECTROMAGNETISM WAS THE RST AND REMAINS THE MOST FAMILIAR THEORY OF ELDS MOST FUNDAMENTAL PHYSICS IS ABOUT ELDS OF VARIOUS KINDS SO ELECTROMAGNETISM HAS PROVIDED A SORT OF TEMPLATE FROM WHICH OTHER MORE ELABORATE THEORIES HAVE GROWN COMBINING ELECTROMAGNETISM WITH A QUANTUM THEORY OF ELDS RICHARD FEYNMAN AND OTHERS DEVELOPED QUANTUM ELECTRODYNAMICS THIS IS THE MOST PRECISELY 2 2 CON RMED PHYSICAL THEORY EVER DEVISED MEASUREMENT AND THEORY AGREE TO AT LEAST 16 SIGNI CANT GURES THE SUCCESS OF QUANTUM ELECTRODYNAMICS HAS INSPIRED THE SEARCH FOR ANALOGOUS THEORIES OF THE OTHER THREE FORCES WITH THE ULTIMATE GOAL OF OBTAINING A THEORY OF EVERYTHING A SUPER UNI ED THEORY OF ALL FOUR FUNDAMENTAL FORCES SUCH AMBITIONS LIE BEYOND THE SCOPE OF THIS BOOK AND WE WILL NOT USE QUANTUM THEORY AT ALL BUT IT IS INTERESTING TO NOTE THAT CLASSICAL ELECTROMAGNETISM HAS NOURISHED ALL THESE DEVELOPMENTS ELECTROMAGNETISM HAS LED TO OTHER PROFOUND IDEAS TOO FOR EXAMPLE EINSTEIN WAS UNCONVINCED BY THE EXPLANATIONS OF ELECTROMAGNETIC INDUCTION FOUND IN THE TEXTBOOKS OF HIS DAY SPECIAL RELATIVITY WAS THE FRUIT OF THIS SCEPTICISM ELECTROMAGNETISM CONTINUES TO SUGGEST CONCEPTS THAT MIGHT BE USEFUL IN OTHER AREAS OF PHYSICS FOR EXAMPLE ASTRONOMERS ARE CURRENTLY TRYING TO DETECT A GRAVIMAGNETIC FORCE THIS IS A MODI CATION OF GRAVITY THAT MIGHT ARISE WHEN BODIES MOVE VERY RAPIDLY IT CAN BE THOUGHT OF AS A GRAVITATIONAL ANALOGUE OF THE MAGNETIC FORCE APPLICATIONS IN SCIENCE THE FORCES THAT BIND ATOMS TOGETHER IN MOLECULES AND SOLIDS ARE ELECTROMAGNETIC IN ORIGIN THIS MEANS THAT OTHER SCIENCES RAISE QUESTIONS THAT REQUIRE DEEP UNDERSTANDING OF ELECTROMAGNETISM IN BIOLOGY FOR EXAMPLE THE MEMBRANE OF A RESTING NERVE CELL HAS A NEGATIVELY CHARGED INNER SURFACE AND A POSITIVELY CHARGED OUTER SURFACE NERVE IMPULSES CONSIST OF LOCALIZED REVERSALS OF THIS POLARITY WHICH SWEEP ALONG THE CELL AT SPEEDS OF UP TO 100 M S 1 FIGURE 3 AS YOU READ THIS PAGE ELECTROMAGNETIC WAVES ENTER YOUR EYE EVEN THE TRANSPARENT OUTER COATING OF THE EYE THE CORNEA IS A WONDER OF NATURE IT IS CONSTRUCTED FROM BRES OF COLLAGEN THE SAME MATERIAL THAT FORMS TENDONS YET IT IS ALMOST PERFECTLY TRANSPARENT TO UNDERSTAND THE ORIGIN OF THIS TRANSPARENCY WE NEED TO KNOW HOW ELECTROMAGNETIC WAVES PROPAGATE THROUGH MEDIA CONTAINING BRES A TOPIC THAT WILL BE DISCUSSED IN BOOK 3 OF THIS SERIES LIGHT FOCUSED ON THE BACK OF THE EYE STIMULATES NERVE IMPULSES WHICH PROPAGATE TO THE VISUAL CORTEX OF THE BRAIN AND IN A COMPLICATED AND POORLY UNDERSTOOD WAY STIMULATE OTHER AREAS OF THE BRAIN ALL OF THIS ACTIVITY INVOLVES ELECTRICAL SIGNALS SUBJECT TO THE LAWS OF ELECTROMAGNETISM STEP OUT OF THE HOUSE AND LOOK UPWARDS WITH CLEAR SKIES YOU SHOULD SEE BLUE SKY OR STARS DEPENDING ON THE TIME OF DAY THE BLUENESS OF THE SKY IS DUE TO THE SCATTERING OF SUNLIGHT IN THE ATMOSPHERE THE STARS REVEAL THEIR PRESENCE BY EMITTING ELECTROMAGNETIC WAVES WHICH PROPAGATE ACROSS VAST DISTANCES OF ALMOST EMPTY SPACE OF COURSE THE WHOLE SUBJECT OF ASTRONOMY HAS PROGRESSED LARGELY THANKS TO OUR ABILITY TO DETECT AND ANALYZE ELECTROMAGNETIC WAVES OF VARIOUS INTRODUCTION FIGURE 3 A NERVE IMPULSE IS A REVERSAL OF POLARITY ACROSS THE MEMBRANE OF A NERVE CELL IT SWEEPS ALONG THE CELL ACCOMPANIED BY A BLIP IN THE VOLTAGE DIFFERENCE ACROSS THE MEMBRANE AND SUSTAINED BY OWS OF SODIUM IONS NA AND POTASSIUM IONS K THROUGH VOLTAGE SENSITIVE CHANNELS IN THE MEMBRANE 3