=00 .,:. PL-MTV nc nrcjj^,(>^j_ LABOP*^^^^ ^ *PPtl60 SCIENCf Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/advancedtheoryofOOfranuoft ADVANCED ELECTRICITY AND MAGNETISM JT^^^ THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS SAN FRANCISCO MACMILLAN & CO., Limited LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO ADVANCED THEORY OF ELECTRICITY AND MAGNETISM A TEXT-BOOK FOR COLLEGES AND TECHNICAL SCHOOLS BY WILLIAM S. FRANKLIN and BARRY MacNUTT ^ Kr.OPERTY OF ELECTRICAL LABORATOPV. ) i FACULTY OF AfPLlED SCIENCE. THE MACMILLAN COMPANY London; Macmillan & Co., Ltd. 1915 A II rights reserved Copyright 1915 By The Macmillan Company Set up and electrotyped. Published February, 1915 PRESS OF THE HEW ERA PRINTING COMPAHY LANCASTER, PA. r PREFACE To make the study of theoretical physics something besides a purely formal mathematical exercise, it is necessary to keep physical things always clearly and vividly in mind. It is com- paratively easy to do this in the study of applied science, but it is extremely difficult to do it in the study of many of the un- familiar topics which are discussed in this book. The character of the treatment in this hook has been determined throughout by the desire to keep the student's mind jammed up tight against physical things! The chapter on potential, for example, is kept within severely concrete limits, and the student who wishes to get an insight into the elementary mathematical theory of potential should read chapter IX of Franklin, Mac- Nutt and Charles* CalciduSj published by the authors, South Bethlehem, Pa., 1913. The authors desire to express their appreciation of the great assistance which Professor R. L. Charles has rendered in the preparation of this book. January 16, 1915. W. S. Franklin Barry MacNutt. TABLE OF CONTENTS PART I Chapter I Pages Elementary Theory of Ferro-Magnetism 1-17 Chapter II Elementary Theory of Electro-Magnetism 18-36 Chapter III Induced Electromotive Force and Inductance .... 37-74 Chapter IV The Magnetic Properties of Iron 75-103 Chapter V Ship's Magnetism and the Compensation on the Compass 104-120 PART II Chapter VI Electric Charge and the Condenser 1 21-136 Chapter VII Theory of the Electric Field 137-164 Chapter VIII Theory of Potential 165-192 Chapter IX Electric Oscillations and Electric Waves 193-273 PART III Chapter X The Election Theory 274-297 vii -j,-^«-5r'°"''. r,»aptiiT» 01 11": r.rj,..ct ADVANCED ELECTRICITY AND MAGNETISM. CHAPTER I. ELEMENTARY THEORY OF MAGNETISM. 1. Ferromagnetism and electromagnetism. — ^There are two groups of magnetic phenomena, namely, (a) the phenomena of ferromagnetism, that is to say, the phenomena which are asso- ciated with magnetized iron and steel, and (b) the phenomena of electromagnetism, that is to say, the magnetic phenomena which are exhibited by the electric current. In developing the theory of magnetism it is best to consider some phases of ferro- magnetism first, because the phenomena of ferromagnetism are more familiar than the phenomena of electromagnetism and because important magnetic measurements are based upon ferromagnetism. The phenomena of* electromagnetism are comparatively ob- scure, and in many cases Imperceptible, except where they are enhanced by the presence of iron. Thus a dynamo or a trans- former would operate if all iron parts were removed, but the effects produced would be in most cases nearly imperceptible. Practically,' the phenomena of ferromagnetism and the phenomena of electromagnetism are inextricably mixed up with each other. 2. Poles of a magnet. — ^The familiar property of a magnet, namely, its attraction for iron, is possessed only by certain parts of the magnet. These parts of a magnet are called the poles of the magnet. For example, the poles of a straight bar-magnet are usually at the ends of the bar. 2 I ADVANCED ELECTRICITY AND MAGNETISM. When a bar-magnet is suspended in a horizontal position by a fine thread, it places itself approximately north and south like a compass needle. The north pointing end of the magnet is called its north pole, and the south pointing end of the magnet is called its south pole. The north poles of two magnets repel each other, the south poles of two magnets repel each other, and the north pole of one magnet attracts the south pole of another magnet; that is to say, like magnetic poles repel each other ^ and unlike magnetic poles attract each other. The mutual force action of two magnets is, in general, resolvable into four parts, namely, the forces with which the respective poles of one magnet attract or repel the respective poles of the other magnet. In the following discussion we consider only the force with which one pole of a magnet acts upon one pole of another magnet y not the forces with which one complete magnet acts on another complete magnet. 3. Distributed poles and concentrated poles. — ^The poles of a bar magnet are always distributed over considerable portions of the bar. This is especially the case with short thick bars. In the case of a long slim bar magnet, however, the poles are ordinarily approximately concentrated at the ends of the bar. The forces of attraction and repulsion of concentrated magnet poles are easily formulated, there- fore the following discussion applies to ideally concentrated poles at the ends of ideally slim bar magnets. N N N N N N N N d S ^ S S S S 4. Definition of unit pole. — Con- sider a large number of pairs of magnets a, b, c, d, etc., as shown in Fig. I , the two magnets of each From such a set it would be possi- * That is, the magnets of each pair are made of identically the same kind of steel, sebjected to the same kind of heat treatment and magnetized by the same means. S S Fig. 1. Pairs of exactly similar magnets. pair being exactly alike.* ELEMENTARY THEORY OF MAGNETISM. 3 ble to select a pair of magnets such that the north pole of one magnet would repel the north pole of the other with a force of one dyne when they (the two north poles) are one centimeter apart; each pole of such a pair is called a unit pole. That is, a unit pole is a pole which will exert a force of one dyne upon another unit pole at a distance of one centimeter. 5. Strength of pole. — Let us choose a slim magnet with unit poles, and let us use one of these poles as a "test pole." Any given pole strength m-^ unit test pole one centimeter / qn dynes / / \< IB dyneei Fig. 2. given magnet pole is said to have more or less strength according as it exerts more or less force on our "test pole" at a given dis- tance. And the force m (in dynes) with which the given pole attracts or repels (or is attracted or repelled by) the unit test pole at a distance of one centimeter is taken as the measure of the strength of the given pole. That is, a given pole has m units of strength when it will exert a force of m dynes on a unit pole at a distance of one centimeter, as indicated in Fig. 2. 6. Attraction and repulsion of magnet poles. — Unlike poles attract and like poles repel each other, as stated in Art. 2. When the two attracting or repelling poles 6 d ne are unit poles their attraction or q^--- -----=0 repulsion is equal to one dyne rw^r'^'^^r"^^^^ when they are one centimeter apart, m'^ 2 units •'-^ and the attraction or repulsion of p- 3^ two poles whose respective strengths are m' and m" is equal to m'm" dynes when the poles are one centimeter apart. One may think of each unit of m' as exerting 4 ADVANCED ELECTRICITY AND MAGNETISM. a force of one dyne on each unit of m" , Thus if w' = 3 units and m" = 2 units, then the force of attraction or repulsion will be six dynes, as indicated in Fig. 3, where each dotted line rep- resents one dyne. 7. Complete expression for the force of attraction or repulsion of two magnet poles. — Coulomb discovered in 1800 that the force of attraction or repulsion of two magnet poles is inversely pro- portional to the square of the distance between them (Coulomb's law). But the force of attraction or repulsion of two magnet poles when they are one centimeter apart is w'w" dynes as explained in Art. 6. Therefore, according to Coulomb's law, the force of attraction or repulsion is — ^ — dynes when the poles are r centimeters apart. That is : in which m' and m" are the respective strengths of two magnet poles, r is their distance apart in centimeters, and F is the force in dynes with which the poles attract or repel each other. Algebraic sign of magnet pole. It is customary to consider a north pole as positive and a south pole as negative. That is, m is positive when it expresses the strength of a north pole and negative when it expresses the strength of a south pole. There- fore, the product m'm" is positive when both poles are north poles or when both poles are south poles, and in this case the force F in equation (i) is a repulsion. The product m'm" is negative when one pole is a north pole and the other pole is a south pole, and in this case the force F in equation (i) is an attraction. Therefore, when F in equation (i) is positive it is a repulsion, and when it is negative it is an attraction. 8. Direction and intensity of a magnetic field at a point. — A magnetic field may be defined as a region in which a suspended magnetic needle tends to point in a definite direction, and the ELEMENTARY THEORY OF MAGNETISM. 5 "north pole" of the needle points* in what is called the direction of the field. Thus the entire region surrounding the earth is a magnetic field; the region surrounding a magnet is a magnetic field ; and the region surround- _ ing a wire In which an electric P^2V current is flowing is a mag- \ netic field. Figure 4 shows a ^\^ compass needle placed at a point p near a large magnet. The dotted arrow shows the direction of the magnetic field at p. Fig-4- ^, , - The dotted line shows the direction in The poles of a magnet are ^hi^h the small magnet at p points. acted upon hy equal and oppo- site forces when the magnet is placed in a magnetic field, (This statement refers to what is called a uniform magnetic field, see Art. II.) large magnet \s\ Fig. Sa, Figure 5a is a photograph of the figure obtained by dusting iron filings on a pane of glass which is placed over two large * If the needle is perfectly balanced. ADVANCED ELECTRICITY AND MAGNETISM magnet poles, a north pole and a south pole, facing each other. The filaments of iron filings show what are called the lines of force of the magnetic field; a line of force being a line drawn so as to be at each point in the direction of the field at that point. The magnetic field in the central region in Fig. 5a is approximately uniform. Figure 5& represents a small magnet held in the approximately uniform magnetic field between two large magnet poles, and the arrows represent the equal and opposite forces which are exerted on the small magnet by the approximately uniform field. N N Fig. 5&. The arrows show the forces which act upon the poles of the small magnet. The force H (in dynes) which a magnetic field exerts upon a unit test pole is used as a measure of the intensity or strength of the magnetic field, and this force -per-unit -pole is hereafter spoken of simply as the intensity or strength of the field. The unit of magnetic field intensity (one dyne per unit pole) is called a gauss » That is, a magnetic field has an intensity of one gauss when it will exert a force of one dyne upon a unit pole. Complete expression for the force exerted on a magnet pole by a magnetic field. — A magnetic field of which the intensity is H gausses exerts a force of H dynes upon a unit pole as above explained, and it exerts a force of mH dynes upon a pole of which the strength is m units. That is: F = mH (i) in which 7^ is the force in dynes which is exerted on a pole of strength w by a field of intensity H, ELEMENTARY THEORY OF MAGNETISM. Uniform and non-uniform fields. — ^A magnetic field is said to be uniform when it has everywhere the same direction and the same intensity, otherwise the field is said to be non-uniform. The earth's magnetic field is sensibly uniform throughout a room. The magnetic field surrounding a magnet is non-uniform. The magnetic field surrounding an electric wire is non-uniform. 9. Direction and intensity of the magnetic field surrounding an " isolated " magnet pole of strength M. — By an "isolated" magnet pole is meant one pole of a very long slim magnet — the other pole being so far away as to be negligible in its action. Fig. 6. Fig. 7. The magnetic field in the neighborhood of an isolated north pole is everywhere directed away from the pole as shown by the radiating straight lines (lines of force, as they are called) in Fig. 6. The magnetic field in the neighborhood of an isolated south pole is everywhere directed towards the pole as indicated in Fig. 7. Consider two magnet poles M and m which are at a distance of r centimeters apart as shown in Fig. 8. The force F with which M repels m is equal to — ^ , according to Art. 7 ; but the force exerted on m can also be expressed as equal to mH where H is the intensity at m of the magnetic field which is due ADVANCED ELECTRICITY AND MAGNETISM. Mth to M. Therefore mH = —j- , whence we have: r H = M (I) in which H is the intensity of the magnetic field produced by the pole ilf at a place which is r centimeters from M. I centimeters 2 F Fig. 8. 10. Action of a magnetic field on iron or steel. — When an iron or steel rod is placed in a magnetic field, the length of the rod being parallel to the direction of the field, the rod becomes a magnet. Thus the iron rod AB m Fig. 9 is magnetized by the field due to the large magnet pole N, the end A becomes a south pole and the end B becomes a north pole. I N Fig. 9. The effect of the magnetic field on the iron rod AB is to magnetize it, the end A becoming a south pole. II. Behavior of a magnet in a magnetic field, (a) Behavior in a uniform field. — The equal and opposite forces which are exerted on the poles of a magnet by a uniform magnetic field tend only to turn the magnet into the direction of the field, the forces do not tend to produce translatory motion of the magnet. Consider a magnet / centimeters long placed in a uniform magnetic field of which the intensity is Jf, the angle between the axis of the