.( ffi LIBRARY OF THE UNIVERSITY OF CALIFORNIA. DEC 9H892 l8g Sd-, s No. IP^@ Vf^Class No. " UNIVERSITY OF CALIFORNIA LIBRARY OF THE UFPflFiTME.NT O c LOWER DIVISIOM Accessions No. ...... Book No WORKS ON ELECTRICITY & MAGNETISM. ELECTRICITY TREATED EXPERIMENTALLY. For the use of Schools and Students. By LTNNJEUS GUMMING, M.A. late Scholar of ' Trinity College, Cambiidge ; Assistant Master in Rugby School. With 242 Illus- trations. Crown 8vo. 4s. Bd. EXERCISES in ELECTKICAL and MAGNETIC MEASUREMENTS, with Answers. By R. E. DAT. 12mo. 3. 6d. A COUESE of LECTURES on ELECTRICITY, delivered before the Society of Arts; Bv GEORGE FORBES, M.A. F.R.S. (L. & E.) F.R.A.S. , M.S.T.E. and E. Assoc. Inst. C.E. With 17 Illustrations. Crown 8vo. 65. The ART of ELECTRO-METALLURGY, including all known Processes of Electro-Deposition. By G. GORE, LL.D. F.R.S. With 56 Woodcuts. Fcp. 8vo. 6s. ELECTRICITY and MAGNETISM. By FLEEMING JENKIN, F.R.SS. (L. & E.) 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LARDEN, MA. n AUTHOR OF ' A SCHOOL COURSE IN HEAT,' IN USE AT RUGBY, CLIFTON, CHELTENHAM, BEDFORD, BIRMINGHAM, KING'S COLLEGE LONDON, AND IN OTHER SCHOOLS AND COLLEGES. NEW EDITION LONDON LONGMANS, GREEN, AND CO. AND NEW YORK : i$ EAST 16* STREET 1891 Ail rights reserved :, r ' _... ^ PREFACE TO THE SECOND EDITION. IN this edition many misprints have been corrected, several sections have been revised and improved, and some additions made on the subject of modern measuring instruments. In answer to certain criticisms the author would add, firstly, that the book is not intended to be a practical companion to the laboratory, but rather to indicate the amount of theory that it is advisable to master before entering upon technical work ; secondly, that it is assumed that the student can see and examine the instruments referred to, the descriptions being intended merely to enable the learner to understand at once the principle of each instrument that comes under his notice. The writer wishes to acknowledge his obligations to Dr. A. H. Fison for help most kindly given with respect to corrections and additions for this edition. W. L. August 1888. 6 >*< * i" t\ f ?o^y PREFACE. IN this Course the writer aims at giving a sound though elementary knowledge of the modern science of Electricity. With a view to rendering the book suitable for use in public schools, it has been thought better to assume no more mathematical knowledge than is usually possessed by the higher boys in a classical school. For reasons of space, no attempt has been made to treat the history of the subject at all completely. Names and dates may be occasionally mentioned ; but for information on this head the reader can consult works in which the historical sequence of dis- covery has been carefully traced. In that part of the subject in which an elementary knowledge of Chemistry is demanded, the student is referred to the elementary text books of Roscoe and of others, in which he can easily read up to the desired standard. But with respect to such knowledge of systems of units and of mechanical principles as is essential for the proper understanding of the present Course, the writer conceives that the case is different. Believing it to be far from easy for the student to collect from various sources just the kind of mechanical knowledge required for the present purpose, the writer had originally included in the Course all the mechanics needed. He has, however, since been obliged, for reasons of space, to curtail ; and for the most part to indicate merely the exact amount of knowledge required. With regard to diagrams, it may be remarked that there have vni PREFACE to be considered two classes of drawings. There are the simple diagrams illustrative of principles, and there are the complete figures representative of actual pieces of apparatus. After some deliberation the writer decided to draw the former specially for the Course ; but the latter have, for obvious reasons, been freely borrowed from other works, wherever, through the courtesy of the publishers, this was permitted. For such use of diagrams the writer is chiefly indebted to Messrs. Longmans, Green, & Co.; and also to Messrs. Elliott, to the publishers of * Nature,' and to others. At the end of the book will be found questions and numerical examples ; these are intended both to serve as an exercise, and also to some extent to supplement the teaching of the text. Some few of the numerical exercises have, with the kind permission of the publishers, been borrowed. It would be impossible for the writer to acknowledge in detail all that is owed to other books, papers, and lectures. But he would here express his thanks to Professor Tait, Professor D. E. Hughes, and Professor Crookes, for most kindly answering ques- tions and giving information on certain points. Also to Mr. Lupton, of Harrow, for kindly giving permission to use some tables given in his * Numerical Tables.' But especially would he acknowledge his obligations to his friends Professor W. H. Heaton, of Nottingham, who most kindly looked over the MSS. and the proofs of the first ten chapters ; and Professor W. N. Stocker, of Cooper's Hill, who, with like kind- ness, looked over the MSS. of the remaining fifteen chapters. To both of these the writer is indebted for many valuable suggestions and criticisms. Also to his friend and colleague, Mr. A. S. Davis, who undertook the work of looking over the proofs of the last fifteen chapters. CHELTENHAM COLLEGE : June 1887. REFERENCES. The writer here gives some references to works which the student may consult if he desires to go deeper into the subject. He has thought that it may be useful to some if he gives also references to various matters discussed in * Nature/ volumes of this paper being found in most libraries that are even partly scientific. The list that follows is, however, very far from being complete. I. Magnetism. Lloyd's Magnetism. [Longmans, Green & Co.] Airy's Magnetism. [Macmillan.] Admiralty Manual of Scientific Enquiry. [Murray.] Elementary Manual for the Deviation of the Compass, by F. J. Evans. Theory of Electricity, by L. Cumming. [Macmillan.] Electricity and Magnetism, by Mascart and Joubert ; trans- lated by E. Atkinson. [De la Rue & Co.] II. Theory of Potential. Elementary work, not employing the notation of the calculus, Theory of Electricity, by L. Gumming. [See under I.] Treatise on Statics, vol. ii., by G. M. Minchin. [Clarendon Press.] Mascart and Joubert. [See under I.] Elementary Treatise on Electricity, by Clerk Maxwell. [Clar- endon Press.] Electricity ; the large work by Clerk Maxwell. [Macmillan.] III. Electrolysis. Faraday's Experimental Researches. Stewart and Gee's Elementary Practical Physics, vol. ii. p. 72, gives a list of references. [Macmillan. J Sprague's Electricity [Spon & Co.] gives useful practical details. There is to be a discussion on Electrolysis at the B. A. meet- ing for 1 887. Probably therefore much useful information will be given in the report of this discussion. IV. Electric Testing, Telegraphy, &c. Handbook of Telegraphy, by Culley. [Longmans.] Stewart and Gee's book. [See under III.] Practical Physics, by Shaw and Glazebrook. [Longmans.] Electric Testing, by Kempe. [Spon & Co.] Telegraphy, by Preece and Sivewright. [Longmans.] Practical Electricity, by W. E. Ayrton. [Cassell.] X REFERENCES V. Phenomena in High and Low Vacua. A Physical Treatise on Electricity and Magnetism, by Gordon [Sampson Low], gives description and figures of experi- ments by Crookes, Spottiswoode, De la Rue, and others. We may refer also to ' Nature,' vols. xx. pp. 174, 199, 228, 250, 438; xxii. pp. 101, 125, 149, 153, 174, 196; xxiii. pp. 442, 461 ; xxiv. pp. 66, 89, 130, 255,346, 569 ; xxv. pp. 539, 594; xxvii. p. 434 ; xxviii. p. 381 ; xxx. p. 230. VI. Dynamos and Motors ; Electric Lighting, &c. Electric Transmission of Energy, by Kapp. [Whittaker.] Dynamo-Electric Machinery, by S. P. Thompson. [Spon.] Modern Applications of Electricity, by Hospitalier, translated by Julius Maier. [Kegan Paul.] Arc and Glow Lamps, by Julius Maier. [Whittaker.] VII. General. Clerk Maxwell's book. [See under II.] Die Lehre von der Elektricitat, von G. Wiedemann. [Fried- rich Vievveg und Sohn, in Braunschweig.] Cours de Physique, par M. J. Jamin. [Gauthier-Villars, Paris.] Qiuvres de Verdet. [Imprimerie Nationale, Paris.] Mascart and Joubert's book. [See under I.] VIII. Miscellaneous Papers in ' Nature.' Hydrodynamical Analogies to Electric Phenomena, vol. xxiv. p. 362 ; xxv. p. 271 ; xxvi. p. 134. Electricity, Medical Use of, vol. xxv. p. 521; xxviii. p. 463. Electric discharge by red-hot balls, and towards a flame, vol. xxv. 475, 523. Electric Light and Horticulture, vol. xxiv. p. 567. Aurora Borealis, &c., vol. xxii. pp. 33, 146; xxiv. p. 613 ; xxv. pp. 53, 319, 368 ; xxvi. p. 571 ; xxvii. pp. 31, 389, 443 ; xxviii. pp. 60, 107, 128 ; xxix. p. 409; xxx. p. 80; xxxi. p. 372 ; xxxii. pp. 275, 348 ; xxxv. 433. Electric Current Meters, vol. xxviii. p. 162 ; xxx. p. 220 ; xxxiv. p. 508. * Contact-Electricity* of Metals, vol. xxxv. p. 142. Magnetism; Hughe? Researches in, vol. xxviii. pp. 159, 183; xxix. p. 459. Earth's Magnetism, vol. xxv. p. 66. Magnetisation, Change in Dimensions of Magnetised Iron, &*., vol. xxii. p. 543 ; xxxii. p. 45 ; xxxiii. p. 597. Saturation of Cores of Electromagjiets, vol. xxxiv. p. 159. Magnetic Field produced by rotation of an Insulator, vol. xxii. p. 89. Floating Magnets, vol. xvii. p. 487. Diamagnetism, vol. xxxiii. pp. 484, 512. CONTENTS. CHAPTER I. GENERAL PHENOMENA OF MAGNETISM. SECTION i Introductory . PAGE 2. First Phenomena observed 3 Polarity ; . i 2 4. Constitution of a long, thin, Magnet . 5. Molecular Theory of Magnetism 6. Induction, General Phenomena. 7. Use of Keepers .... 8. Methods of Magnetisation . ,.-..' . . . 2 . .' '. .'; ... 3 .. i ... 7 . . . . . 9 . . . 10 CHAPTER II. MECHANICAL AND MAGNETIC UNITS. 1. Introductory. . . . . . . . 12 2. Fundamental and Derived Units . . . . . . . 12 3. Velocity and Acceleration . . . . . ' ^. ., " 13 4. Force . . . . . . . . . 13 5. Parallelogram of Forces, &c. . . . . . . . . 13 6. Moments and Couples . . . ...,.,. . . 14 7. Unit Magnetic Pole . . ....... .14 8. Magnetic Fields, and Unit Field . . . .... 15 9. Magnetic Moment of a Needle . .. . . -.-. . 15 10. Magnetic Moment of Practical, not Ideal, Magnets . . . . 16 11. Magnetic Curves . . . . . .-.'.. 17 12. Magnetic Induction takes place along the Lines of Force . 19 CHAPTER III. MAGNETIC MEASUREMENTS. THE EARTH'S MAGNETISM. 1. Coulomb's Torsion Balance .... 7" .. 2 2. Use of Torsion Balance ' at constant angle* . . . . . 21 3. Method of Oscillations .... 24 xii CONTENTS SECTION PAGE 4. Laws of Magnetism . .'. . . . . 26 5. Proof of Law II. by Torsion Balance ...... 27 6. Proof of Law II. by Method of Oscillation . . . . . 27 7. Measurements as affected by Induction ...... 28 8. Earth's Magnetism. General Ideas . . . . . . . 28 9. Compasses . .- . , * V' 29 10. Modification of Earth's Lines of Force by the Presence of Iron Masses . . . . . . . . . . . 30 11. The Earth's ' Magnetic Elements '. . . . . . .30 12. Measurement of Declination . . . . . . 31 13. Resolution of Earth's total Field into two, or three, Components . 34 14. To find the Inclination, or Dip . . . . . . . . 36 15. Measurement of the Earth's Magnetic Elements . . . -37 16. The ' Method of Deflexions ' 38 17. Magnetometers. Changes in the Earth's Field . . . .40 .CHAPTER IV. THE SIMPLER PHENOMENA OF ELECTROSTATICS. 1. Introductory . . . . . . . . . . 42 2. Electrical Excitement ......... 42 3. Dryness needed ; not High Temperature . . . . . . 43 4. Attraction and Repulsion Phenomena ...... 43 5. Conductors and N on- Conductors . . . . . . 44 6. Electrics and Non-Electrics ........ 45 7. Two sorts of Electrification . . ' . . . . 45 8. The two sorts of Electrification are always produced together . . 46 9. Equal Quantities of the opposite Electrifications are always pro- duced simultaneously . . . . . . . . 47 10. The 'Fluid' Theories of Electricity . . . . .48 n. The three Laws of Electrostatics -49 12. Law II. The Force varies as Q x Q' . . . . -49 13. Law III. The Force varies as . . . . -, . . 51 14. First Ideas as to Induction ........ 52 15. First Ideas as to Distribution . * 55 1 6. Faraday's Ice-pail ; illustrating the Laws of Distribution and of Induction ........... 57 17. ' Electrophorus ... 59 18. Frictional Electric Machines . . .*- i '-. . .60 19. Miscellaneous Experiments with the Electrical Machine . . . 63 CONTENTS xiii CHAPTER V. INTRODUCTORY CHAPTER ON POTENTIAL. SECTION PAGE 1. Quantity of Electrification . . . ,\ . , V -65 2. Electrical Level, or Electrical Potential , ..'.." ' -.:" '. . . 66 3. Measurement of differences of Electrical Level by Work . ,. -67 4. Elementary Ideas on ' Capacity ' . . . ,'..... 68 5. Lines of Force, and Equipotential Surfaces . .' ,''''. . 69 6. Induction ; from a ' Potential ' Point of View . . . . . 70 7. Necessity of distinguishing Sign of Charge and Sign of Potential . 74 8. Further on Capacities . ' . f 74, CHAPTER VI. ELEMENTARY DISCUSSION OF CONDENSERS. 1. General Ideas. Apparatus used . . . .... -76 2. Experiments with the two Condenser Plates . . . . . 78 3. Discussion of the Terms ' Bound ' and ' Free ' = , . ., . .81 4. Conditions Affecting the Magnitude of the * Bound ' Charge . . 82 5. An Isolated Body considered as the Limiting Case of a Condenser . 83 6. Alternate Discharge . . . . . . . . . . 83 7. Leyden Jars 85 8. The Unit Jar 86 9. Cascade arrangement of Leyden Jars . . ".'.. f . 88 10. Nature of the Leyden Jar Charge . . v- . . 90 11. Various Effects of the Discharge .. .-.'; . . . 91 12. Induction Effects of the Discharge . .. . ' . . . 94 13. Wheatstone's Spark-board . ....... . . .96 14. The Condensing Electroscope . . .. r . . . . 98 15. Various Forms of Electrical Discharge . . . .-.,.. 98 CHAPTER VII. INDUCTION MACHINES. 1. Some further Propositions in the Theory of Potential* . ... ioa 2. Application to Induction Machines . . . . . . .102 3. Sir W. Thomson's Water-dropping Accumulator . . . . . 102 4. Varley's Induction Machine . . . .. ....". .103 5. Sir W. Thomson's Replenisher . . . . ' .. . . . 104 6. The Voss Machine ' . . . .. ,. . . . 105 7- The Holtz Machine . .... .. . .. . . . . 108-' XIV CONTENTS CHAPTER VIII. ATMOSPHERIC ELECTRICITY. SECTION PAGE 1. Franklin's, and Other Early Experiments . .'' . . . 113 2. Lightning Conductors . . . ... > ..113 3. Return Shocks . ' . V . . . . ,.' . . 115 4. The High Potential of Thunder-clouds . . .' . . .115 5. Potential at a Point in the Atmosphere . . .,,. . H7 6. Methods of Measuring the Potential at a Point in the Atmosphere . 1 18 7. Results of Observations ...... JI 9 8. Sheet- Lightning and other Phenomena ... . . 120 CHAPTER IX. SPECIFIC INDUCTIVE CAPACITIES. I Definition . . . , . * . 121 2. Variation with Time . , . 3. Cavendish's Method 4. Faraday's Method .. . .. 5. Modern Methods ... . . . ., . . . 122 . . ... . .122 ... ... 124 . . .125 CHAPTER X. ELECTROSTATIC POTENTIAL. 1. Introductory .,.%..-.... 127 2. Definition and Measurement of Work . .. .. . .127 3. Dimensions of Work . . . V. . ... . . 128 4. Energy and Conservation of Energy . . .... .128 5. Work against a Constant Force . ., . . . . . . 129 6. Work where the Force varies as the Distance . . . ;' ,129 7. Work where the Force varies as the Inverse Square of the Distance, or as L . ' . . . . . . . . . 129 8. Potential, and Difference of Potential . ' . . . -131 9. Application of 7 to the Measurement of Electric-Potential . .131 10. Equipotential Surfaces . . . . , ' T 33 11. Lines of Force are Perpendicular to Equipotential Surfaces. . -134 12. Field-strength; and Rate of Change of Potential . . . . 135 13. The Mapping Out of Lines of Force ; Simple Case . . .., . . .136 14. General Case . ,, . . . . ... .. 137 15. Total Number of marked Lines of Force . . . . . 138 16. Tubes of Force. ' Fa is Constant ' .... .. . 138 17. Statement of some further Theorems on Lines of Force ~. . . 139' 18. The Potential of an ' Isolated ' Body . . . .... .141 LOWER DIVISION CONTENTS XV 19. Potential of an Isolated Sphere . . . . v . . . .141 20. Capacity of an Isolated Sphere . . . . . . . .142 21. Distribution; from the Potential Point of View . Y . . .143 22. Two Spheres of Different Radii .- . , '. . . . 143 23. Potential, and Density, distinguished . '. '.',... . . .144 24. Force on a + Unit, acting Perpendicularly to a Conducting Surface . 144 25. Important Case of a Spherical Condenser . . . '. . . 145 26. The Plate Condenser v . "' . . 147 27. Formulae for Capacities, &c. . . . . . ... 148 28. Energy of Charging and Discharging . . . . . .150 29. Examples in Energy of Discharge . . . . . ..151 30. Energy of Discharge in the Cascade Arrangement of Leyden Jars . 152 31. Electroscopes and Electrometers ....... 153 32. Electrometers. The < Attracted-Disc ' Form 153 33. Sir William Thomson's Quadrant Electrometer . . . . . 155 34. Uses of the Quadrant Electrometer . . . . . .158 35. Examples in Energy of Discharge, &c. . . . . . . 159 36. General Consideration of Electrostatic Fields of Force . . .161 CHAPTER XI. THE PHENOMENA OF ELECTRIC CURRENTS. BATTERY-CELLS AND BATTERIES. 1. Introductory . . . . . . . . . . . 163 2. Galvani's Experiment . . . . . . . .164 3. Volta's Experiments and Views . . . . . . . . 165 4. Volta's Pile ; from Volta's p6int of View . . . . . .167 5. Volta's Cell, and the Couronne des Tasses ; from Volta's point of View 169 6. The Contact ' and ' Chemical' Theories 170 7. Theory of the Simple Volta's Cell . . 171 8. Digression on the Galvanometer . . . . . . . 173 9. The Solution of Zinc in the Voltaic Cell 173 10. Polarisation . . . . . . . . . . 175 11. Constant Batteries 176 12. Remarks on Cells and on Batteries 180 CHAPTER XII. THE CHEMICAL PHENOMENA ACCOMPANYING THE PASSAGE OF THE CURRENT. 1. Introductory . . . . '''". . . . . 182 2. Heating Effects ; a Brief Account ". * . " . . . V .182 3. Chemical Effects ; General View Y V '. .' . . '. . 183 a XVI CONTENTS SECTION PAGE 4. Grothiiss's Hypothesis. Nature of Electrolysis . . ,' . .186 5. Primary and Secondary Decompositions . . . .. . ' . . . 188 6. Simultaneous Decompositions . . . .-'. . .189 7. Faraday's Laws of Electrolysis . . . . . . . . 189 8. Further on Faraday's Laws of Electrolysis . . . . .190 9. Electro-chemical Equivalents . . . , . . ..191 10. Electro-plating . . . . . . , . . . 191 11. Polarisation of the Electrodes . . .. -, . . . . 194 12. Secondary, or < Storage,' Cells . . . . . . . 196 13. Plante's Secondary- Cell v ........ 197 14. Faure's Accumulator . . ^ , . .. , r . . 200 CHAPTER XIII. OHM'S LAW. I. General Ideas as to the Scope of Ohm's Law , . 2. Statement of Ohm's Law . 3. Resistance further Discussed . . . - . . 4. The Exact Conditions on which Resistance Depends 5. Conductivity . . . . 6. Application of Ohm's Law in a Simple Case . 7. Graphic Representation of Ohm's Law . . . . . 201 . - . .203 . . 204 . 204 . . 206 '. . .207 . ; . . 209 9. Divided Circuits ....... 10. 'Shunts' . . ' . " . * . " , * . . r . . 211 21,1 II. Fall of Potential through the Circuit . . . 12. Kirchhoffs Two Laws . . ' . " . '. . 1 1. Maximum Current with a given Battery . . 215 . . . . 216 . 2IQ CHAPTER XIV. MEASUREMENT OF RESISTANCES AND OF E.M.F.S. 1. Preliminary, on the Units Employed . ', . .. V . 221 2. Resistance Coils, and Resistance Boxes, . . . . . - . . . 222 3. Wheatstone's Rheostat . . . . . . . . . . . ' .. . . 224 4. Wheatstone's Bridge ; General Principle . . . - . . . 226 5. Slide-form of Wheatstone's Bridge ....... 228 6. Wheatstone's Bridge ; Resistance Box Form ... . . 229 7. Resistance of a Galvanometer . . . . . . . 229 8. Resistance of a Battery- Cell . . . . . . . . 230 9. Measurement of E. M. F. ' . , . . . . .231 10. Electrometer Methods ; Open Circuit . . . . . 232 11. Volt-meter Galvanometers . . . . . . . . 233 12. Method of Opposition . * . ' ....... 234 CONTENTS xvii SECTION PAGE 13. Latimer-Clark's Potentiometer . . ' . . . 234 14. Table of Resistances in Ohms (or o>) . . . .... 235 15. Table of E.M.F. sin Volts . . . . . . . .237 CHAPTER XV. JOULE'S LAW, AND CONSERVATION OF ENERGY. 1. General Survey. . . . . . .... 238 2. Units of Heat, Work, and Activity . . . . . 239 3. Energy of the Electric Current . . . . . . . 240 4. Joule's Law ........... 241 5. The Heating of Uniform Wires , . 243 6. Distribution of Heat in the Circuit . . . . . . 244 7. Heat Evolved with various Arrangements of n Cells . . . 245 8. Case of no Back-E.M.F. in the Circuit 246 9. Case of a Back-E. M. F. e in the Circuit .... . . 246 10. Numerical Examples . . . . . . . . 247 11. Failure of a Smee's Cell to Decompose Water . ... 248 12. Partial Polarisation in the foregoing Case ..... 249 13. Connection between E.M.F.s and ' Heats of Combination '. . . 249 CHAPTER XVI. THERMO-ELECTRICITY. 1. Introductory. ... . . . . . . . 252 2. The Simple Thermo-Cell . 253 3. The Thermo-Pile 254 4. Thermo-Electric Series ......... 254 5. Thermo-Electric Powers ........ 256 6. The Neutral Point .......... 257 7. Thermo-Diagrams .......... 259 8. Peltier Effect ; Observed Facts 263 9. The Thomson Effect ......... 264 10. Theory of the Simple Thermo-Cell ....... 265 n. Theory of the Peltier and Thomson Effects . . .. . 268 CHAPTER XVII. GALVANOMETERS ; WITH A PRELIMINARY ACCOUNT OF THE MAGNETIC ACTIONS OF CURRENTS. 1. Magnetic Field about a Simple Rectilinear Current . . . . 271 2. The + and - Directions of the Lines of Force .... 273 3. Simple Form of Galvanometer . . . . . . . 275 4. Relation of Strength of Field to Current-Strength . . . 275 xviii CONTENTS SECTION PAGE 5. The Tangent Galvanometer . . . '. . . . . 276 6. The Sine Galvanometer . . . . ... . . 279 7. The Multiplying Galvanometer . . . . . . . . 280 8. Astatic Galvanometer ; Two Needles ...... 280 9. The Controlling Magnet ' Method 282 10. Sir W. Thomson's Mirror Galvanometer . ... . . . 283 11. The Differential Galvanometer . . V 285 12. The Ballistic Galvanometer . t , . . . . . 286 13. Sir W. Thomson's ' Graded ' Potential Galvanometer . . . . 289 14. Sir W. Thomson's Graded Current Galvanometers . 290 15. Weber's Electro-Dynamometer . . . . . . 291 16. Some General Observations on Galvanometers ... . 292 17. Galvanometers for Practical, or Commercial, use . ~ . . . 293 18. Calibration of Current-meters and Voltmeters .... 293 CHAPTER XVIII. ACTIONS BETWEEN CURRENTS AND MAGNETIC POLES. MAG- NETIC EQUIVALENT OF A CURRENT. ACTION BETWEEN CURRENTS AND CURRENTS. 1. Action of an Infinite Rectilinear Current on a Magnetic Pole . . 294 2. Action of an Element of a Current on a Pole ..... 294 3. The Absolute System of Electro- Magnetic Units 297 4. Summary of Electro-Magnetic Units (see 3) . . . . . 298 5. The Dimensions of the Derived Units . . . . . . 299 6. Field due to a Circular Current . . . . . . . 302 7. Magnetic Shells ; and the Fields due to Them . . . . . 302 8. Magnetic Potentials due to Magnetic Shells ..... 304 9. Magnetic Equivalent of an Electric Circuit . . . . . . 306 10. This Equivalence is for the External Field only .... 307 1 1. Principle of Sinuous Currents ........ 307 12. Reaction of a Pole on an Element of Current ' \ * . . . 308 13. Action of a Pole upon a Closed Circuit . . . . . . 308 14. Action of a Pole on an Incomplete Circuit . , . . .310 15. Action of the Earth's Field on Currents Completely or Partly Mobile 311 1 6. Actions between Currents ; Ampere's Laws . . . . .311 17. Continuous Rotations of Currents .-.., . . . .313 1 8. Ampere's Laws of the Actions between Elements of Currents . .313 CHAPTER XIX. LAWS OF THE MOVEMENTS OF CURRENTS AS DEDUCED FROM THE CONSIDERATION OF MAGNETIC FIELDS AND POTENTIALS. 1. Magnetic Fields and Potentials . . * ../, . .-, .315 2. Movements are from Higher to Lower Potentials . . . * 315 CONTENTS xix SECTION PAGE 3. Potentials on Poles and on Circuits - . . . . : . . . 316 4. General Law of Movement of (Magnetic Shells or of) Electric Circuits ;.. .. .. '. \. .. . -..'<' ' . . . 318 5. The Case of a Uniform Field . . .... . . . 318 6. The Case of a Field not Uniform ...;. . . . . . .318 7. The Case of Incomplete Circuits . . . . . . .319 8. Reconsideration of Ampere's Laws . . ".'-. ' .,'" . . 321 9. Cases of Continuous Rotation . . . . . . . 322 10. Potentials Due to Circuits . . . . .' ..... 322 11. Potentials on Circuits . . . ^ .... . . . 323 CHAPTER XX. SOLENOIDS, ELECTRO-MAGNETS, DIAMAGNETISM, AND ELECTRO-OPTICS. 1. Cylindrical Magnet built up of Circular Laminae ."... . 324 2. The Ideal Solenoid . . . . . -,. ' . ' . . . 324 3. The Practical Solenoid . . . . . . > . . . 324 4. Ampere's Theory of Magnetism . . . . . .' , - . . 326 5. Solenoid, and Hollow Cylindrical Magnet, Contrasted . . . 327 6. Matter Placed in a Uniform Magnetic Field of Force . * '. . 328 7. Movements of Small Bodies in a Non-Uniform Magnetic Field . 330 8. The ' Setting ' of a Long Body in a Uniform Magnetic Field . . 330 9. A Long Body in a Non-Uniform Field . . . ... 332 10. Solenoid With, and Without, an Iron Core . .-..-. 332 11. Electro-Magnets .......... 333 12. Paramagnetic and Diamagnetic Phenomena . . : . , . 334 13. Pseudo-Diamagnetic Phenomena . . . , . . 335 14. Relative Magnetism or Diamagnetism . . . . . 335 15. Is there Absolute Diamagnetism ? . . . ...-'-. . 336 1 6. Rotation of "the Plane of Polarisation in a Magnetic Field . 337 17. Other Electro-Optical Phenomena . . . . . . . 340 18. The Electro- Magnetic Theory of Light . . . . . 340 CHAPTER XXI. ELECTRO-MAGNETIC INDUCTION. 1. General Account .of Induction Phenomena . . ; , 343 2. General Reason for ' Induced Currents ' . , ; . . , . 346 3. More Exact Reasoning, in a Simple Case . . . . . 347 4. General Expression for Induced E.M.F. -..,. . *'*:'' . 349 5. Induction where there is no Initial Current .- ." .- . - . 351 6. Direction of the Induced Currents ; Lenz's Law . ., . . . 352 7- Constant Induced Currents -. .. .;. -. ... ... . 355 8. Changes that Give Induced Currents . . . . . . . 357 XX CONTENTS SECTION PAGE 9. Coefficient of Mutual Induction, or of Mutual Potential . . . 359 10. Self-Induction. The Extra Current ' . . . . . 360 11. Induced Currents of Higher Orders . ... . . . 362 CHAPTER XXII. ARAGO'S DISC, RUHMKORFF's COIL, AND OTHER CASES OF INDUCTION. 1. Induced Currents (' Eddy Currents ') in Solid Metallic Masses moving in a Magnetic Field' . . . . . . . . . 363 2. Arago's Disc and Magnetic Needle ....... 363 3. Continuous Current Collected from Barlow's Wheel . . . . 364 4. Induction in the Earth's Field . . . ,. . - r . . 366 5. Induction Coils ; General Plan ........ 368 6. Practical Difficulties to be Overcome ...... 369 7. Ruhmkorffs Coil , '. ..'.. . ' *. ; . . 370 8. The Part Played by the Condenser .. . . 4 . . .372 9. Condition of the Secondary Circuit when Closed . . . . 373 10. Secondary Circuit with Air-Break . . - .'. . . . 374 11. Electrostatic Condition of the Open Secondary Terminals. The Charging of Leyd en Jars *'..* *, 374 12. Various Phenomena of the Secondary Discharge .... 376 13. High and Low Vacua . ' V . . . ' . . . 377 14. Discharge in Low Vacua .' . . * . . . . 377 15. Discharge in High Vacua . . .. . ..." - . -* . . 379 CHAPTER XXIII. DYNAMO-ELECTRIC MACHINES. 1. General Nature of a * Dynamo ' . ... , .. . 383 2. General Account of Induced Currents . ... . . 383 3. Clark's Machine . . . . . ...... 385 4. The Simple Commutator . . . ."'.'. . . . 386 5. Siemens's Armature ......... 387 6. The Self-Exciting Principle .,'..>,. . . . . 387 7. ' Continuous Current ' Machines . . ,.,.,,-'. . . 387 8. The 'Gramme.' Construction of Armature ... . . . 388 9. The Gramme. The E.M.F.s Induced in the Ring . ' ; . . 390 10. The Gramme. The Collecting Brushes . . . ' . 392 11. Curve of Potential Round the Collecting Axis. . . , . 392 12. The * Lead ' that Occurs when a Current is Running . . 393 13. Armatures Wound 'for E.M.F.' and 'for Current' . . . . 394 14. The Siemens- Alteneck Armature ....... 395 15. The Brush Machine . - V . . 395 CONTENTS xxi SECTION PAGE 1 6. Magneto Machines . . . 395 17. Separately- Excited Machines ........ 396 18. Series-Excited Machines * . , . : '.-'' . . . 397 19. Shunt-Dynamos . . . -. --. . , . . . 398 20. Other Methods of Winding > '. . % . - , ( ' . '.. . 399 21. Alternate Current Machines . . - ."'.'. . .399 22. The Ferranti Alternate Current Machine . . '.'". 401 CHAPTER XXIV. DYNAMOS AND MOTORS. 1. Magnetos, .or Separately-Excited Machines, as Motors . . . 402 2. Series-Dynamos as Motors . . . , . , . 403 3. General Remarks on Dynamos and Motors . * , 404 4.. Formulae for Activity, &c. Maximum Activity . ... . . 405 5. Efficiency ..... ..... . 408 6. Representative Curves . . . . , .... 408 7. S. P. Thompson's Diagrams ......... 409 8. Transmission of ' Power ' from a Distance . . . . . 409 9. Electric Railways and Tram- Cars ; Telpherage; &c. . . .411 10. Distribution of Potential in the Circuit of a Dynamo and Motor . 412 11. Work done per Second upon a Dynamo as Related to the Velocity v of Rotation . . . . . '.'.,'. . 415 CHAPTER XXV. VARIOUS APPLICATIONS OF ELECTRICITY ; TELEGRAPHS, TELE- PHONES, MICROPHONES, ELECTRIC LIGHTING. 1. Introductory . . . . . -.-.-. . . 416 2. General Principle of Telegraphy . . .. . . . .416 3. Telegraphic Alphabets . . . . . . ' . ..419 4. The Needle System of Telegraphy . ' ..... . ... . 420 5. The Morse System .......... 421 6. Relays . . . . . . * '. . . . . 422 7. Transmission through Cables, under Water . , f , . 422 8. Earth Currents. Condenser System of Working .... 424 9. Insulation of Wires . . . ; . . 426 10. Duplex Telegraphy . . . . . . . . 426 11. Telephones. Introductory . . , , , . 428 12. The Bell Telephone . ,~. . . ' . ., . 429 13. Telephones with External Source of Current , . . , . . 431 14. Microphones . . . . . . , . . * . .431 15. Properties of Selenium. The Photophone . ..... 432 16. General Account of Electric Lighting >,.. 433 XX11 CONTENTS SECTION 17. The Incandescent ' Lamp ' 18. Arrangement of Incandescent Lamps in Parallel 19. The Economy of Incandescent Lighting . 20. The Arc Lamp . . . 21. The Reverse E.M.F. of the Arc 22. * Equivalent Resistance ' in an Arc Lamp 23. Series Arrangement of Arc Lamps 24. Further Remarks on the Use of Arc Lamps . 25. Current-meters or Ammeters . . ; 26. Voltmeters . !. V : . ' '. 27. ' Damping ' in Ammeters and Voltmeters . 28. Wattmeteis . ... 29. Integrating Instruments. Coulomb-meters . 30. Integrating Instruments. Energy-meters 31. Testing of Coulomb-meters and Energy-meters . 32. General Principle of the System . . QUESTIONS AND EXAMPLES . ; ; ; . ANSWERS TO QUESTIONS . . . , ; . INDEX 434 436 . 437 438 440 441 442 443 443 446 446 447 448 449 45 450 xxiii xl , xlv SOME ABBREVIATIONS USED. Magnetic pole-strength /u- Magnetic moment . . m and /a/ Field-strength H and I Work and Energy W Electric Quantity Q Electrostatic Capacity . . . . K Specific-inductive-capacity cr Electric and Magnetic density . . . p Potential V Difference of Potential . AV Gold-leaf electroscope . . g.l.e. Current-strength C Resistance R and r Equivalent-resistance R' Battery-resistance ..... B Galvanometer-resistance . . . . G Shunt-resistance S Electromotive Force . E.M.F., E, and e /Joule's equivalent J [This is the factcr which reduces calories to ergs ; and = 4*175 x io 7 , or 42 x io s V nearly.] Note. In cases where the above symbols have other meanings, the context will obviate ambiguity. Thus V may sometimes mean -velocity ', H may mean heat ; and p may mean specific resistance. . <,,,,, . gijVsw i ", O-S'.'.',' $jti'& '-' ELECTRICITY. CHAPTER I. GENERAL PHENOMENA OF MAGNETISM. i. Introductory. The subjects of magnetism and electricity e in reality not two, but one ; all magnetic, electro-magnetic, voltaic phenomena to use terms with which most of our readers will have some acquaintance belong to one great branch of science for which we have not yet one comprehensive name. Perhaps it might be more logical were we to give at once a general survey of all the above-named classes of phenomena before pro- ceeding to a more detailed discussion of each. But, as in this course no previous knowledge has been assumed, it has been thought better to avoid any chance there might be of confusing the mind of the student by the presentation of a multitude of strange facts, and hence we shall first discuss the main phenomena of magnetism. A study of these will serve as a training to the beginner, and he will incidentally become acquainted with many facts and conceptions that will prove to be of great value in the study of the other branches of our science. 2. First Phenomena observed. Accustomed as most of us are to the use of a 'magnet,' there is still something very startling in the simple elementary experiment with a ' magnet ' or a piece of lodestone. When a piece of lodestone (magnetic oxide of iron, Fe 3 O 4 , magnetised by the influence of the earth) is held above a piece of iron or steel of not too great weight, the piece of iron or steel will move up to the lodestone, against the force due to gravity. And, more remarkable still, the lodestone can convert pieces of hard steel into permanent ' magnets,' it being B 2 u ELECTRICITY en. i. merely necessary to make a convenient bar of steel and to stroke it repeaterlly' fr^rr 1 end to end with the lodestone, taking care that the strokes' are alike 'in all respects.' We shall henceforth employ such steel magnets instead of lodestone. 3. Polarity. If we examine our magnetised bars of steel we shall find that, though the lodestone passed along every portion of them equally, the ends always possess far greater magnetic powers than do the middle portions. Further than this, we find that in each magnet these two ends are different. If we magnetise a bar A B, and another A' B', in such a way that A and A', B and B' are respectively corresponding ends (and we can do this by stroking the one bar from A to B, the other from A' to B' with the same portion of one piece of lodestone), then we find that A repels A', B repels B', but that A attracts B' and A' attracts B. We find also that if such bars be suspended they will take up a definite position with respect to the earth, setting themselves ina direction called magnetic north and south, the similarly treated ends A and A', B and B', lying in similar directions. We may say that the ends of the bar have acquired remarkable properties, these properties being, in a sense, of opposite natures at the two ends respectively. It is usual to call the ends where the magnetic properties are displayed poles^ and the bar is said to have acquired magnetic polarity. We may here point out a possible source of error. We have seen that like poles repel, and unlike poles attract, one another ; and further that all needles or bars, that are magnetised and suspended, turn so as to point in a northerly and southerly direction. That end of the bar that points towards the geographical north pole of the earth is usually called a north pole, or north-seeking pole ; the other end is called a south pole, or south- seeking pole. But from what we have seen it is evident that the north-seeking pole of the bar must be of opposite polarity to the ' north pole ' of the earth towards which it turns; and so with the south-seeking pole of the bar. It is better, therefore, to speak of the ' north pole ' of the earth, but the ' north-seeking ' pole of a magnetised bar. 4. Constitution of a long thin Magnet. Much light is thrown upon the nature of this Polarity by the following simple experiment : A strongly magnetised knitting-needle is taken, and the CR. i. GENERAL PHENOMENA OF MAGNETISM $ polarity of the ends, and neutrality of the middle, tested by use of filings and compass-needle. It is then broken in half. We now find each half a complete magnetic needle ; two opposite poles having apparently started into existence in the pre- viously neutral middle. Each half is then again broken, and so oral The original needle, and the condition wherWHr broken, are shown in the accompanying figure. It would seem that all these intermediate poles were existing in the original needle, but that they neutralised each other as far as any external action went. As far as experiment goes there is nothing against the sup- position that if we had a magnetic bar very thin, in fact, a single row of molecules of steel, we might continue this process of breaking up until we should find each molecule a complete little magnet with poles at its ends (if we may use the term ends with respect to a molecule which may be spherical), and a neutral region at the middle. Experiments, (i. ) Magnetising a bar of steel with lodestone, we find that iron filings will cluster chiefly at the ends. (ii. ) Take a series of unmagnetised knitting-needles ; testing them by seeing that either end acts the same on the ' north ' end of an ordinary compass- needle. Lay them down side by side, and magnetise them successively with a piece of lodestone or with a steel magnet ; marking with gummed paper the ends that must, by the process of magnetisation, be similar. It will then be found that (a) similar poles repel, dissimilar attract, one another ; (b} the suspended needles will all set themselves with the marked ends turned in the same direction. 5. Molecular theory of Magnetism. The above experiment suggested, very early, that probably magnetism is molecular. The view adopted was that when a long thin needle was magnetised, its molecular constitution was somewhat as represented in the accompanying figure. N s is the magnetic needle, supposed to B 2 ' CF CALIFORNIA 4 ELECTRICITY CH. i. consist of a single line of molecules. Each of these molecules is a small magnet, and they are arranged so that north-seeking poles are opposed to south-seeking poles all along : there being left at the one end^attorth-seeking pole, and at the other end a south- seeking pole, uncompensated. FIG. i. It is assumed, as justified by such experiments as that alluded to in 4, that the south-seeking pole of one molecule can neutralise, as far as external manifestation of magnetic properties goes, the opposed north-seeking pole of another molecule, while the north- seeking and south-seeking poles of the same molecule, ''thougrr.yery close to one another, do not so neutralise one another. ^ The state of neutrality was formerly considered to be a 'higgledy-piggledy' arrangement, or rather absence of arrange- ment, producing on the whole external neutrality. Thus the end of a magnetised bar would present to external bodies a whole set of molecular north-seeking or south-seeking poles, while the end of an unmagnetised bar would present a mixed surface of north-seeking and south-seeking poles whose total external action would be nil. Professor D. E. Hughes has taken up the molecular theory, and has reduced it to order, much extending it. In fact, he has done so much that the chief questions left unsolved are the funda- mental ones : * What is the polarity of a molecule ? and, Why do similar poles repel and dissimilar attract one another?' The question of diamagnetism also needs some further research. The main drift of Professor Hughes's theory, each portion of which he has supported by experiment, is somewhat as follows : (i.) Each molecule (possibly each atom) of every substance possesses to a fixed and unalterable degree a property called polarity. The poles of the molecule are probably fixed in that molecule, and can be made to lie in a changed direction only by rotation of the molecule. (ii.) The opposite poles of consecutive molecules do, when the molecules are so rotated that these ' poles ' lie one over against the other, neutralise each other with respect to external action. en. i. GENERAL PHENOMENA OF MAGNETISM 5 (iii.) In neutrality there is always some symmetrical arrange- ment by which the molecules neutralise one another with respect to any external action ; either by the molecules being paired off with opposite poles together, or by a whole line of molecules neutralis- ing each other's poles in one long chain, or by some equivalent arrangement. One of the great features of Hughes's theory W the symmetry of arrangement that is considered to exist under all circumstances. The accompanying figures indicate two possible cases of neutrality. Since the molecule, whatever its real form, is practically for mag- netic purposes a short line with poles at the ends, we have here re- presented the molecules as lines ; indicating thereby their magnetic, though not their actual shape. ft IHHHHf \ IHHIKHMHi 7 ^ nf Aft 1 ). FIG. iii. (iv.) When the bar is magnetised to the greatest possible degree, the arrangement will be such that there is a chain of mole- cules in which the north-seeking pole of the molecule at the one end of a chain, and the south-seeking pole of the molecule at the other end of the same chain, will be left unneutralised ; while between these there is complete neutralisation. And, further, these chains must be arranged side by side so that the free north- seeking poles are all at one end of the bar and the free south- seeking poles at the other. In the above two cases this might be effected in fig. ii. by a rotation of the molecules so as to form one line as in fig. i., while in fig. iii. it would be only necessary to break the chain at any point. Thus there is a limit to possible strength of magnetism. The utmost that can be done is to arrange the molecules in the 6 ELECTRICITY CH. i. most advantageous position. The strength of magnetism will then depend on the ultimate properties of the molecules. (v.) Usually, or perhaps in all cases, the setting of the mole- cules so as to give evident magnetism is not completed. The molecules are partly rotated away from their position of what we may call ' short-circuiting ' or neutralisation, and we get all inter- mediate cases^between perfect neutrality and perfect magnetisa- tion. A piece of iron when magnetised to its fullest extent is said to be saturated. (vi.) Arrangements of the molecules, in which they do not pair off, or in any other way neutralise one another, and so give zero external action, are unstable. In fact, if only there be given free movement to the molecules, so that they can act on each other without hindrance, they will always ' short-circuit ' with one another and give us zero external magnetic action. (vii.) The rearrangement of the molecules so as to give ex- ternal magnetism is resisted both by their action on each other and by the mechanical difficulties in the way of rotation of the molecules. This mechanical rigidity, that opposes the ' magnetisa- tion ' of a bar and also opposes the molecular tendency to a return to the neutral, short-circuited, condition, is called by the inexact name of coercive force. Any action (as heating, ham- mering, and the like) that gives freer play to the molecules will diminish the coercive force of the bar. In general, therefore, it is useful to hammer a bar while it is being magnetised, but bad to do so when it is no longer under the magnetising influence. (viii.) The degree of molecular rigidity depends upon the chemical nature of the metal and upon its temper. Thus, some specimens of ' soft iron ' will hardly retain any trace of magnetism when the magnetising influence has been removed, while some specimens of 'hard steel' will remain unaltered for a long time. (ix.) Temporary and Residual Magnetism. In general, if we magnetise a bar, it will readily lose part of its magnetism, but will retain the remainder unaltered for a long time. It is supposed that the molecules can, without much difficulty, rotate part of the way back to their original positions of neutrality ; but that at a certain point their freedom of movement is limited by the ' molecular rigidity ' (or coercive force) spoken of above. CH. i. GENERAL PHENOMENA OF MAGNETISM 7 It is found that when iron or steel is raised to a white heat, it is not attracted by a magnet ; in fact, it ceases to be magnetic. This phenomenon has not been satisfactorily explained by any theory ; and, until further investigation has been made, we cannot say whether it makes for or against the above molecular theory. Experiments to illustrate molectilar theory. (i.) We can magnetise a tube of steel filings, and then render it neutral by shaking it up. As the filings are not free to traverse the tube, but merely turn about where they are, this illustrates the case of neutrality by the pairing off of the molecules. (ii. ) Magnetise a strip of watch-spring, and test it. If this watch-spring be now made into a complete circuit, we get almost complete external neutrality ; while the evident magnetism is again restored when we break the circuit. (iii. ) A poker may be magnetised by being held along the lines of the earth's magnetism (see Chapter III. 8) and in that position hammered. (iv.) The magnetism of a steel knitting-needle may be reversed if it is held with south-seeking end downwards along the lines of earth's magnetism, and in that position kept at a bright red heat for some time ; being then allowed to cool in the same position. (v. ) It was shown by Joule that a bar when magnetised increases in length. This might well be the case if each pair of molecules (see fig. ii. above) does in fact rotate from a parallel, to an end-on, position. (vi. ) A tube is filled with water in which is suspended magnetic oxide of iron in a finely divided condition ; the whole being nearly opaque to light. If the tube be magnetised in the direction of its length, it is found that it is now less opaque to light in this direction. This would tend to show that the particles of the magnetic oxide have arranged themselves end-on in the direction of magnetisation. 6. Induction, General Phenomena. The matter of induc- tion will be clearer to the learner when he has learnt something of ' Fields of Force,' in Chapter II. At present we shall only describe the actual facts observed. When a piece of iron is placed near the pole of a magnet, the end of the iron that is nearest the pole acquires a polarity oppo- site to that of the said pole ; while the end of the iron that is furthest away acquires a polarity similar to that of the magnet- pole. This action of a magnet, in making iron near it also magnetic, is called 'Induction.' It is more powerful as the ' inducing ' magnet is more power- ful, and as the iron is nearer to the magnet. 8 ELECTRICITY CH. I. FIG. i. It is stronger and less permanent as the ' coercive force ' of the iron is small (or the molecular freedom great), weaker and more permanent as the ' coercive force 7 is greater. Experiments in Induction. (i. ) NS is a powerful magnet ; and ns, n's , n"s" y &c., are pieces of very soft iron arranged near NS as shown. The letters of the accompanying figure indicate the observed polarities of the pieces of soft iron. One may test these polarities either by means of a small compass-needle applied to the ends of the iron pieces, or in some such way as that indicated in the next experiment. (ii.) The writer believes that the following experiment is due to Professor Guthrie. NS is a powerful permanent magnet and v^l^' ^^r^-'-- ~~ ^fyf' ' '/j'f ' ' I ' v'V^ 7 '"' FIG. ii. Now iron filings do, by induction, become small magnetic needles when they are placed in a magnetic field. Hence, if we -CH. II. MECHANICAL AND MAGNETIC UNITS place a magnet under a sheet of glass and scatter over the glass iron filings, tapping the glass to allow of arrangement, we shall find traced out for us many of these lines of force that lie in the plane of the glass. The accompanying figures show us the general aspect of the curves thus obtained. In fig. ii. we have respectively the cases of a bar-magnet ; a horseshoe-magnet ; two bar-magnets with unlike poles opposed ; and the same with similar poles opposed. 12. Magnetic Induction takes place along the Lines of Force. It is along these lines of force that we must consider our molecules, regarded as small magnetic needles, urged to direct themselves. Hence, if these molecules were perfectly free to move, any mass of iron would by the rearrangement of its molecules become magnetised along the lines of force of the inducing field. But as the molecules are subject internally to mechanical con- straint as well as to each other's action, the resulting magnetisation is determined by the combined influences, internal and external. If the bar or needle lies so that its greatest length is along the lines of force, and if the form of the bar is symmetrical about this axis, then the magnetisation should be whether weak or strong along the lines of force. But if the bar be unsymmetrically situated with respect to the lines of force, the resulting magnetisation may be quite unsym- metrical and be oblique to the lines of force. c 2 20 ELECTRICITY CH. III. CHAPTER III. MAGNETIC MEASUREMENTS. THE EARTH'S MAGNETISM. i. Coulomb's Torsion Balance. In considering the subject of magnetic measurements we shall first describe the torsion balance. This instrument deserves notice as the earliest by which exact magnetic and electrostatic measurements were obtained, and some study of it will be in- structive. In practice it has, however, been now super- seded by other instruments. The figure represents one form of the instrument. A rectangular or cylindrical glass case is provided, either with a graduated scale round the sides, as here shown, or, what is better, with a plane mirror at the bottom, on which is marked a circle graduated in degrees. In what follows we shall = assume that the latter method of graduation has been adopted, and that the centre of this graduated circle is called O. Above the mirror is sus- pended (when the instrument is used in magnetism) a magnetic needle a b, in such a way that its axis of suspension is immediately above the centre O of the graduated circle. By looking down from above we can tell over what degree the needle is lying. By moving the eye until the needle and its image coincide, we avoid errors due to the fact that the apparent position of the needle over the scale varies according to our point of view. CH. in. MAGNETIC MEASUREMENTS 21 The needle is suspended by a fine wire or thread of glass, so hung that its prolongation would pass through O, the centre of the graduated circle. This thread is fixed at its upper end to a brass piece e d (called a torsion head} that caps the glass tube shown. This piece is so constructed that the thread can be twisted without being displaced laterally, and the angle of twist can be measured. A sight of the instrument will make clear how this is contrived. Now to twist a wire or thread of glass without displacing it laterally requires a simple couple. It can be proved experimentally that as long as the thread or wire is not permanently altered by a twist, the couple required to twist it is directly proportional to the angle of twist. Or if we twist the wire through 10, 20, 30, 170, &c., we are exerting couples proportional to 10, 20, 30, 170, &c. We cannot measure the couples in absolute C.G.S. units unless we know more about the length, radius, and material of the thread. But we can, without this knowledge, compare couples. If the top of the wire be twisted through ft one way, and the bottom through a the other way, then the total angle of twist will be + a. 2. Use of Torsion Balance * at constant angle.' We can use the torsion balance to compare the strength of magnetic fields and of magnetic poles ; or even to measure them in C.G.S. units if we are acquainted with the 'con- stants ' of the instrument we are using. In the accompanying diagram we are supposed to be looking down upon the whole instrument. The small circle represents the graduated circle of the torsion head ; and, in the case given, the torsion wire has been twisted either through an angle a, or through (n x 360 4- a), where n is some whole number. The large circle is the graduated circle on the mirror ; and, in the case given, the needle has been deflected through ah angle B from zero. The opposite directions of twist of the needle and of the 22 ELECTRICITY CH. m. torsion head respectively, that must exist where we are dealing with repulsions, are indicated by the opposite directions of gradua- tion. In this figure the needle n s answers to the needle a b of i. Supposing now that we wish to compare the strength of two magnetic poles. We start with needle and torsion index both at o. There is a hole in the glass top to the instrument in such a position that we can lower the one or the other pole of the magnets we are comparing into the place occupied by the pole of the needle n s when this lies over o. The needle is swung to one side, and the first pole, which we will call /u t , lowered. As we always take care to lower a pole similar to that pole of the needle whose place at o it takes, we have the needle deflected until the moment of the force of repulsion about the axis of suspension of the needle just balances the moment of the couple due to the twisting of the wire. As a rule we shall find it advisable to turn the torsion head, and so to twist the wire, until k the needle is forced back to some angle 6 of deflexion from zero, this angle being much less than the original angle of deflexion. One reason for this is that if we cause ft to be small, we may neglect the restoring couple due to the earth's field. Let us suppose that the torsion head has been turned through (n { x 360 + a^ . Then the total twist on the wire will be (n l x 360 + aj + 0) ; and the couple that it exerts will be mea- sured by k (, x 360 + G! + 0), where k is some constant that depends upon the nature and dimensions of the wire. Now this couple is balanced by the moment about O due to the action of the pole MI upon the needle ns ; and this moment will be measured by the product \i\h. In this product ^ represents the pole strength of the first magnet ; while h is some quantity depending upon the magnetic moment of the needle n s and upon the angle 6 of deflexion, and so will be constant while is con- stant and while n s remains unaltered. Since the two are in equilibrium, we have Hi h = k (! x 360 + oj + 6) or >!!=(! x 360 + a, +6). CH. in. MAGNETIC MEASUREMENTS 23 Repeating the process with the second magnet pole p. 2 , arid turning the torsion head through such a number of degrees that remains constant, we have A respectively. Now in electricity we have, roughly analogous to the above, the question of electrical potential,. difference of electrical potential, and quantity of electricity. We take as a zero of level (whether it is quite arbitrary or not we do not here discuss) the electrical potential of the earth ; and we reckon potentials as + or when above or below our zero respectively. In gravity levels the test of the same level and different levels is CH. v. INTRODUCTORY CHAPTER ON POTENTIAL 6/ to see whether such a fluid as water will, or will not, flow from the one spot to the other when free to move. We call that the higher level from which the water flows. In electrical levels (or potentials) two points are at the same potential or different potentials, according as + electricity is not, or is, urged from the one to the other point when free to move. That is called the higher potential from which the + electricity is urged. Thus in gravity a spot has a level if water will run from the sea-level to it ; in electricity a point is at potential if + elec- tricity will flow from the earth to it. In the dual nature of electricity, and in the opposite move- ments of + and electricity, we perceive the limitations of our rough analogy. 3. Measurement of Differences of Electrical Lsvel by Work. In Chapter X. we shall discuss this matter of measure- ment of potential more fully. Here we wish only to show in what way the measurement is made. The measurement of gravity levels may be made by means ot a tape, caused to hang vertically by being weighted at the end ; or by some equivalent method in which the action of gravitation gives us either a vertical line to measure along, or a horizontal plane from which the vertical can be deduced. In all such cases the measurement is made directly in feet. But there is another way that is theoretically possible. We shall in Chapter X. consider the meaning and measure- ment of work. Here we will only say that in lifting weights from a lower to a higher level we do work ; that the work done is pro- portional to the weight raised, and also to the vertical height through which it is raised ; and that it is independent of the route, whether direct or roundabout, by which the weight is raised from the lower to the higher level. Now if we could in some way accurately register work done, then, by carrying some standard weight from one place to another, we could, by the amount of work done, estimate the difference of level of the two places. There has, however, been elaborated no exact method of this sort, because the former method is so very simple and direct. In electricity, on the contrary, there is known no method analogous to the direct plumb-line measurement. The only F 2 68 ELECTRICITY CH. v. conceivable method of measuring the difference of electrical level between two points is by measuring the work done in moving a + unit of electricity from the one point to the other. If we assume that the reader understands in what unit work is measured, we can say that we have now shown how to give an exact numerical meaning to that symbol ' V ' which is used to express the potential of a point with respect to the earth as zero. Notes. (i. ) We are said to do + work when we move against the lines of force, or expend energy ; we are said to do work when we move down the lines of force, or have energy expended on us. (ii.) The reader will understand from the above section, that the electrical potential of a point in space (or of a body) is + , zero, or , according as we do + , zero, or work respectively in moving a + unit of electricity from the earth to the point in space (or body). The potential is higher as this work is greater. The + unit of electricity was defined in i of this chapter. Any work done in moving it from one place to another is due to the repulsive and attractive actions of other electrical quantities on this + unit. [As regards sign, see Chapter IV. 7.] 4. Elementary Ideas on ' Capacity.' A certain quantity of water will fill a certain vessel to a definite level, this level depend- ing on the quantity of water and on the dimensions of the vessel. The more the water, and the less the horizontal section of the vessel, the higher the level to which the latter will be filled. In electricity, when we charge a conductor with a quantity Q of electricity, we raise the conductor to an electrical potential V. This potential will be greater or less according to a property of the conductor which we may naturally call, by analogy, its electrical capacity. We call the capacity greater or smaller according as the potential V, to which a given charge Q of electricity will raise it, is smaller or greater respectively. Now we have already indicated how Q and V are defined and measured. Since capacity (K) is a new term we may define it, and express it numerically, by the relation which is equivalent to saying tfie capacity of a conductor is directly proportional to the quantity of electricity required to raise it to a CH. v. INTRODUCTORY CHAPTER ON POTENTIAL 69 certain potential, and inversely proportional to the potential to which it is raised by a certain quantity. Of the units in which V and K are measured we shall treat more fully in Chapter X. We see, however, from the above definition of K, that A conductor has unit capacity when unit quantity raises it to unit potential, or to unit electrical level above the earth. 5. Lines of Force, and Equipotential Surfaces, We have in Chapter III. sufficiently explained what are meant by lines of force. The reader can easily, mutatis mutandis, define for himself lines of force in a gravitation field of force, or in an electrical field of force. Thus in an electrical field a line of force is a line along which a particle charged with + electricity, and free from all other forces, would move under the influence of electrical forces ; it being assumed that the movement is indefinitely slow, so that there is not any ' centrifugal ' desertion of the line of force when this is curved. It is along these lines of force that electrical level is measured; just as it is along the lines of gravitation force (i.e. vertical lines) that gravitation levels are measured with the plumb-line. It is in movement of electricity along the lines of electrical force, as in movement of masses along the lines of gravitation force, that we do work. Surfaces over which we can move our + unit of electricity without doing work are called equipotential surfaces. In the case of gravitation these are clearly horizontal surfaces ; a mass moved over these surfaces always cuts at right angles the lines of force of gravitation, and hence no work is done, or these horizontal surfaces are equipotential. In electrostatics let us consider the simple case of a single isolated charged particle. Since a + unit would be urged straight from it or straight towards it, according as the charge on it is 4- or , it follows that the lines of force are straight lines radiating from the particle as centre. What are the equipotential surfaces, i.e. level surfaces, or surfaces of no work ? They must be those over which our + unit always cuts the lines of force at right angles, thus doing no work. That is, they must be spheres having the particle as centre 70 ELECTRICITY CH v. and the lines of force as radii. In more complicated systems of electrical charges we may have lines of force curved about in any way whatever. But still the equipotential surfaces, over which our -f unit moves without doing any work, will be those that cut at right angles the lines of force. From what we have said the reader will see that wherever are lines of force, there is rise or fall of potential ; and conversely. A region where are no lines of force must be all at one potential ; since there can be no work done in moving our + unit against zero force, and therefore there can be no differences of electrical level. And conversely, where a region is all at one potential, there are no lines of force. 6. Induction ; from a ' Potential ' Point of View. We will now consider briefly what light the above considerations throw upon the matter of electrostatic induction. Definition of a conductor. We must first, however, define what we mean by the often-used term conductors. We mean bodies on which electrical charges are perfectly free to move, bodies which can resist no electrical stress due to differences of electrical potential. By our definitions then it follows that when any conductor is left until the electrical charge has had time to arrange itself, it will be all at one potential ; for were two parts at different poten- tials, the electrification would readjust itself until the two places were at the same potential. This readjustment is so rapid that we can consider it to be practically instantaneous. The reader may object that one might conceivably have a con- ductor of which two parts remain at different potentials for the reason that there is no electrical charge to be readjusted. The answer to this is that experiment shows us the universal presence of unlimited quantities of + and electrifications in all bodies, these being equal in amount if the body be in what we call an uncharged condition. If two parts of the conductor be at different potentials, then there is unlimited + electricity ready to flow from the higher level to the lower, and unlimited electricity ready to flow from the lower to the higher level ; and a flow takes place until all is at one potential. Hydrostatic analogy to electrostatic inductions. We may with advantage discuss, in connection with the matter of electrical induction, a hydrostatic analogy. We shall consider the sea-level CH. v. INTRODUCTORY CHAPTER ON POTENTIAL /I as analogous to the electrical level or potential of the earth ; a hill- side as analogous to a region through which there is a fall of 'electric potential, or what we may term an electric hill : and a trough of water as (very roughly indeed) analogous to an electrical conductor. In a still rougher sense we may sometimes take a tower as analo- gous to a + electrical charge \ and a 7^// as analogous to a elec- trical charge. We may now note the following facts. (a) The water in the trough will always be at one level, or water will have no more tendency to flow in one direction than in the other, in whatever position the trough be placed. (b) If the trough be placed upon a hill-side, the whole will be above the level of the sea ; and water would flow from any part of the trough down to the sea, if connected therewith by a pipe. (c) Further, the water will rise at one end and will sink at the other ; still remaining level. It will be shallower at the end that lies uphill, and deeper at the end that lies downhill. (d) If instead of a trough we have a trench of unlimited depth, and connected this with the sea-level, water will flow out until all is at the sea-level. The water will then be at the sea-level, but will be deep below the surface of the hill. (t') If the hill-side now sank, with the trench still cut in it, to the sea-level, the water in the trench would be now below the sea- level ; and water would flow from the sea into the trench until the level was again raised to sea-level. (/) Let there be a trough of water whose surface is at the sea- level. Now let this trough be raised up on to a hill-side and there be put in a sloping position. The water will dispose itself as in (c) above. Now let a partition be put in the middle of the trough, parallel to the ends, and let the whole be replaced in its original position. It is obvious that in the one half we shall now have the surface above, and in the other half the surface below, the sea-level. On removing the partition the original state of things is restored, and the surface is once more all at sea-level. (#) Let us build a tower on the sea bottom so that its top reaches the sea-level. If the sea-bottom were to rise up, the top of the tower would now be above sea-level. (h) Let us dig a well on the hill so that its bottom reaches sea-level. If the hill sinks to sea- level, the bottom of the well will be below this. /2 ELECTRICITY CH. v. We will now consider the facts in electrical potential, or elec- trical level, that are analogous to the above. Both by the word- ing of the statements and by the correspondence of the lettering (a\ b', c, &c.), the reader will be able to see how the two sets of facts answer the one to the other respectively. We must, however, .again repeat that, though such analogies are interesting, too great stress must not be laid upon them ; above all we must not argue from analogy. Let us suppose that there is a room whose walls &c. are all connected with the earth, and are therefore at the same electrical level, or potential, as the earth ; this we take for our arbitrary zero of potential. Further, let there be a +ly charged body A sus- pended in this room. There will be a fall of potential, or an electrical hill, from this body A, down to the walls, &c. Such was the case, if only we change the sign of the charge, in Chapter IV. 14. If any insulated uncharged conductor B be placed between the walls and the body A, this body B will also be at a + potential ; it will take work to bring a + unit of electricity from the earth up to B, in consequence of the presence of A. So that A has given rise to an electrical hill running down to the walls ; and all insulated un- charged bodies placed, as it were, on this hill, will be at a + poten- tial ; though of course at a lower potential than is A, since it will take less work to bring a + unit of electricity up to B than up to A. The corresponding electrical facts are as follows. (a) A conductor such as B will always be at one electrical potential over all, wherever it be placed. (b 1 ) When the insulated conductor B is placed between A and the walls, i.e. on an electrical hill, the whole is above the electri- cal potential of the earth ; and -f electricity would flow from any part of this conductor to the earth, if it be connected therewith by a wire. (c'} Further, we shall find electricity at the end nearer to A, and -4- electricity at the end nearer to the walls. This readjust- ment, or separation of + and electricities, will be just such as will leave B all at one potential ; or will be such that a + unit of electricity will move on B in any direction indifferently, and this in spite of the presence of A (see Chapter IV. 14, Experiment (i )). (d 1 ) If we connect B to earth by a wire, + electricity will cir. v. INTRODUCTORY CHAPTER ON POTENTIAL 73 flow out of B until it is all at zero potential. It will then be charged with electricity, though all at zero potential. Since B is all at one potential, this flow of electricity to earth will take place from any point on it indifferently (see Chapter IV. 14, Experiment (ii.)). (e f ) If A be now discharged, or reduced to the level of the earth's potential, our electrical hill disappears. It is now found that B is at a potential ; and + electricity will flow from the earth into it until it be again at the earth's potential. (/') Let an uncharged insulated cylinder, capable of division about its centre \see Chapter IV. 14, Experiment (iii.)), be placed between A and the walls. It will now exhibit a + charge at one end, and a charge at the other. If the two halves be separated and A be removed or discharged, we shall find these halves to be above and below zero potential respectively. On again connect- ing the two halves the whole resumes its original condition of zero potential. (g') When a body is in a region of potential let us give it such a + charge as will raise it to zero potential. If the region rises in potential, the charge given to the body will cause it to be now above zero potential. (/i 1 ) So when a body is in a region of 4- potential let us give it such a charge of electricity as will reduce it to zero potential. If the region now sink to zero potential, the body with its charge will be below zero potential. From what we have said above it is clear that all cases of 'induction can be regarded as electrical redistributions that take i)lace on conductors when placed in regions of varying (or unequal} potential : these redistributions being such as to maintain the con- ductor at the same potential all over. Note. Charging the gold-leaf electroscope by induction. In Chapter IV. 14 (iv.) we indicated how we could charge the gold-leaf electroscope induc- tively. It is now time to explain more exactly what occurs. Let the electroscope be uncharged, and let a + ly excited rod be approached. The electroscope will be in the field between the rod and the earth, and it will be at a + potential. There will, therefore, be a field of force between the leaves and the wall ; and + electricity will be ready to flow from any part of the instrument to earth. The leaves will, therefore, move from each other towards the walls, or will ' diverge with + electricity. ' If now we touch the instrument with the hand, we reduce it to zero potential, and the leaves collapse ; it now has a charge sufficient to keep it at zero potential 74 ELECTRICITY CH. v. in the presence of the + charged rod. If the hand be removed, and then the rod, the instrument will be at a potential in virtue of its - charge, there being now no region of + potential to be counteracted ; and the leaves will 'diverge with electricity.' The approach of a charged rod will cause its potential to be still greater, and the leaves will diverge more. But the approach of a + charged rod will again cause the region about the electro- scope to rise in potential. Hence, according to the strength of charge of the + ly excited rod and the nearness of its approach, the electroscope will become less , or zero, or even + in potential ; and its leaves will diverge less, collapse, or diverge with +,' in the three cases respectively. 7. Necessity of distinguishing Sign of Charge and Sign of Potential, Where we have an isblated charged body, there the potential of the body is + or according as the charge is + or in sign. For we do + or work respectively in moving our test + unit of electricity from the earth to the body. But if we have more than one charged body we cannot predict the sign of the potential of a body from its charge. Thus if we have a pail charged +, a ball hung inside it may be at a + potential even if it be itself charged ; for, owing to the presence of the pail, it may require + work to bring our test + unit of electricity from the earth up to the ball. So again, in Chapter IV. 14, Experiment (ii.), the conductor C D was at zero potential although it had a + charge on it. 8. Further on Capacities. When a body is charged with a quantity + Q, there is a complementary quantity Q on the surrounding surfaces. If these surroundings are so far off that this charge Q exerts no perceptible force on our test-charge of a ' + unit ' as we carry it from the earth up to the body, then the work done on our ' + unit ' as we so carry it will be independent of the surroundings ; it will depend only on the charge Q of the body, and on the shape and size of the latter. And the potential of the body, measured by this work, will also depend only on its shape, size, and charge. In such a case the body is said to be isolated, and it has a definite capacity. But if the surroundings be not so far removed as to make the above condition hold, then the work done on our ' -f unit ' will depend in part on the position of the surroundings. In such a case the potential of the body depends not only on its shape and size and on the charge given to it, but also on the position of CH. v. INTRODUCTORY CHAPTER ON POTENTIAL 75 surrounding surfaces. The body is then not isolated ; and we must consider the whole system of body and surroundings. Note. A little consideration will show that if a body previously 'isolated' be brought near (e.g.} to the walls of the room, most of the complementary Q will now be on the part of the walls near to the body. This will have the effect of lessening the work required to bring our + unit up to the body. In other words, the nearer approach to the walls will increase the body's capacity- ELECTRICITY CH. VI. CHAPTER VI. ELEMENTARY DISCUSSION OF CONDENSERS. (For further, see Chapter X.) i. General Ideas, Apparatus used. In this chapter we consider the alteration of the capacity of a conductor as the dis- tances and positions of surrounding conductors alter. Or, more accurately, we consider the electrical capacity of the system of conductors taken as a whole. We shall use the following apparatus. FIG. i. (i. ) The sliding condenser plates. (ALpinufs condenser. ) A and B are two discs of metal, having well-rounded edges. They are FIG. ii. CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 77 insulated by being fixed on to elbow-bent rods of glass. These rods are set in stands that run, guided by rails, along a board three or four feet in length. (ii.) A form of Peltier electrometer. Insulated inside a glass shade, the method of insulation being indicated in the accompany- ing figure, is a frame of wire abed. Balanced on a steel point is a magnetic needle n s, provided at the ends with gilt pith- balls. The apparatus is so placed that the needle, as directed by the earth's field, rests with the pith-balls just lightly touching the sides of the brass frame. When the knob K is connected with a conductor at a + or potential, the frame and needle all become of a + or potential with respect to the earth ; there is an electric field, and the needle is urged away from the frame towards the walls. Or, in ordinary two-fluid language, if we connect K with a conductor that we have charged 4- or , we shall have the needle urged away from the similarly charged frame towards the walls of the room ; and it finally rests in such a position that the couple due to the electrostatic field balances the couple due to the earth's magnetic field. (iii.) We shall also use a charged ebonite rod with which to test the sign of the charge that may be at any time causing the needle of the electrometer to be deflected ; this testing having been explained earlier. We may add that the electrometer can be made more or less sensitive by weakening or strengthening the magnetic field acting on the needle, by means of a controlling magnet, as explained in Chapter XVII. 9. (iv.) Source of electricity. For charging the plates we shall use an electrophorus or frictional machine ; and shall assume that it is so regular in its action that it is of a constant potential. That is, the machine will continue to give a charge to a conductor, cease to do so, or receive a charge back from the conductor, according as this latter is below, up to, or above, this fixed poten- tial of the machine respectively. ?8 ELECTRICITY en. vi. The analogy would be a water-supply kept at a certain fixed level. 2, Experiments with the two Condenser Plates. In the following experiments the two plates A and B are connected with the electrometers E A and E B respectively, and are insulated, unless the contrary is stated. Experiment. .(i.) Let B be at some distance, say -5 metre, from A, and let both plates be initially uncharged. Now charge A with (say) + electricity, until it will receive no further charge from the source. It will be found that FIG. i. the needle E A is deflected with + electricity. At the same time the usual proof-plane test will show that on the face of B turned towards A there is electricity ; while E B is deflected with + electricity. In two-fluid language we should say that the * charged body A induces a charge on the face of B turned towards it, repelling the complementary + to the further side of B and to E B (see Chapter IV. 14). In potential language we should say that there is on B and E B such a redistribution of the total charge (whose algebraic sum is zero) as to leave this system at one potential. Since the whole system is between a + charged body A and the earth, it must be at a + potential, since it would take work to bring our + unit from earth up to B. Hence + electricity tends to run from all parts of the system to the earth ; and hence the needle of E B is urged down an electric field between the brass frame and the walls. Experiment. (ii. ) Put B to earth, by touching any part of it. It will be seen that E B returns to zero deflexion, while EA is deflected less than it was before, and further that A will now receive more charge from our source. In two-fluid language we can say that we have allowed the repelled + to pass away. In consequence of this, A has further inductive action on B j so that there is more on B than CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 79 before, and more + passes to earth. Corresponding to this larger charge on B, there is drawn over to the surface of A that faces B more of the + charge on A ; and, as a consequence, E A is less deflected, and the whole will receive further charge. (The reader may here look at the diagram given at end of this section. It represents roughly the effect that the presence of B has on the distribution of the charge on A.) In potential language we may say that we have put B and E B to zero potential. There is therefore no field of force between the brass frame of E B and the walls, and hence the needle has no couple deflecting it. Now, there was in (i.) on the side of B turned towards A a certain charge. . This by itself would have made B of a poten- tial ; but, owing to the presence of the + charge on A, B was yet at a + potential. In (ii.) we have B at zero potential, since it is to earth. It seems clear, then, that there must now be a larger charge on B than there was before, since it is as close to the + charge of A as it was before and is yet at zero potential. For a similar reason to that indicated in Chapter V. 8, note^ the presence of this larger charge near A will lessen the work required to bring a ' + unit ' up to A. In other words the poten- tial of A will be in this case (ii.) lower than it was in case (i.). Experiment. (iii. ) Re-insulate B, and approach it nearer to A. We shall find that E B again diverges with 4- , while E A falls a little in its deflexion. And further, as we should expect from the falling off of the deflexion of E A , we can give A more charge from our source. In two-fluid language we have a further separation of -f and on B, the + being repelled to the regions (such as E B ) remote from A. There is also in A a further concentration of the charge on the face turned toward B, and consequently a weakening of the charge in E B . We should thus expect to be able to give more charge to the whole, A and E A . In potential language we have moved B higher up the ' elec- trical hill ' that exists between A and the earth, or into a region of higher potential than before. If, before, its charge of electricity was such as to keep it just at zero potential, this charge will be insufficient to keep it at zero now that it is nearer to the + charged A. It will therefore again become of a 4- potential, or + electricity 8O ELECTRICITY CH. vr. will be ready to flow from it to the earth. The electrometer needle is again in an electric field, and is again deflected. At the same time the potential of A will be lowered for a reason similar to that given in case (ii.). Experiment. (iv.) Next put B to earth. We find that EB is no more deflected, E A falls slightly. The discussion of this will be similar to that of Experiment (ii.). Experiment. (v. ) Again insulate B and remove it further from A. The needle EB is deflected with this time, while E A is deflected more strongly with + . In two-fluid language the removal of A and B from each other causes some of the + and electricities, concentrated on the op- posed faces of A and B respectively, to pass away from these faces to the other portions of the conductors, and so to E A and E B . In potential "language the potential of A will be raised, for a reason similar to that given in cases (ii.) and (iii.). There will be a stronger field between the wire frame of the electroscope and the walls, and the needle will be more deflected. With respect to B, the charge that kept it at zero potential when in the nearer presence of A, is too much for this purpose when A is removed to a greater distance ; it renders B in sign of potential ; and the charged needle of the electrometer is in an electric field running down from the walls to the frame, and therefore moves up this field away from the brass frame. It is important to note that by charging A when B is near, and by then removing B, we can get in A a charge of a much higher electrical potential, or level, than that of the source from which we charged A initially. Experiment. (vi.) When A Bare very close and B is to earth, the charge that can be given to A [roughly measured by counting the number of sparks given] will be seen to be very large indeed as compared with the charge that can be given to A when isolated. The proof-plane, if it can be used, will show that this charge is mainly on the face of A turned towards B ; while on the opposed face of B is an opposite charge of equal magnitude (see 4). If B be pushed still nearer, a brilliant blue spark will be seen to span the gap between A and B, and a strident ' crack ' will be heard. If a person holding B touch A, this discharge will pass through him and will give him a shock. Such an experiment must, however, be tried with caution. CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 8[ The accompanying figures show roughly the cases of A charged when isolated, and of A and B acting as a condenser, respectively. The crowding of the signs + and is meant to indicate the con- centration of charge. 3. Discussion of the Terms 'Bound' and 'Free.' We have seen that when a body is charged with a certain quantity, say + Q, of electricity, there is induced : , . , . , FIG. ii. FIG. iii. on surrounding objects an equal quantity of opposite sign, or Q (see Chapter IV. 16). The distribution of this quantity + Q on the conductor depends on its form and on the relative positions of surrounding bodies. It is found to be part of the essential nature of * electricity ' that a + or charge does not exist alone ; it is always the one or other side of an electrical field. Now the charge on the plate A (see figure at end of 2) in- duces on the whole an equal and opposite charge on the walls, &c., and the greater part of this will be on that surface of B which is opposed to A ; since B is by far the nearest part of the earth that is presented to A. If we now cut off B from the earth and insulate it, without otherwise disturbing the system, and then put A to earth, what will happen ? To begin with we may say roughly, but as we shall see in 6 not quite exactly, there will pass away from A that part of the charge that can, by so passing away, get at and neutralise the cor- responding part of the induced charge. If B were still to earth, then all the charge would pass from A, and the two sides of the field (the equal and opposite + and charges) would close in and leave no field or charge ; the + and would entirely ' neu- tralise ' each other. But B has, we suppose, been insulated again before A is put to earth. Hence, to speak roughly, that part of the charge on A which answers to the charge induced on B, will have no tendency to pass away, since it cannot thereby get at the charge on B, which is the opposite side of its field. But if we stood on B, insulated from the earth, and then touched A, we should withdraw from A all that part of the charge answering to G 82 ELECTRICITY CH. vi. the charge induced on B ; while on A would remain that part of its charge which has its corresponding induced charge on the walls and ceilings, &c., from which we are now insulated. When B is insulated, therefore, we can divide the charge on A into two portions ; one part can be withdrawn by a person in connection with the walls, ceiling, &c, and unconnected with B, but cannot be withdrawn by a person connected with B and insulated from the walls, &c. ; the other part can be withdrawn by a person con- nected with B and unconnected with walls, c., but not by a person insulated from B. The former is free with respect to any person connected with the walls, &c., the latter is bound with respect to such a person. ,-^r This is the origin of the terms bound and free ; the reader will see in how relative a sense they must be understood. 4. Conditions affecting the Magnitude of the ' Bound Charge. When B is very close to A we may neglect the * free ' charge on A as relatively insignificant (see 3). The ' charge ' of the condensing system, composed of A and B, will then mean the ' bound charge ' residing on that face of A that is turned to- ward B ; the inseparable accompaniment of an equal and opposite charge on B will be understood, but not generally mentioned. The magnitude of the charge can be experimentally shown to depend on several conditions. (i.) On the size of the plates A and B. If these are equal in area, and if the distance between them be very small as compared with the diameter of either, then the charge will be (approximately) directly proportional to the area of the plates. (ii.) On the distance between the plates A and B. The nearer the plates, the greater the charge. And, with the condition just given, the charge is inversely proportional to the distance between the plates. (iii.) On the difference of potential between the plates A and B. We have not yet explained exactly in what units we measured electrical level or potential ; but we have sufficiently indicated the nature of the unit employed. For further, the reader must wait for Chapter X. 9, 19, 25, &c. W T e may, hovvever, here state that the charge is directly proportional to the difference of the potentials of A and B. (iv.) On the nature of the dielectric (see note). If the charge is CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 83^ of a certain magnitude Q when the dielectric is air, then cceteris paribus it will be m ' - m' <^ m nt 3 m* - m* ^> m* ' "" < m* m 5 m* J^ S* CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 85 Note. As regards the exact meaning of m, the reader will understand the following explanation better when he has read Chapter X. Let us consider the plate A only. There is on the inside surface next to B a bound charge Q ; and on the outside surface, opposite to the relatively distant walls, a relatively insignificant charge Q'. The charge Q raises the inner surface, and the charge Q' raises the outer surface, to a potential V A , which is necessarily the same for the two surfaces. If then K is the capacity of the inner surface, and K' the capacity of the outer surface, we have the relation V A - ^ = ^ We may call K the ' bound capacity ' of A, or the capacity K. .K. of the condensing system, its magnitude depending on the shape and size of A and also on the position &c. of B. We may call K' the ' free capacity ' of A or rather of its outer surface ; its magnitude depends on the shape and size of this outer surface, but on nothing else provided that the walls &c. are so remote that the outer surface of A may be considered to be ' isolated ' with respect to them. From the relation given above we easily obtain the result that / Remembering then that Q + Q' is the total charge on A, and that the K * charge on B equals Q in magnitude, we see that in - -. K + Iv Experiments* (i.) With the plates A and B, each connected with a Peltier electrometer, the alternate discharge can be shown ; though the arrangement is not one adapted to give indication for more than a few discharges. As we touch A we receive a spark, and EB, previously undeflected because at zero potential, will be deflected with electricity, B and E B being now at a potential, while E A will fall to zero deflection. If we now touch B, E B falls to zero and E A is deflected with + , and so on. (ii. ) A Leyden jar (see next section) is mounted on an insulating stand. While it is being charged the outer coating is put to earth, and then the whole is left insulated. We can, by alternately touching the knob connected with the inside, and the outside coating, draw off sparks of + and electricity in turn?. The signs of these alternate charges can readily be tested as in Chapter IV. 7. Leyden Jars. When it is required merely to store a large charge at the potential of the source, and when there is no need to alter the distance between the plates, the condenser usually takes the form of a jar of glass having inside and outside coatings of tin-foil. A knob gives connection with the inside of the jar ; this knob should, if possible, pass up from the inside without any connection with the neck of the jar, in order to insure good insulation. Since the charge, cczteris paribu s, is proportional to the differ- 86 ELECTRICITY ence of potential of outside and inside, it should be our object to make this difference as large as possible. As a rule the highest available potential is that of our source, and with this we connect the knob ; the lowest available is the zero potential of the earth, and so we put the outside coating to earth. A larger charge would be obtained if we could put the outside to a potential. When the outer coating is to earth, the charge of the jar is determined by the formula Q=K.V where K is the capacity of the jar, and V is the potential of the inner coating. Glass is used for several reasons. It is easy to make glass jars ; glass has a high specific inductive capacity (see Chapter IX.); and the jar can be made thin, or the distance between the plates small and therefore the capacity of the jar large, without a discharge taking place through the glass. For the calculation of K, as well as for the units in which V is measured, the reader is referred to Chapter X. 8. The Unit Jar. If we can fix the capacity of a jar and the difference of potential between its two coatings, then we have fixed the charge in that jar. A unit jar is a small Leyden jar of some convenient shape ; usually so made as to admit of easy cleaning and drying, and to insulate well. Connected with the outside is a knob B, and connected with the inside is a knob A, whose distance from B can be regulated by sliding the piece A C along the graduated rod DE. It is proved experimentally that the distance between A and B across which the jar will just discharge itself is directly proportional to the difference of potential between A and B. CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS S/ (It depends also on the curvature of the knobs, and on the medium between A and B ; but we suppose these conditions to remain constant.) Hence, if we keep the distance between A and B fixed, the difference of potential between A and B at the moment when the jar discharges itself in a spark across the gap A B will also be fixed. Let us now put the outer coating of the jar to earth, put A at a fixed distance (that we will call i) from B, and charge the jar by placing the knob D near the prime conductor of a machine ; and let us suppose that it takes a difference of potential mea- sured by v\ for a spark to pass from A to B. Then on charging the jar we shall shortly perceive a brilliant discharge to pass from A to B ; this discharge leaving the jar uncharged. At the moment of this discharge there was on the inner coating a charge + Q, determined by the relation QIK(V A -V*) or, by hypothesis, Q, =K v } . There was, on the inner surface of the outer coating, a charge Q! ; and there had passed to earth a quantity + Q t . Hence, every time that the jar discharges itself, the + Q, and the Q], that are opposed to each other, neutralise each other; while there has passed to earth a quantity + Q,. If we put A and B at a distance apart twice as great as before, or at a distance 2, then we have Q 2 = K . v 2 K . 2 v\ = 2 Q,. Let us now see how the unit jar is used. We connect the outside coating of our small unit jar with the inside of a large Leyden jar whose outer coating is to earth. If we now work the machine, then each time that the unit jar discharges itself we know that a certain quantity + Q x of elec- tricity has passed away from the outside of the unit jar into the large jar. And so, by counting the sparks of discharge of the unit jar, we are able to give the large jar a charge represented by Qi, 2 Qi> 3Qi,&c. Comments and notes. (i. ) We ought to have the unit jar so small as com- pared with the jar to be charged, and the knobs A and B at such a distance apart, that a considerable number of discharges of the unit jar may take place by the time that we have given the required charge to the large jar. For we cannot stop charging the moment that a spark passes, and therefore there will always be a fraction of + Q, in excess given to the large jar. 88 ELECTRICITY CH. vi. (ii. ) We have said that, at each discharge of the unit jar, a quantity -f Q has passed into the large jar. This is not accurately true. The outside of the unit jar is a small conductor at a considerable distance from surrounding bodies, and the inside of a large jar is a relatively large conductor having the outer coating of the jar very near it and opposed to it. These two form one con- ductor at one potential, since they are connected ; but the capacity of the first is negligibly small as compared with that of the second. The charge + Q, distributes itself between these two portions of the conductor in the ratio of their capacities. Hence, we may say that very nearly the whole charge, that would have passed to earth if the outside of the unit jar had been to earth, will have passed into the large jar. (iii.) It requires a certain difference of potential v for a spark to pass between A and B. If then A is connected with the prime conductor of the machine, and B with the inside of a large jar, we cannot charge this latter up to the full potential of the prime conductor, but to a potential less than this by the amount v. 9. Cascade arrangement of Leyden Jars. P, Q, and R are three Leyden ars mounted on insulated stands, arranged 'in cascade ' as indicated in the figure. These jars must be equal in capacity. To insure this we may take one jar of many that are presumably equal, and then by comparison choose two or more others equal to it. If we give a certain charge to our jar chosen, and measure the potential of the inside (when the outside is to earth), this potential ought to fall to one-half when the knob is connected with that of an uncharged equal jar. This method is employed in Chapter IX. 4. But for rough experimental purposes it will be sufficient to choose the jars as follows. Let an electrical machine be worked for some time so that it has got into a constant condition. Then charge the jars in question with a unit jar interposed, as shown in the last section. If the jars refuse further charge after the same number of discharges of the unit jar have taken place, then their capacity will be approximately equal. CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 89 Having arranged our three (or more) jars as shown, let us work the machine. When no more charge will pass, then the inside of the first jar will be at a potential V t ; which will be that of the prime con- ductor, or something less, according as the knob is or is not in contact with the prime conductor. The outside of the first jar and the inside of the second jar (in contact with the former) will be at some lower potential V 2 . The outside of the second jar and the inside of the third will be at a still lower potential V 3 ; and the outside of the third will be at zero potential, which we may call V or zero-V, as we please. Now, as shown in the last section, if there be in the first jar a ' bound ' charge + Q, then there has passed into the second jar also a quantity + Q, and from the outside of this into the third jar also a quantity + Q, if we neglect the trivial ' free ' charges on the outside of the first jar and on the knob of the second jar, and so on. That is, in the cascade arrangement we have of necessity equal charges in the jar. But by Chapter VI. 4 (explained further in Chapter X.) we have ( for first jar Q = K . (V l - V 2 ) { for second jar Q = K . (V 2 - V 3 ) 1 for third jar Q = K . (V 3 - V ). So that, since Q and K are the same for each jar, we see that V, - V 2 = V 2 - V 3 = V 3 - V = (V, - V ). Hence the total fall of potential that is possible, viz. the fall measured by Vj V (or by V b since we take V as our zero), has been broken up into three equal falls. (This is somewhat like the breaking up of a large waterfall into three smaller ones.) Now let us consider the charge Q' that is given to one of these jars when its outside coating is to earth and its inside coating is at the potential Vj of the prime conductor ; i.e. its charge, when it has been charged in the usual way. By our formula we shall have Q' = K . (V, - V ). But by what has preceded we see that this equals 3Q. Hence 9O ELECTRICITY CH. vi. We see then that the sum of all the charges given to the jars, when these are arranged in cascade, equals the charge given to a single jar when this is treated in the usual manner. The above reasoning can readily be extended to any number ;z jars. .For further on the 'cascade arrangement' the reader is referred to Chapter X. 30. 10. Nature of the Leyden Jar Charge. It is now time to investigate, as far as we are able, what is meant by the expression 'an electric charge.' All that we have hitherto observed indicates that neither ' + charges,' nor ' charges,' can exist by them- selves ; but that there must always co-exist two equal charges of opposite sign. In fact, it would seem that in all electrostatic phenomena there must be a + charge and an equal charge, separated by some insulating medium called a dielectric. It is pro- bable that in the case of two conductors thus separated by a dielectric it is the surfaces of the dielectric in immediate contact with the conductors that are in the condition that we have called ' + charged ' and ' charged ' respectively ; though it is usual, and perhaps more convenient, to speak of the conductors them- selves as so charged. Various experiments tend to show that the dielectric is, under these conditions, subjected to a stress ; and to this it yields to a greater or less degree, becoming deformed or strained. The conductors appear to play somewhat the following part. They mark out the portion of the dielectric that can be converted into an electric field and can be put under a stress ; and they allow this stress to be imposed or taken away with rapidity. If the stress be continued for a sufficient time, it is found that all solid dielectrics become strained (or distorted) to an appreciable distance from the conducting surfaces ; and that then the strain cannot at once be removed, or the ' condenser ' cannot be at once discharged. Some of the stress remains and must be allowed to be relieved gradually. This gives rise to the ' residual charges ' discussed below. Experiments.- (i.) Leyden jar with mcveable coatings. A is a Leyden jar so constructed as to be separable into three portions ; B the glass jar, D the inside coating, and C the outside coating. The jar is put together, charged as usual, and then placed on an insulating stand. The inside is then removed, a small charge coming away with it, and CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS a small charge being ' set free ' on the outside coating. Then the glass is lifted out of its outer coating and set down on the table, and finally the outer coating may be handled and removed. In all this we notice but very slight discharges on touching D and C respectively. FIG. i. On carefully putting the whole together again (on the insulating stand), a discharging rod (fig. ii. ) will show us that the whole charge of the condenser has been (approximately) unaffected by the above process. Hence, since the coatings B and C were separately put to earth by handling, the charge of the condenser must have resided in the glass. We could not, how- ever, discharge the glass by itself ; it was necessary to have the metal coatings as distributors. (ii.) Residual charge. The penetration of the charge, from the surfaces inwards, is shown by the following. If a Leyden jar be discharged and then left for a time, a second small discharge can be obtained, and so on. That the interior of the glass is actually strained or distorted is shown by the fact that its optical pro- perties, when it is under the electrical stress, undergo an alteration similar to or identical with that produced by mechanical stress. When the electrical stress is excessive, the material may give way and a hole may be made ; through this a discharge takes place, since the air which then separates the two charges is far weaker to withstand the stress than is glass. 11. Various Effects of the Discharge, The Leyden jar gives us a means of observing the effects of electrical discharge. There is nothing essentially different between the discharge between the prime conductor and the floor and walls, where was collected the induced charge of opposite sign, and the discharge between the inner and outer coatings of the Leyden jar. Only the latter ELECTRICITY CH. VI. gives us a means of storing up quantities, ready for discharge, enormously greater than we should get did we discharge the prime conductor only. I. Mechanical effects. Piercing card, &^c. In several of these experiments we make use of a discharging table. Its construction is evident from the figure. One rod, A, is connected with the outside C of the jar, or battery of jars, and the other, B, can be connected with the inside D of the jar FIG. i. or battery by means of an insulated discharging rod E. The battery is charged, a piece of card placed between the points of the rods A and B, and then connection made. If the pierced card be examined, it will be seen that the hole has been apparently formed by an explosion from inside, due to the sudden expansion of air, or to sudden vaporisation of water in the inside of the card. Whatever the cause, it may help the reader to remember that ' the passage of a spark ' is an expression hardly authorised by our knowledge ; the discharge may take place from + to , or from - to + , or may be a series of alternate discharges, or may be of some dual nature not represented by any of the above suppositions. II. Magnetic effects. Experiment. (i.) In fig. ii., A B is a wire along which the discharge can be effected. If we, for convenience, agree to say that by the ' direction of a discharge ' we mean that direction in which the + charge would move in order to meet the charge, then in fig. ii. the discharge is from B to A. This expres- sion, ' direction of discharge,' is not here assumed to have a real physical meaning, but is employed in order to save confusion. n s is a piece of steel placed at right angles to the wire B A, and under- neath it. After discharge it will be found that the steel is magnetised in the way shown. That is, if we swim with the + electricity and face the steel bar, the induced north pole will be found at our left hand. CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 93 Experiments of this sort, and similar ones made with steel filings on glass, show us that round a wire bearing a discharge are lines of magnetic force ; they are circles round the wire as axis, and their + and direction is clearly given by the italicised rule above. (Compare also Chapter XVII. I and 2.) FIG. ii. FIG. iii. The reader can verify this rule when he has determined the sign of the charge on the prime conductor of his machine. (ii. ) We may (see fig. iii.) intensify the effect by coiling a wire round a glass tube in which lies the steel needle. The above rule will be found true, and it will be seen to be equivalent to saying that if we face one end of the spiral coiled round the tube, the needle will have been made n or s at that end, according as the discharge so viewed is against or ^v^th movement of the hands of a watch. A reference to Chapter XVII. 2 will make the rule clear. This experiment is one of the many that link together the electrostatical and electrodynajnical divisions of our subject. III. Heating effects. Experiment. (i. ) The brilliant light of the discharge, and the fact that discharges over the dust of various metals scattered over a glass surface give the well-known spectrum of each metal respectively, indicate that in the dis- charge there is an elevation of temperature sufficient to volatilise metals and other bodies, and to raise the vapours to brilliant incandescence. By experiments easily contrived we may reduce a platinum wire to fused drops ; or may volatilise gold-leaf between two well dried, and therefore in- sulating, cards. Experiment. (ii.) Ignition of gunpowder. If we place gunpowder on the small table of the discharger, between the points of the discharging rods, we find that the discharge merely scatters it. But if we interpose between A and C a substance, such as wet string, which 94 ELECTRICITY CH. vi. conducts worse than does a metal, we find that the discharge will ignite the powder. We conclude that this retards the discharge and gives sufficient heating effects without such violent mechanical disturbance. IV. Chemical effects. The subject of chemical decompositions may be left until we come to it under the head of ' Phenomena of electric currents,' in Chapters XL and XII. It will suffice here to say that if we keep to the convention given above, as to which we shall call the direction of discharge, then the chemical phenomena given later on in Chapter XL could be repeated with similar though less striking results if we substituted a series of discharges for the continuous currents employed in that chapter, 12. Induction Effects of the Discharge. Note. This section may be omitted until the student has read Chapter XXL We will first give a general account of the phenomena that are the subject of this section, and will then describe how these phenomena may be observed. If a discharge be effected along a wire B A, it is found that at the moment of discharge there passes along a wire C D, placed parallel to B A and near to it, an in- duced discharge. Further examination, in which we are aided by results of experiments with the more manageable currents given by voltaic batteries, shows us that the phe- nomenon is not a simple one. It is found that in C D are induced two rushes of electricity ; one answering to a discharge of + electricity from D to C, one in the contrary direction. It would seem that these two induced rushes of electricity are equal in total amount of electricity \ but that the rush from C to D, or that whose direction is the same as that of the inducing discharge from B to A, is the more violent in character. If we interpose a sufficient gap in the circuit C D E, we find- that the direct induced discharge (or that passing from C to D) will leap over the gap in the form of a spark ; while the inverse induced discharge (or that passing from D to C) will not pass, but will be suppressed. ELEMENTARY DISCUSSION OF CONDENSERS 95 Comparing with the results obtained in Chapter XXL, where we employ electrical currents over whose increase and decrease we have complete control, it would seem that as the discharge from B to A begins, there is induced a rush in the contrary direction from D to C ; while as the discharge from B to A fades away, there is induced a rush from to D in the same direction. It would seem also that the direct induced rush from C to D is the more violent of the two, for the reason that the cessation of the inducing discharge in B A is more abrupt than its commence- ment. The inducing current is called the primary ; the induced is called the secondary current. Experiments. In the figure, X represents a screen of card or of glass, on the further side of which is fixed a ' primary ' spiral of wire through which a Leyden jar discharge can be passed. On the nearer side of the screen is fixed FIG. ii. another, 'secondary,' spiral of wire, parallel to the other but separated from it by the glass or card. The ends of this last wire are connected with the ends of a wire coiled round a tube, C D. Then, by placing steel needles inside the tube C D, we can [by the results obtained in the last section] examine the direction of the current induced in the secondary spiral. Thus, if the end P of the needle were found to be a north pole, it would mean that the current passed in the general direction from t>6 ELECTRICITY CH. VI. P to Q ; since then, if one swam with the current and looked at the needle, the north pole would be to the left (see the. rule given in last section). We shall find that when there is no break in the secondary circuit there is a weak magnetising action making the end P the north pole. This implies a total magnetising action as of a current in the opposite direction to that in the primary circuit. Without further experiment this would indicate merely a weak inverse current induced in C D. Other experiments show us that as regards quantity of current in C D there is really a zero total, though the inverse current is the more effective as regards magnetising powers. But if we make a small break in the secondary circuit, such as will just allow a bright spark to pass, we find a relatively powerful magnetising effect, making Q a north pole. This indicates that now there is in CD a powerfully induced direct current. Other experiments, which we do not describe here, are needed to bear out fully our statements as given above. 13, Wheatstone's Spark-board. This was a piece of ap- paratus designed to investigate whether the spark of -discharge had any appreciable duration, whether the discharge passed from one coating to the other or from both simultaneously, and with what velocity it traversed such conductors as copper bell-wire. The apparatus is sketched in the accompanying diagram, in which the reader is supposed to be viewing it from above. FIG. i FIG. ii. In fig. i., L is a Leyden jar. The discharge takes place through the circuit YabcdefQ ; so that sparks appear at the three breaks ab,cd, and ef. Between b and c, d and e, are coils of wire of the same known thickness and length, say 1,000 metres of common copper bell- CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 97 wire. Between a and P, f and Q, there snould be wire of the same thickness and length respectively. In fig. ii., m is a small plane mirror revolving on a horizontal axis that lies parallel to the line of the three sparks a b, c d, and ef. Above it is a horizontal screen of ground glass, marked out in degrees in a manner indicated later on. The mirror can be made to revolve rapidly, its rate of revolu- tion being indicated by clockwork or by the musical note produced by its striking a fine metal wire. If the discharges take place very rapidly, then a person looking down from above will see some of the sparks reflected in the mirror as it passes through the position proper for reflection of the sparks to the eye ; others of the sparks will not be seen, owing to the mirror being unsuitably situated at the moment of their occurrence. The rotation of the mirror will of course distort, by ' drawing out ' in the direction of the motion any spark that lasts an appreciable time ; i.e. that lasts while the mirror has moved appreciably. Now it is found that, for moderate rates of revolution of the mirror, we see on the screen X Y merely three points of light, FlG. iii. FIG. iv. lying parallel to the sparks of which they are the images, as represented in fig. iii. But when the mirror revolves with sufficient rapidity, we see these points ' drawn out ' into three lines of equal length, covering a number of degrees on the glass screen that is definite for each definite rate of revolution of the mirror, as indicated in fig. iv. Further, we see that the bright lines answering to the sparks H 98 ELECTRICITY CH. vi. ab and efhzve their ends lying in a direction parallel to the line of sparks, while the bright line answering to the spark cd is displaced as shown. This shows (i.) that the duration of the spark is such that the mirror turns through an appreciable angle while it is reflecting ; and (ii.) that, whereas the sparks ab and ef occur simultaneously, the spark c d occurs later than either. The fact that the spark has duration was perhaps obvious beforehand; we cannot conceive of 'no duration.' But this apparatus gives us a measure of -.the duration of a spark under any particular conditions of distance between a and b, and nature of the intervening gas. But the fact that the lines answering to the sparks a b and ef are still similarly situate shows that these occur simultaneously, and that the discharge is of the dual nature indicated before, and that when we speak of its * passing from one coating of the jar to the other ' we are but using a convenient convention. 14. The Condensing Electroscope. Before concluding this chapter on condensers, we must describe the ' condensing electro- scope,' a piece of apparatus in which the principle of experiment (v.) of 2 in this chapter is made use of. If the reader will turn to the figure to Chapter XI. 3, he will there see the apparatus in question represented. On the plate N of a gold-leaf electroscope is placed another plate M, separated from the first by a thin insulating film of shellac. When M is put to earth, the plate N considered by itself will have a capacity very large as compared with its capacity when 'isolated.' Hence, if it and the gold leaves are raised to a very low potential when M is to earth, the charge that it has received will suffice to raise it to a potential many times greater when M is removed. Thus the leaves, which did not stir when at some small difference v from zero, may diverge widely when this has been multiplied to (say) 500 v. We can, in fact, multiply any potential v by the fraction, capacity of N, when M is on it, and is to earth capacity of N when isolated 15. Various Forms of Electrical Discharge. We have many times alluded to the passage of a ' spark ' or ' disruptive electrical ^, and at distance r B it will be - 2 , dynes ; v A) v R) r A and r B being measured in centimetres. What work then will be done in moving the particle from A to B ? This is evidently no simple matter of arithmetic, for we cannot by any arithmetical means find the average force as we did in 6. We might work out the problem here without going into any advanced mathematics. But, as the calculation is one which belongs essentially to the integral calculus, and is a very easy matter to settle with the aid of this powerful mathematical weapon, we prefer in an elementary Course to give merely the result. This result is that Work done in moving the _ -p / I I \ .^ particle from A to B ' V B r A ) * If A be so remote from O that the force at A is quite inap- preciable as compared with that at B, or so remote that z ^ and VA) *- are practically zero as compared with JL- and -L-, then ( r jj VK) VB) we may consider A to be at an infinite distance from O. We thus obtain from our formula the result that Work done in bringing the particle F from an infinite distance up to B r B * ' The reader is here warned to remember (i.) That F is the force acting on the particle when at unit distance from O. CH. x. ELECTROSTATIC POTENTIAL 13! (ii.) That r refers to the distance from O up to which the particle is brought, and is not the distance through which it is moved. 8. Potential, and Difference of Potential, If the reader will turn back to Chapter V. he will there see explained the general meaning of potential, and the measurement of the same with respect to some chosen zero. The explanation was, however, incomplete, because we had not then considered the question of Work. Referring to the diagram given in 7, we now understand that there is between A and B a certain difference of potential ; and that it is measured by the work done on some sort of unit particle in bringing this from A to B. If the field of force be one due to gravitation, and be con- sidered to be practically uniform and acting with a force of 981 dynes on i gramme mass, or with i dyne on | T gramme, then we do one erg work in raising our mass of -g-| T gramme through i centi- metre vertically. Or we could theoretically measure in centimetres the difference of gravitation potential QV level between two points by finding the work done in ergs in raising a particle of F | T gramme mass from one point to the other. If the field of force be electrical, then we take as our unit particle a + unit of electricity ; and we measure the difference of electric -potential between the points A and B by the work that is done on this test-particle in moving it from A to B. If the work is found to be negative, we say that the final posi- tion has a lower potential than the initial position. Note. The reader must observe that all fields of force can be considered from the potential point of view, and that we may speak of gravitation, potential as well as of magnetic- or electric-potential. 9. Application of 7 to ths Measurement of Electric-Poten- tial. Referring to the diagram given in 7, let us now consider the case of electrostatic forces. Let us take as our test-particle (see above) a + unit of elec- tricity ; and let there be situated at O a quantity of electricity measured by Q units, this giving us an electrical field of force. By the laws of electrostatics, summed up at the end of 13, Chapter IV., the force exerted upon our + unit by a quantity of K 2 132 ELECTRICITY CH. x. Q units will at unit distance be measured by Q dynes ; and, at any other distance r from the centre O, by -^ dynes. LetV A and V B represent the numerical value of the potentials at A and B respectively ; these being measured by the number of ergs work done in bringing our + unit from infinity up to the points A and B respectively. Then the formulae given in 7 tell us that The absolute potential V A v of the _ O point A r A * The absolute potential V B of the __ O point B r s The difference of potential, V B V A , _ Q / I i \ between the points A and B \r* r A An example or two will make the meaning clearer, (i.) Let there be + 24 units of electricity at the point O ; let r^ be 4 cms., and let r A be 6 cms. Then at unit distance (or i cm.) from O our + unit will be repelled with a force of 24 dynes ; at B with a force of 2i 4 = * dynes ; and at A with a force of ^ = - dynes. O The potential of B will be greater than that of A by an amount measured by V B -V A=24 (<->)=, Or it will take 2 ergs work to move our + unit from A up to B. The absolute potential of B will be measured by ~ = 6 ; and 4 ? i" that of A by 24 = 4 ; or it will o 6 A take 6 ergs to move our + unit from infinite distance up to B, and 4 ergs to move it from in- +10 finity up to A. (ii.) Let there be several quantities of electricity, as + 20 at O, 16 at P, + 10 at Q. These will give rise to a somewhat complex field of force. But still the results arrived at in 7 and in the present section enable CH, x. ELECTROSTATIC POTENTIAL 133 us to find easily the potentials of any points as B and A in the field, and the difference of potentials. For the total work due to this system of electric charges will be the algebraic sum of the works due to each charge separately. Thus the absolute potential of B will be measured by of Ab + - Hence the difference of potential, V B V A , is easily found by subtraction. (iii.) Where there are a whole series of quantities q^ q^ q z . . . situated at distances r^ r. 2 , r z . . . respectively from a point B, then the potential of B as due to these quantities will be ^J. + i + fl -f . . . ; which is conveniently represented by r \ r i r i 2 $. Here, of course, attention must be paid to the signs of the r -& quantities ^, &c. And V B - V A = S-?_- S -?. >B r A (iv.) Where the quantities are distributed continuously over a conductor, or over any surface, this adding up of -^ + . . . &LC. r \ becomes in general a matter for the application of the integral calculus. Note. It may occur to the reader that if r be zero, or if we touch the charge Q, then the potential becomes infinite, since V oo . In answer to o this we may remark that it is physically impossible to be at zero distance from any finite mass, or electrical or magnetic quantity. We can be at zero distance only from a geometric point. Hence, we cannot have Q finite and r zero simultaneously. 10. Equipotential Surfaces. (i.) If we so move our + unit of electricity as not to move up or down the lines of force, then we do no work against the electrical forces. Hence, by definition, all the positions thus arrived at must have the same potential. Let us consider the case of a simple field of force due to a single charge of Q units situated at the point O (see 7, figure). If we move our + unit over any sphere that has O as centre, we I 34 ELECTRICITY en. x. do no work ; and hence every point on such a sphere has the same potential. Thus if we consider the sphere that his O as centre and that passes through the point A, it-is clear that all points on this sphere have the same potential as A ; a potential that is measured by -i ergs. It is necessary to do work to the amount of Q ( [ ) >1s ^"A/ ergs to move our + unit of electricity from any point on the sphere passing through A to any point on that passing through B. (ii.) Equipotential surfaces in general. Where there are many centres of force, there both the lines of force and the equipotential surfaces will be complex, and will not have the simple forms that they had in the last case. But in all cases certain general results hold good. The equi- potential surfaces are always perpendicular to the lines of force (see n); they may be mapped out into a series of 'marked' surfaces such that we do one erg work in moving our + unit between two consecutive surfaces ; these marked surfaces will be closer together where the field is stronger, further apart where the field is weaker, and at a constant distance where the field is uniform. ii. Lines of Force are Perpendicular to Equipotential Surfaces. This has been already proved implicitly ; or rather it is a consequence of the very definition of an ' equipotential sur- face.' But it may be well to give a formal demonstration. Let P F be a line of force ; and let A P B be a section of the equi- potential surface that passes through P ; and let P T be a section of T the tangent-plane to this surface, at the point P. ^ B Then shall P F be perpendicular to the tangent P T. For if possible let it lie as P F' ; making with P T an angle less than 90. We can now resolve the force P F into two components, along the tangent P T and perpendicular to it respectively. For the former component we have P F cos F' P T ; and this is not zero, since F' P T is not 90. Since then there is a tan- gential component, we should do work in moving our + unit along CH. x. ELECTROSTATIC POTENTIAL 135 the surface near P. But this is contrary to hypothesis, since A P B is an equi potential surface. Therefore, F P must be perpendicular to A P B. 12. Field- strength.; and Rate of Change of Potential* The diagram represents a portion of a field of force ; this por- tion being so small that we may consider it to be uniform, or the lines of force to be parallel (see 13, coroll.). Let A and B be any two points in this field, and let them be separated by a dis- tance of A B cms. If A B be not perpendicular to the lines of force, i.e. if A and B be not on the same equipotential surface, there will be a force on a + unit of electricity acting along the line A B. We may represent the force on a + unit that acts from B to A by the symbol FA ; this will be measured in dynes, and may be + or as it acts from B to A or from A to B. [See i (i.) ; and compare Chapter II. 8.] By the definition of * field-strength,' c., it is clear that F" measures that component of the field-strength that has the direction B A. Then, by the definition of work, we have The work done in moving = F B x A B er(rs our + unit from A to B But, by the definition of potential, this work measures the difference of potential between A and B ; or we have (V B V A ) = F* x A B. Whence F = V,- V AB Now since A B is measured in centimetres, this last expression represents the change of potential per unit length, or the rate of change of potential, along A B. So we may translate this formula (F* = B ~" A ) into words A Jj / somewhat as follows. 136 ELECTRICITY en. x. In a field of force, the component in any direction of the force acting upon a + unit, i.e. the component in any direction of the total field- strength, is measured by the space-rate of change of potential in that direction. When the potential is constant there is zero field-strength, and conversely. It is very important to remember that a field of force implies varying potential, and conversely. Examples. (i.) In the inside of a charged vessel we may havea potential of very great magnitude ; but it is constant throughout, and we have zero field of force. (ii.) Half-way between two equal charges of the same sign we have a certain potential ; but there is a point of zero force. (iii.) Half-way between two equal charges of opposite signs we have zero potential, but not zero field of force. 13. The Mapping Out of Lines of Force; Simple Case. Since we can draw a line of force through every point in the field such a line being defined as the direction in which a + unit of electricity is urged it follows that the lines of force are infinite in number. But a little further consideration will show us that here, as in the case of equipotential surfaces, we can mark a certain number out of this infinite crowd of lines, in such a way as to represent the various strengths of different parts of the field. Let us as before consider the simple case of a single charge of Q units collected at the point O (see fig. 7) ; and let us con- sider first a spherical surface, described about O as centre, with radius i centimetre. At any point on this sphere the force exerted on a + unit will be Q dynes ; or the strength of the field will be measured by Q. The reader must remember that in our present case the lines of force, which are infinite in number, radiate from O ; and that they cut at right angles the surfaces of all spheres which have O as centre. Now let us mark out (we may suppose that we paint them red] such a number of lines of force that exactly Q of them pierce each i square centimetre of this spherical surface. Then the number of lines piercing unit area placed perpendicularly to the lines of UNIVERSITY OF CALIFORNIA CH x. force does, over this spherical surface, represent numerically the strength of the field. At 2 centimetres distant from O these marked lines will have spread out over a sphere of four times the area ; and therefore there will be \ Q lines piercing each i square centimetre of the sphere whose radius is 2 centimetres. And, in general, at any distance r from O the lines will have so thinned out that the number piercing each i square centimetre of the sphere, whose radius is r, will be ^. But , -, ... -^, also represent the strengths of the field at distances 2 centimetres, 3 centimetres, r centimetres, from O respectively. Therefore these marked lines will represent numerically the strength of the field at any point, by the number of them that at that point cut i square centimetre held perpendicular to them. The reader should carefully study this method of marking out the lines. He will see that it consists in marking out such a num- ber that they do over one equipotential surface (viz. over the sphere of unit radius) represent, by the number piercing each i square centimetre, the field- strength at all points over this surface. He will see also that the result found to hold is that these lines so chosen will indicate the field-strength at any distance from O, in just the same manner. Of course where the force on + unit is less than i dyne, or the field-strength is less than unity, there will be less than i line piercing each i square centimetre. We must then take several square centimetres and divide the number of lines piercing them by the number of square centimetres taken. Corollary. Uniform field. Hence it follows that in a uniform field the marked lines of force are parallel and equidistant from one another. 14. General Case. We cannot in an elementary Course prove that the same result holds in the general case ; but at least the above discussion will prepare the reader for the following statement. It may be well to point out that ' equipotential sur- face,' and ' surface lying perpendicularly to the lines of force ' are synonymous expressions. 138 ELECTRICITY CH. x. * If we mark out such a number of the lines of force that ovet any one equipotential surface . they measure, by the number piercing each i square centimetre, the field-strengths all over that eqtti- potential surface, then will these same marked lines in any part of the field measure the field-strength by the number piercing i square centimetre held perpendicularly to the lines of force at the place in question." 1 It is to be noted that when we use the expression * number of lines of force,' we refer to these marked'Xvnes. 15. Total Number of marked Lines of Force. In the simple case given in 13, since we mark Q lines of force for each i square centimetre of the sphere of i centimetre radius, and since such a sphere has an area of 4 TT square centimetres, it follows that we have a number 4 TT Q of such marked lines. And if there be any system of quantities of electricity + or , and if the whole system of quantities be surrounded by an envelope, then the total number of 4- lines piercing this envelope will be 4 TT 2 Q ; where S Q is the algebraic sum of the quantities. 1 6. Tubes of Force. -. Hence, when A and B K. are concentric, the presence of the insulated spherical screen B makes no difference in the potential of external points. (ii.) Next, let B be no longer concentric with A ; but let the distance from P to the centre of A be R, while to the centre of B the distance from P is r. Then without the screen we have V P = ^ as before. But, with the screen, we have R now Vp = ^> ; since it is only the free charge on the screen that gives a field at the external point P. (iii. ) If any charge Q' be given to B, we have merely to add it to the induced charge Q that has been inductively given to its surface ; and the potential at P becomes in the two cases respectively. 18. The Potential of an 'Isolated' Body. Potential is a property of a point in space, as explained earlier. But any equi- potential surface has one potential, the same for all points on it. Now any continuous homogeneous conductor (as a brass sphere or cylinder,- a tin vessel, &c.) forms an equipotential surface ; for no differences of potential could exist on a surface where electrical readjustment takes place instantly. Hence we can speak of ' the potential of a conductor.' The potential may be a hard matter to calculate, owing to the continuous nature of the distribution of electricity over the body, which makes the expression 2^- of 9 (iii.) in general a matter for the integral calculus. But we can easily deal with the case of a sphere. 19. Potential of an Isolated Sphere. The figure represents in section a sphere of centre O and radius R. Let it have on I4 2 ELECTRICITY CH. x. it a charge Q, which we will consider to be divided into a series of small charges q lt q^ q& &c., distributed over the surface of the sphere. What is the potential of the sphere? Or, what will be the work re- quired to bring our + unit from infinity up to the sphere ? Now the sphere has the same poten- tial all over it, and inside it. Hence we may find the potential of the most con- venient point, viz. of the centre O. But by 13 this is V, V ,Q Hence, for the sphere we have V = . R Now let us consider what we mean by 'an isolated sphere.' There is. on the walls, &c., at a distance (let us suppose) of r centi- metres from the centre O, a charge equal to Q. Hence the true potential of O and of the sphere will be v=Q-2 R r' But if r is so large as compared with R that the latter term may be neglected, then we have V = 5^. R This shows clearly the meaning of isolated ; and also. shows how, for the same charge, the potential will be smaller as r becomes smaller, or as the walls close in (see Chapter V. 8). 20. Capacity of an Isolated Sphere. By the definition of capacity given in Chapter V. 4, we have for an isolated sphere But we have just found that V=. Hence it follows that R K = R. Or we arrive at the result that The capacity of an isolated sphere is measured numerically by its radius expressed in centimetres. CH. x. ELECTROSTATIC POTENTIAL 143 Note. The reasoning just employed is of very general use. Suppose that we find by calculation that V = / . Q for an isolated body ; or V = h . Q for a condenser ; where / and h involve only one or more of the following quantities, viz. dimen- sions of the plates, distance between them, and nature of the dielectric. We may then argue that, by the definition of capacity given in Chapter V. 4, and Chapter VI. 4, it follows that K = /, or K = /i, in the two cases respectively. 21. Distribution; from the Potential Point of View. In hydrostatics it is frequently very useful to consider the distribu- tion of a liquid to be such as will cause every part of the free surface to be at the same level. The more immediate cause of the distribution of a liquid is the action of gravitation-forces ; and a level free surface rather follows from this. But still the former view is often convenient. So, in electrostatics, the cause of distribution is the direct force acting on each elementary portion of the charge. But here also it is often convenient to regard the distribution on a con- ductor as such as will cause each part of it to be at the same electrostatic potential. This last must be the resulting condition, since otherwise we could not have equilibrium. We will take one case as an example. ' What must be the general distribution on an elongated cylinder that it may be all at one V ;' or, we might say, 'that it may take the same work to bring up our + unit to any part of it ? ' If the distribution were uniform, it would certainly require more work to bring up our + unit to the central portion, into the presence of the whole charge, than to bring it up to the end, out of the way of most of the charge. But, if the charge be more crowded at the ends, we see how it may now require the same work in the latter case as in the former. Thus, the well-known distribution on an elongated cylinder is such as to cause the whole to be at one potential, though it more directly results from the equilibrium of the tangential com- ponents of the electrostatic forces. 22. Two Spheres of Different Radii. Let us consider two spheres of different radii, Rj and R 2 . (i) If they have the same charge Q, then v Q . v __ Q . 1 ~ P~> 2 ~~ P~~ > K { K 2 or Xl = % 144 ELECTRICITY CH. x. (2) If they have the same potential V, then v Qi - v QI T?~~ > P ' K! K<> or 21 = % Q 2 R,' These results should be translated into words, and noted carefully by the student. 23. Potential, and Density, distinguished. By density of charge, we mean quantity of charge per unit area. We use the symbol p to signify the numerical value of density of charge. We have, by definition, where Q is the quantity on the area S ; all of course in C.G.S. units. Now we have seen that, on any conductor but a sphere, the density is not constant all over, while the potential is. At a point the density may be very great, though the potential be small. We will illustrate the difference and the connection between density and potential by the case of spheres. (i.) Consider two spheres of equal radius R. Then, for each, V = ~~ ; p = ^ . Hence, at the same potentials the densities 4 TT R* are equal, and both V and p are proportional to O. (ii.) But now consider two spheres of different radii R t and R.,, and let them have the same potential V. Then for the two respect- ively we have Q x = V R x ; Q 2 = V R 2 ; Q, VR, V and .< -IT p -tr St2 Whence p t and p 2 are not the same, and Pl = ^? . 24. Force on a + Unit, acting Perpendicularly to a Conducting Surface. It is for many purposes very important to know how great is the force on + unit at a surface, urging it normally to the surface. The resultant force must evidently be normal, since any tangential force would imply that the conducting surface was not at one potential. We will take the case of a sphere. Let the diagram represent a CH. X. ELECTROSTATIC POTENTIAL 145 sphere of radius R, and total charge O ; and let P be a point on the sphere. It is required to find the force at P acting on + unit. Now by the result of calculation, stated in 17 (4), we learned that the action of the charged sphere on a point external to it is the same as if the whole charge O were collected at the centre E. This is true quite up to the surface ; or is true when we take our point P to be just on the external surface. Hence on a + unit at P there is acting a normal force equal to ^- r dynes. But, if p be the uniform density of charge over the sphere, then Q Hence, for the force F acting on + unit at the surface, we get F = 4 IT p in dynes. Now if the radius of the sphere increase without limit, so that the surface at P becomes ultimately plane, no change will occur in the expression just found, provided that p remains constant. Hence the formula is true for a plane surface on which the density is p. It is indeed true for any surface, p being the density at the point considered. This general result is, in more advanced Courses, proved independently, without reference to a sphere. 25. Important Case of a Spherical Condenser. We will now show how to find the capacity of a spherical condenser. As in the case of an isolated sphere we found two expressions for the potential of the sphere, and by equating them found the capacity of the sphere ; so here, in the case of two spherical surfaces forming a condenser, we shall find two expressions for the difference of potentials of the two surfaces, and then by equating them shall find an expression for the capacity of the condenser. The figure represents the ideal spherical condenser, in which there are two concentric spherical conductors, the inner being totally enclosed by the outer. In the practical con- denser we must have an opening through the outer conductor, and an insulated wire and knob connected with the inner ; as in Faraday's condenser, Chapter IX. 4. L 146 ELECTRICITY CH. x. Let C be the centre. Let the radius of the inner conductor be R A cms., and the charge on it be + Q ; then there will be on the inside surface of the outer conductor a charge of Q, by Chapter IV. 16. Let the inner radius of this outer conductor be R B cms., and its outer radius be r cms.; and let there be on the outer surface of B a charge Q'. As we wish to find an expression for the difference of potential (V A V B ) between the two surfaces, we will find V A and V B sepa- rately first. The reader must note that Q is the bound charge of Chapter VI. Now V A is the potential of any point on, or in, A, since A is a conductor containing no insulated charged bodies ; and, there- fore, V A is the potential at C. .Hence V A = Q + =2 + 2 _K A K B r by the results of 19. And V B will be the same as the potential of a point just on the outer surface of B. But, by 17 (4), this will be the same as that which would be due to all the charges collected at the common centre C. That is, v _ B ~ Q + Q'_Q' B ~ ~r~ "7 Hence we get as one expression for the difference of potential between A and B. But, by the definition of capacity, we have also that (V A -V B ) = | since Q is the 'bound ' charge Chapter VI. 4. Hence, equating, we have The quantity Q', and the outer radius r, have both disappeared from the expression. And this is what we should expect, since O' has opposite to it on the walls, &c., a charge equal to - Q', and these two charges with the intervening air as a dielectric form another ELECTROSTATIC POTENTIAL (relatively insignificant) condensing system with which we are not con- cerned in our present discussion. The above calculation is based upon the supposition that the dielectric is air. For our quantity Q is measured in terms of a unit which repels another unit at i cm, distance with i dyne force in air ; and our expressions for measurement of potential depend on this condition. Now, if we have any other dielectric of specific inductive capacity Now it is clear that p = ^ ; since p is assumed to be o uniform. Hence we have V A V B 4 TT p t or But by definition we have (V A V B ) = -^=. rL Whence In the above we have assumed air to be the dielectric. But if we have another dielectric of specific inductive capacity = n, then the formulae become (y) 4 7T / 27. Formulae for Capacities, &c. For convenience we here put together the chief formulae that have occurred up to the present point in this chapter, adding also one or two more. The following remarks will apply to some of them, if not to all. (a) All the measurements are in the C.G.S. system of units. (b) Electric quantity is measured by reference to force exerted when air is the dielectric. When there is another dielectric of specific inductive capacity v, the force exerted is -th that exerted through air \see 3 25, end). The capacity of a condenser with this dielectric will by definition be a times that of a similar air condenser. CH. x. ELECTROSTATIC POTENTIAL 149 ( 2 , (iii.) For two discs, that surface of either on which is the bound charge being S, and the distance between them being /, 150 ELECTRICITY CH. X. This formula applies to any case of parallel surfaces where the dimensions are very large as compared with /. In fact, the formula (i.) above can be transformed into this when R, and R 2 are very great as compared with (R 2 RI). 28. Energy of Charging and Discharging 1 . When electricity falls from a higher to a lower level, work can be done ; we lose a certain quantity of electrical potential energy, and we must have an equivalent in heat or mechanical work, &c. If conversely we raise electricity from a lower level to a higher, we expend this work and gain the above electrical potential-energy which is an equivalent (see 4). There is no more important matter in our present science of electricity than this question of the application of the law of * Conservation of energy ' to electrical charge and discharge. We will consider the matter under three heads. Case I. The case where an electrical quantity Q passes between two conductors of fixed potentials V l andV^. Now by definition and measurement of potential, we do on each + unit an amount of + or work that is measured by (V^ V 2 ) ergs. Hence, on the quantity of Q units we do + or work measured by If one potential be V and the other be zero, the work is Q . V ergs. Hence, in this case the energy E of charge or discharge is measured by or E = Q. V ergs) ' Case II. The case, more usual in electrostatics, where the potential alters during charge or discharge. We will consider the simplest case of charging a conductor, that is originally at zero potential, with a quantity Q, up to a potential V. Here the potential of the conductor rises in arithmetical pro- gression with the charge given, since potential^ - chai * e , capacity and capacity is constant. The result, therefore, is that the same work is done as if the whole charge were raised to half the final potential V. Hence the work of charging, or energy expended, CH. x. ELECTROSTATIC POTENTIAL 151 will be E = or = since V = . Jv This same expression gives the energy of discharge. Analogies. (a) So, it we reckon the number Q{ foot-pounds-weight done in building with P Ibs. of bricks a tower of H feet high, it being assumed that the tower is cylindrical or rises in arithmetical progres- sion with the amount of bricks used, we get as a result the same work as if all the bricks were raised to half the vertical height H ; i.e. we get work = \ P . H . f o ot -pounds -iv eig Jit. (b] A similar expression is obtained for work done by water flowing out of a cylindrical reservoir of an average height of i H above the lower level. Case III. Case of discharge of a condenser. In the ordinary condenser we have two plates of potentials V t and V 2 respec- tively. In this case the discharge will cause the equal and opposite charges + Q and Q to disappear, leaving the whole condenser at some potential v due to the ' free ' charge. Taking the two charges + Q and Q separately, we have for the former energy = J Q (V t - v\ and for the latter energy = \ ( - Q) (V 2 - v). Adding the two we get E = iQ.(V l -V 2 ) ...... ( 7 ) which is the same as result (/3), if the one coating of the condenser be at zero potential. The reader should notice that in the case of the Leyden jar charged to potential V of inner coating, the outer coating being to earth, we have for the discharge of the jar the expression | Q V. 29. Examples in Energy of Discharge. (i.) * A conductor is charged with 8 units to potential 7. What is the energy of discharge? ' Here E = QV = |.7.8 = 28 ergs. I5 2 ELECTRICITY CH. x. ' And what was its capacity ? ' (ii.) 'A conductor of capacity 10 is charged with 4 units. What is the energy of discharge ? ' (iii.) 'A sphere of radius 4 cm. is charged with 12 units. Find the energy of discharge.' Here K = 4 (see 19) Two harder examples are worked out in 35. 30. Energy of Discharge in the Cascade Arrangement of Leyden Jars. We must now ask the reader to turn back to Chapter VI. 9, and to notice the arrangement there described and the notation employed. We found that if there were n equal jars in cascade, the differ- ence of potential between the inside and outside coatings of each jar was _th that of a single jar charged from the same source and having its outside coating to earth, the charge of each jar being also -th that of the single jar fully charged. It follows that, in the case of the single jar thus fully charged, we have for the energy of discharge E the value E = lQV^ .... ..,, . (i.) while, for all the n jars of the cascade, we have for the total energy of discharge E' the value E'=a x i.9.-= * . JQV= --E . (ii.) n n n 2 n This result may be better understood if we give an analogy. Analogy. If instead of one tower 100 feet in height we build 10 towers each having ^th of the height, and th of the number of bricks of the large one, a little thought will show us that the total work done in the second case will be j^th of that done in the first case ; for in the second case the same total weight of bricks will have been raised through T a 5 th of the former average height. CH. x. ELECTROSTATIC POTENTIAL 153 31. Electroscopes and Electrometers, In general these in- struments give their indications by movements. These move- ments are due to the action of a field of force on electrical quantity, the extent of the movement being such that equilibrium is finally arrived at between electrical forces on the one hand and gravita- tion-, magnetic-, or torsional-forces on the other hand. We can have movements when there is only one charged body, e.g. when this body is situated nearer to one wall of a room than to the other. Here there is a field of force between the body and the walls, and this field is strongest in one direction. Now, be- sides the effect of a field on a charged body situated in it, we know that the two sides of a field tend to close in ; or we may say * the lines of force tend to shorten.' Hence the charged body will move so as to close up the strongest portion of the field due to it. (a) In a gold-lea] electroscope each leaf is screened from one wall by the other leaf. Hence, there is no wavering between two walls about equally distant, but each leaf moves off towards the wall from which it is not screened. This view, or one like it, is more in accordance with present ideas than is the view of ' repulsion between the two leaves.' (b) Bohnenberger's electroscope. Here a single charged leaf moves along the field of force that lies between two brass knobs ; these brass knobs being maintained at different potentials by being connected with the two poles respectively of a dry pile. In both cases here given we indicate differences of potential by the extent of the movement ; since in case (a) the field is stronger according as the leaves are at a larger or smaller difference of potential from the zero potential of the walls, and in case (b) the movement of the single leaf in the fixed field of force depends on the potential up to which this single leaf is charged. 32. Electrometers. The ' Attracted-Disc ' Form. In electro- meters we aim at measuring differences of potential, or absolute poten- tials with respect to the earth as zero. Consider the plate condenser of 26, and make the same assumptions as to uniformity of field and of p as were there made. 154 ELECTRICITY CH. x. It can be shown that the total stress F between the plates is given by the formula S (V. - V 2 ) 2 8 TT P This expression therefore gives in dynes the total force with which one plate is urged towards the other ; and, if we measure F by counterbalancing 'weights/ each gramme weight being about 981 dynes force^ we can express (Vj V 2 ) in absolute units. But in the above apparatus there is error ; for the assumptions made do not hold near the edges of the plates, and this error is not one to be readily allowed for. Hence, in the actual attracted-disc electrometer, Sir W. Thomson adopted a contrivance called a guard- ring. The moveable disc was cut out in the centre of a larger disc. Then, since the fixed and moveable portions were connected and at one potential, and since there was no perceptible break between them, it followed that with respect to the moveable por- tion of the one disc, and the central portion of the other fixed disc that was opposite and parallel to the former, the assumptions as to uniformity did hold. In fact, only the central portions of the discs were used. In the figure we have the arrangement indicated. The two plates are h h and g ; and the formula applies to the moveable portion /"of the one plate and the portion opposite to this of the other fixed plate. The -^ moveable portion / is at the end of a light lever whose fulcrum, or axis of suspension, is a wire m n. At the free end of this lever is a hair /; this is viewed by a lens, and serves to indicate when the plate /is flush with the guard- plate h h. The fane required to keep /in its place gives us F. Whence, by knowing the dimensions, &c., of the instrument, we measure CH. X. ELECTROSTATIC POTENTIAL 155 33. Sir William Thomson's Quadrant Electrometer. The principle of the quadrant electrometer can be illustrated by the Bohnenberger's electroscope. If the gold-leaf were kept at a constant high potential, while the difference of potential between the two knobs was the variable quantity, then the amount of deflexion of the gold-leaf would in- dicate the difference of potential between the two knobs. The next figure gives a sketch, taken looking down directly from above, of the essential parts of the quadrant electrometer. Let us take a cylindrical box of thin brass about one inch high and about five inches in diameter. Let us cut this into four sectors as indicated in the figure, and let us connect opposite sec- tors respectively by wire, and sup- port all four sectors on insulating glass legs. We thus have four hollow brass sectors ; in the interior of each of which, as an approxi- mately ' closed vessel,' there is, excepting near the edges, a con- stant potential. The opposite pairs of these are connected. By u is represented a light aluminium needle suspended by two parallel silk fibres this being called bi-filar suspension. Hence there will be called into play, when the needle is deflected from its position of rest, a couple tending to restore it. The value of this couple for any given angle of deflexion can be calculated. This needle swings horizontally in the interior of the sectors ; and matters must be so arranged that it may come to rest exactly along the line of one of the slits. The diagram repre- sents the needle thus at rest, unacted upon by any electrical forces. This needle is maintained at some constant potential 1 V, by- being connected with the inside coating of a charged Leyden jar. If the one pair of sectors c and b be at a potential V l5 and the other pair a and d at some lower potential V 2 , there will be a field of force running across the gap or slit from c to a ; and from symmetry, an equal field will run across the gap from b to d. 155 ELECTRICITY CH. X. Moreover, this field will be confined to a region only somewhat wider than these gaps ; the region in the interior of each sector will be practically of a uniform potential, or will not be a field of force. From the symmetry of the whole, the needle will be acted upon by a pure couple due to the electrical field. And, from its peculiar shape, that amount of the needle which is in the field of force (i.e. the portion lying under the slit) will remain constant, since only very small deflexions are employed. We thus contrive that there shall act on the needle, for any given values of V,, V 2 , and V, a constant electrostatic couple whatever be its deflexion ; provided that this deflexion do not exceed a certain maximum depending on the construction of the instrument. This constant couple will deflect the needle until the restoring couple, due to the twisting of the two parallel silk fibres, is suffi- ciently great to give equilibrium. If the potential V of the needle be known, and if the ' constants ' of the instrument be known, then the difference of potential (Vj V 2 ) can be calculated from the observed deflexion. In the next figure we have a sketch of the ' Elliott-pattern ' quad- rant electrometer ; a comparatively simple form of instrument. One of the quadrants is represented as removed, so that the needle may be seen. This needle is maintained at a high potential in somewhat the following manner. From it there hangs a platinum wire, dipping into a glass vessel that contains strong sulphuric acid. Outside this vessel is a coating of tin-foil, so that it is in fact a Leyden jar. Its capacity being very great as compared with the capacity of an isolated body, it serves to maintain the potential of the needle approxi- mately constant for a considerable time. From the needle rises a light stem, bearing a small mirror. CH. x. ELECTROSTATIC POTENTIAL 157 The usual 'lamp and scale ' arrangement gives us, in the reflected spot of light, a very sensitive means of noticing and measuring the deflexions of the needle. Complete form of the quadrant electrometer. In the complete form of Sir W. Thomson's quadrant electrometer there are many details of construction that we shall not discuss here. But we must mention two of the most important of these. (i.) The gauge. Connected with the inside of the Leyden jar (and so with the needle) is an attracted-disc electrometer. If the weights or spring be so arranged that the hair (see 32) is in the proper line of sight when the jar, needle, and disc/ therewith con- nected, are at some fixed potential V, then any alteration in this potential will be at once detected by the movement of the hair. This is one detail. (ii.) The replenisher. The other detail is a replenisher (see Chapter VII. 5), in which one armature is to earth while the other is connected with the Leyden jar. We can thus remedy any error in V by turning the replenisher the one way or the other, so as to raise or lower the potential of the jar. Formula for quadrant electrometer. It can be shown that the following formula holds for the absolute quadrant electrometer. -I_V (Vi _ Va) where L is the electrostatic couple acting on the needle, balanced by and equal to the mechanical restoring couple ; V the potential of the needle ; V x and V 2 the potentials of the two pairs of quadrants re- spectively, and h a constant depending on the construction of the instrument. If the needle be charged to a relatively very high potential, then we may neglect 1 + 2 in comparison with V ; and we have ' L-A.V.^ - V 2 ) This relation may be written as follows. We see then that we could measure the difference of potential : - V.J by observation of the deflexion ; provided that we knew 158 ELECTRICITY CH. x. the values of h, V, and L. In practice, however, these quantities are not calculated ; but the instrument is graduated by means of a ' standard voltaic cell,' or by comparison with an absolute electrometer of the more simple ' attracted-disc ' form. In instruments devoid of $t& gauge and the replenisher, as well as in the above form, we may write . V l - V, = b . tan 8 so long as the potential V of the needle remains constant. And, for small deflexions, we have v i - V 2 = ?>.B \nearly\. Here 8 is the deflexion ; and the constant b may be determined on each occasion of use by means ot a ' standard voltaic cell.' 34. Uses of the Quadrant Electrometer. (i.) General use. The quadrant electrometer is mainly an instrument for the measurement, or comparison, of potentials and potential-differences. We have indicated in the above how it is possible to use simpler forms also, as well as the ' standard ' pat- terns of the instrument, for such purposes of measurement. It may be added that usually one pair of quadrants is put to earth, so as to be at zero potential ; the potential of the other pair of quadrants being then measured with respect to the earth as zero. (ii.) Use as an electroscope. The instrument can also be used for all purposes in which we desire merely a delicate electro.$w/. 35. Examples in Energy of Discharge, &c. (a) In all cases where it is required to find the energy of discharge between two electrical systems A and B, the following method will be found to be a good one. First, find the energies of discharge of A to earth, and of B to earth, separately; and add these quantities Secondly, find what will be the energy of discharge to earth of the combined system A and B after these have been connected. It will then follow, by conservation of energy, that the difference between these two results will give the required energy of discharge between the two systems when they are connected with one another. Let us take an example. ' There are 3 Leyden jars A, B, and C, equal in capacity, having their outer coatings connected with earth. A is first charged. Its knob is then connected with the knob of B. It is then disconnected from B, and connected with C. Finally, the knob of B is connected with the knobs of A and C. Find the energies of the several dis- charges.' (i.) The initial discharge of A to earth would give energy to the amount of * Q V ; and the final discharge of A and B together to earth would give energy measured by - Q . , or - Q V. Since energy 224 cannot have been lost, it follows that the difference between these two quantities must measure the energy w 1 of discharge between A and B when these two were connected. Hence 160 ELECTRICITY CH. x. (ii.) The discharge of A to earth would now give - . -^ . , or * Q V. And the subsequent discharge of A and C together to earth 8 would give - . _~ . , or -- Q V. Hence, energy / 2 of discharge 224 10 between A and C is given by the difference, or (iii.) The discharge of B to earth, and of A and C to earth, would give together (1 . 2- . Y + I . S . Y\ = .3 Q V. And the discharge of all three together to earth would give - . Q . -, or - V. Hence 2 36 the energy / 3 of discharge when B is connected with A and C is given by the difference, or It is to be noticed that all the discharges added together give the energy 1 Q V, which was that of the original discharge of A to earth. If this result had not followed there must have been some error in the above work ; since ' Conservation of energy ' demands that no energy be lost. (/3) 'A sphere A of 9 cms. diameter is connected by a thin wire (i.e. one of negligible capacity) with another sphere B of the same diameter. Round this latter, and concentric with it so as to form a spherical condenser, is a larger sphere of 10 cms. internal diameter, connected with earth. A charge of 33 units is given to A. Find (i.) in what proportion this charge is distributed between the two systems ; and (ii.) the energy of discharge of the two systems separately.' iLet V A be the potential of A, and therefore also of B. C A be the capacity of A, and C B that of B. Q A be the charge on A, and Q B that on B. W A be the energy of discharge of A, and W B that of B. 9 2 condenser, C B = i -T- ( ~ -i ) = 45- (i.) Then C A = radius = ? . And, by the formula of a spherical CH. x. ELECTROSTATIC POTENTIAL l6l 9 Hence Q A =* -^ of 33 = of 33 = 3 units, 2 + 45 and Q B = the remainder = 30 units. (ii.) Again V A = V B = 9.A-2?-*. '-A ^B 3 Hence W A = IQ A V A = Lx 3 *?- i, 1 lQ B V B = - x 30 x -= 10. 36. General Consideration of Electrostatic Fields of Force, It is probable that among all the important conceptions that are due in the first place to the genius and insight of Faraday, none has done more to place the physical theory of electrical and of magnetic phenomena upon a sound basis than his recognition of the part played by the medium across which the forces act. The former view of ' action at a distance,' and the purely geometric conception of lines of force, were essentially mathematical ideas ; to the physicist they were both unreal and unsuggestive. Taking our present case of electrostatic fields, we may explain the modern view, based upon Faraday's conception, somewhat as follows. There is no such thing as ' a + charge ' or ' a charge ' by itself; but on the contrary, wherever electrostatic phenomena occur there is an electrostatic field, on the two sides of which occur equal + and charges respectively. The whole system consists of these equal and opposite charges separated by a dielec- tric which is the seat of the field of force. The lines of force have a real physical meaning. Whether they are lines along which the dielectric undergoes a kind of tension, or whether they are lines along which the molecules of the dielectrics are ' polarised ' by a separation of + and charges in them, is not yet known. But it seems certain that the electrostatic potential energy, implied by the existence of an electrostatic field, resides in the dielectric that separates the two sides of the field ; somewhat as mechanical potential energy resides in a bent spring, a stretched piece of elastic, or a strained elastic solid. Thus, when we speak of ' an isolated body charged with + Q. M 1 62 ELECTRICITY CH. x. units to a potential V, whose energy of discharge is measured by \ Q V,' we really mean that there is on the body a charge + Q, on the walls or other surrounding surfaces an equal and opposite charge Q, and that in the strained dielectric resides energy to the amount of \ Q V. Dielectrics are bodies in which this electrostatic strain can be maintained ; while conductors are those in which no such strain can "be kept up, and in which therefore lines of force and fields of force cannot exist. In reality, bodies are not capable of sharp division into these two classes, since the yielding to electrostatic stress is only a matter of time. But in general we make use of bodies at the two extremes of the series ; so that the foregoing definitions will serve very well for usual cases. It is part of the above view to say that tubes of force are always terminated against surfaces so charged that at the two ends of each tube there are equal and opposite charges + q and q respectively (see 17). It is also part of it to say that conductors act as screens for lines of force cannot penetrate a conductor, and hence if an elec- trostatic field exist on each side of the screen, these two fields are independent of one another. If lines of force were material, the phenomena of the electro- static field would suggest that they act as stretched elastic threads, which moreover repel one another. Thus the two sides of a field are urged towards one another as if by tension of the lines of force connecting them. CH. XI. CHAPTER XI. THE PHENOMENA OF ELECTRIC CURRENTS. BATTERY CELLS AND BATTERIES. i. Introductory. In Chapters IV. -X. we have dealt mainly with the phenomena connected with the actions between charged conductors, and with the strains of the intervening medium. We have, in fact, considered mainly electrostatic fields of force, or fields in which our test-charge of electricity is urged along lines of force. We did, indeed, examine to some extent the results of dis- charge ; but in general the readjustment of electrical equilibrium was so rapid that it was not easy to investigate the phenomena accompanying discharge. With such machines as the Holtz we might, however, have done so ; but not so conveniently as with the help of other apparatus to be described in the present chapter. We shall find that, during discharge, new classes of phenomena arise. In particular there are observed chemical, and heat, phenomena ; and a new field of force, viz. a magnetic field (or a field in which a magnetic pole is urged), springs into existence. In our next division of the subject we enter into the consider- ation of the phenomena accompanying electric discharge. As a rule we shall have very small AV..J as compared with those hitherto employed, but very large quantities of electricity, and an even and continuous flow. W 7 here this is not .the case, atten- tion will be drawn to the fact. The above remarks will indicate that the popular terms * statical electricity ' and ' current electricity ' must not be under- stood in their literal sense of two kinds of electricity ; but must be considered to refer to two classes of phenomena that require different conditions of A V and of quantity for their investigation. M 2 1 64 ELECTRICITY CH. xi. 2. Galvani's Experiment. As stated in the preface, we shall not enter into any historical account of this or of any portion of our subject. But we must mention that experiment of Galvani's which, perhaps more than any other, started Volta on his very fruitful line of inquiry. In the figure, Z C is a compound bar of zinc and copper, as indicated by the two letters employed. The zinc end is put into contact with the lumbar nerves of a frog's hinder quarters. It is found that whenever the copper end touches the muscles of the legs tnere is a sudden convulsion of the limb. This experiment was first performed in 1786. Galvani considered that the metals acted merely as conductors to complete the circuit of 'animal electricity,' and followed put this idea in further investigations. As we do not intend to discuss in the present Course the relations of electricity to physiology, we shall not say more as to Galvani's views. Volta, on the other hand, fixed his attention on the metals, and considered that they by their contact caused the current which the nerves and muscles of the frog's leg conducted. We shall have more to say as to his views, and as to modern modifica- tions of them. For the present we will merely follow his experi- CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 165 ments and the explanations that he gave ; using, however, some terms, such as * potential,' which he did not use. 3. Volta's Experiments and Views. For clearness we will first state Volta's views, and will then give some of the experiments by which he supported them. The student can see how far each statement is supported or unsupported by the experiments given. Volta's views. (i) When two heterogeneous substances are in contact they are found to be at different potentials. (2) As a rule any A V. s between liquid and liquid, or metal and liquid, are negligible as compared with the A V . s between metals and metals. (3) Metals can be arranged in a certain series (such as zinc, tin, lead, iron, copper, silver, platinum, graphite}, with respect to which the following facts hold : that any metal in contact with another occurring later in the list will have a higher potential than this latter ; and that in a series of several metals in contact the A V between the first and last is the same as it would be were these metals directly in contact. Thus, if we make the potential of graphite zero, and find the potentials of all the other metals with respect to this body when in contact with it, then we can calculate the A V between any pair in contact by simple subtraction of their potentials with respect to graphite. (4) This law, however, does not hold good with respect to a series in contact when composed partly of metals and partly of liquids. Thus, whereas in the series copper\zinc\gold\copper there is no A V between the terminal metals, because it is the same as if the initial and final metals (viz. copper\copper) were directly in contact, yet in the series copper\zinc\dilute acid\copper there may be, and as a matter of fact there is, a A V between the terminal metals. Experiments illustrating Volte? s views. In these experiments Volta em- ployed the condensing gold-leaf electroscope, whose principle is explained in Chapter VI. 14. But far more satisfactory results can be obtained by use of any form of quadrant electrometer (say the ' Elliott ' form of Chapter X. 33), in which one pair of quadrants are to earth ; such an instrument indicates quantitative results with an accuracy sufficient for lecture experiments. (i.) The figure represents a simple experiment performed with the con- 1 66 ELECTRICITY CH. XI. densing gold-leaf electroscope and a compound bar composed of a piece of zinc and a piece of copper soldered end to end. We may use the quadrant electrometer, putting one pair of quadrants to earth, and connecting the other pair with an insulated terminal. If the zinc be held in the hand, while we touch the lower condensing plate of the gold-leaf electroscope (or terminal of the electrometer) with the copper, it will be found that the condensing plate (or insulated pair of quadrants) is now at a potential. (For use of .electroscope and electrometer see Chapter X.) FIG. i. In this experiment it is assumed as sufficiently proved by various con- vergent pieces of evidence that the zinc, held in the moistened hand, is practically at the zero-V of the earth ; and that the contact of the copper with the brass of the electroscope gives no A V. Hence, it is argued, the electroscope or electrometer indicates the A V due to the contact of the zinc and copper alone, and shows that zinc is + to copper. (ii. ) An electroscope is provided with an upper condensing plate made of zinc. This is, as usual, provided on its under surface with an insulating layer of lac varnish. Both plates are as usual carefully discharged by means of a Bunsen's flame, until no movement of the leaves is observed on raising or lowering the upper plate. CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 167 FIG. ii. . Then a copper wire is made to connect the upper (unvarnished) surface of the zinc plate with the under surface of the copper plate. It is found that the zinc and copper have now assumed a A V in which the zinc is + to the copper. This A V is of course (see Chapter X. ) magnified, and so rendered evident, when the wire is removed and the upper condensing plate is lifted away. Here there is no doubtful con- tact of metal and wet hand. (iii.) If in experiment (i. ) we hold the copper end of the compound bar, and touch the plate of the gold-leaf electroscope with the zinc, we get no result. This supports Volta's conclusion (3). So if we make a bar of zinc \ gold \ iron \ copper ; and use it as in (i.), we get the same result as in (i. ). (iv. ) If we hold the copper end of the bar and insert a piece of wet paper between the zinc and the gold-leaf electroscope, we shall find that, on removing the bar and raising the upper condensing plate, the lower plate is at a -- V. Volta's view would be that the copper held in the moistened hand is practically at zero-V ; the zinc at a + V ; the wet paper at practically the same V as the zinc ; and the brass of the electroscope at practically the same V as the paper, and therefore at the + V of the zinc. The reader should examine these experimental results in connection with the statements made above as to Volta's views. 4. Volta's Pile, from Volta's point of view. By the application of the method of experiment (iv.) above, it is easy to obtain A V . s high enough to give shocks and to produce small sparks. We lay down a copper disc and put it to earth. On this we lay a zinc disc, which will be at some higher potential that we will call v. On this we place a somewhat larger disc of flannel, moistened with salt and water ; this will also be 1 68 ELECTRICITY CH. xi. practically at v. On this another copper disc, which will be also at v. Then a zinc disc on this will be at 2 v ; and so on. We may conveniently solder together each pair of copper and zinc discs. If there be n such pairs of plates, then we have the bottom copper at zero-V, and the top zinc at n x v. If we solder a copper terminal to this top zinc, it is to be noticed that this wire will be at n i . v, or at the same potential as the n ih copper. This arrangement is called ' Volta's pile.' About fifteen or twenty such pairs will, when newly set up, give a perceptible shock. After a time the surfaces of the metals are altered from slow chemical action, and the pile must be taken down and cleaned in order to give the initial results. Volta's view was that the action is due to the series of con- tacts ; but that when, owing to oxidisation of the surfaces, the sub- stances in contact are altered, we do not have, nor can we expect, the same results as initially. Experiments. (i.) We put the bottom copper, and one pair of quadrants of the electrometer, to earth. Then with an insulated platinum wire connected with the other pair of quadrants we touch the 2nd, 3rd, 4th, &c., copper plates in succession, putting the platinum wire to earth between each observa- tion. The deflexion of the needle will indicate the rise in potential up the pile. (ii. ) We can make a similar observation on the momentary currents sent through a delicate galvanometer of high resistance. But this assumes some previous knowledge of the galvanometer. Other forms of Vollds pile. There are other forms of Volta's pile used ; of these we will mention one, viz. Zambonfs, As a rule the drier the pile, the more lasting is its activity, but the more easily is .it exhausted for the time. In Zamboni's pile the arrangement answers to fixing together a zinc, wet paper, and copper disc, as one element For each element of his pile consists of a disc of paper that is ' silvered ' on one side (i.e. coated with tin or lead}, and on the other side coated with manganese dioxide. Such a pile gives a very high A V for very little space occupied. It can be advantageously employed to give the requisite A V in the Bohnenberger's electroscope, where there is no completion of the circuit and therefore no running down of the pile. CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 169 The Gymnotus. In certain animals, notably in the Gymnotus or ' electric eel,' there have been found organs that appear to be analogous to a voltaic pile. Severe shocks can be received, cur- rents obtained, and other electric phenomena manifested. These manifestations seem to depend partly upon the actual condition of the organ, partly upon the general health and vigour of the animal, and partly upon its volition. The study of the relation of these electrical phenomena to what we may call ' vital ' phenomena is exceedingly interesting ; but we do not intend to include in the present Course anything that belongs of right to the domain of physiology. 5. Volta's Cell, and the Couronne des Tasses, from Volta's point of view, In order to obviate the 'running down' of the power of his pile, resulting from the alteration of the surfaces of the metals, Volta substituted dilute acid for wet paper. His * pile ' then took the form of a number of cells or elements joined end-on, forming what was called the Couronne des tasses. We here give a sketch in section of this arrangement. The zincs and coppers are plates connected by copper wires ; they are immersed in vessels of dilute sulphuric acid. In the figure we have supposed the copper wire that is soldered to the zinc plate to be to earth and so at zero-V. The rise of V through the arrangement is indicated. The reader will see that, according to this view, the A V of a single ele- ment is practically the same as that due to the first copper \ zinc junction. When the terminals are connected a current flows, and energy is given out in the circuit. When pure water is used in the vessels we find the A V between the terminals to be approximately the same as if dilute acid had been used. But if we complete the circuit for a few moments, and again break it, we find that this A V has almost or quite disappeared. The 170 ELECTRICITY CH. XT. battery has 'run down,' as did the pile. Examination will then show that the surface of the zinc has been oxidised, while the copper is covered with a film of hydrogen. From VoltcHs point of mew we should say that we now have a new series of bodies in contact ; viz. copper \ zinc \ zincic oxide \ water \ hydrogen \ copper. The cell is now very different from its original condition ; we could not predict whether there would be a A V between the terminals, and in fact we find that there is none. This view of the matter is doubtless unsatisfactory ; but it is of interest to notice that at least the ' running down of the battery ' does not confute Volta's view. But if we use dilute acid, the surfaces are continually renewed by chemical action ; this action taking place far more freely in the present case than in the case of the pile. 6. The ' Contact ' and < Chemical ' Theories, Volta fixed his attention mainly on the A V that, as it seemed, accompanied the contact of the dissimilar metals zinc and copper. His followers exaggerated a certain one-sidedness that existed in his views ; and the Contact school, as they were called, considered that the chemical solution of the zinc played a subordinate part in the action of the cell, serving mainly to keep the surfaces clean and so to keep the same series of bodies in contact. In fact the word contact was the key-note to their theory of the Voltaic cell. They considered the A V between the terminals of the ' open ' cell (i.e. of a cell in which the terminals were insulated) as the algebraic sum of the different A V . s due to the different contacts ; of which, in the ordinary Volta's cell, the only one of importance was that where the copper wire was soldered to the zinc. The * Chemical school ' of physicists considered the cell when the circuit was closed and a current was running. They pointed out how the strength of the current that flowed was proportional to the vigour with which the chemical action proceeded ; and how the power of the cell depended on having one plate as much acted upon, and the other plate as little acted upon, as possible. Faraday was the great exponent of this view. In modern phraseology, the ' Chemical school ' insisted on the chemical action as the source of the energy of the cell. They, in their turn, were for the most part too one-sided ; and many denied that dissimilar metals in contact did exhibit a dif- ference of potential at all without chemical action. In the next section we shall attempt to show the position of CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 171 modern theory and of modern knowledge in this matter ; and shall conclude by giving a view of the Volta's cell, taken as a whole, which can hardly involve any serious error. But we should add that the whole question is still to a con- siderable extent unsettled. 7. Theory of the Simple Volta's Cell, As is usual in such cases there is truth in both the above extreme views ; though the error of the extreme followers of the Volta school was perhaps the greater, inasmuch as they neglected the source of energy in the current. There seems no doubt but that a difference of potential does in general accompany the contact of dissimilar bodies ; though it is quite beyond our present knowledge to say that the contact causes the difference of potential. . . But the phenomenon is not an easy one to observe ; nor is the amount of A V as easy to measure as would at first appear. When zinc and copper are in contact with each other, they are also in general in contact with the air or other medium surround- ing them. It has been asserted that while the zinc and copper are nearly at the same V, the layers of (badly conducting) air or other gas in contact with the two metals respectively will be found to be at a considerable A V from each other ; and that this A V alters both in magnitude and in sign with the nature of the gas. It is said that in many electrostatic methods we really measure the A V of these layers of gas, and not of the metals themselves. That while zinc really has under all circumstances at ordinary temperatures a very small V with respect to copper, yet when the usual experiments are performed in air we find zinc to be apparently very strongly + to copper, and when performed in an atmosphere of hydrogen sulphide we find zinc to be apparently very strongly to copper. The A V . s between the metals and the gases are considered to result fro n surface chemical action ; and the minute A V . s between the metals themselves to result from some molecular action prob- ably unaccompanied by any chemical change. These latter true A V s between the metals have been measured by methods that will be discussed in the chapter on thermo-electricity. We have given here a view the difference of which from Volta's view shows how necessary it is at present to be very cautious in 172 ELECTRICITY CH. X[ . making any dogmatic assertions as to the respective A V. s at the different places of contact ; though at the same time we see that even this view allows the Voltaists to have been right in asserting that some A V does accompany mere contact. The Chemical school, though they did not use modem expressions, and had not fully grasped the laws of energy, at least understood that we cannot get work done without an equivalent disappearance of energy. They regarded the energy of the current, that ensues when the circuit is closed, as derived from the chemical consump- tion of zinc in the cell ; and they saw that the energy of the cur- rent depended upon the intensity of the chemical affinities and on the amount of the chemical action. Perhaps at present it is safest to regard the cell as a whole* Wherever it be that the A V . s occur, at any rate we may safely say that each cell tends to keep its two terminals or poles at a certain A V ; and, if the circuit be completed and a current be allowed to pass, it is beyond dispute that the energy of the current is supplied by chemical action in the cell. We may regard the cell as a contrivance in which electricity is pumped up from a lower level to a higher by the expenditure of chemical energy ; the electricity running down through the external circuit to the lower level again, and so on continuously. In the simple diagram here given the cell is represented by the vertical line CZ. This indicates that in the cell the elec- tricity is raised from a lower to a higher level by expenditure of chemical energy ; and that then the electricity flows down from the higher level C to the lower level Z through the external circuit X. We may here add that it is usual -to call the zinc the + plate, and the copper the plate ; while the copper wire soldered to the zinc plate is the pole, and that soldered to the copper plate is the + pole. In fact, that plate or pole is called ' -f ' from which the current flows to the other plate or pole respectively. The current flows from zinc to copper through the liquid, and from copper to zinc through the external circuit. cn. xi. THE PHENOMENA OF ELECTRIC CURRENTS 173 The phenomena of the ' open ' cell are, we see, electrostatic ; the A V of the terminals may be observed with a quadrant elec- trometer. When the circuit is completed we get a flow of elec- tricity ; the phenomena accompanying this flow will be surveyed in Chapter XII. 8. Digression on the Galvanometer. In Chapter XVII. the student will find discussed the construction and theory of the galvanometer. We shall, therefore, deal with it very briefly here. The figures of Chapter XVII. indicate that when a current passes along a wire held parallel to a magnetic needle, above or below it, the current produces a magnetic field that tends to set the needle in a position at right angles to the wire. As the needle is deflected, the earth's couple, tending to restore it to its original position of rest in the plane of the magnetic meridian, becomes greater ; while the disturbing couple due to the current becomes less. Hence it will rest at some angle of deflexion from the magnetic meridian ; and the magnitude of this angle can be employed to calculate the strength of the current. By ' strength of current ' we mean quantity of electricity passing across any cross-section of the wire in one second of time ; as will be further explained in Chapter XIII. It is found that the galvanometer indicates the strength of current as just defined. It makes no difference in the indications whether this quantity per second flow quickly through a thin wire, or slowly through a thick one. 9, The Solution of Zinc in the Voltaic Cell. We will now examine the nature of the chemical action that goes on in the Voltaic cell, considering first the open cell, and then the closed circuit. / In the open circuit. If we immerse a plate of ordinary commercial zinc in the dilute acid of the battery cell (we usually mix strong sulphuric acid with water in the proportion by weight of one of the former to ten of the latter), we notice the usual dis- engagement of hydrogen from the surface of the zinc. But if we immerse either very pure zinc, or ordinary zinc whose surface has been amalgamated with mercury, no such action takes place. Here the zinc and copper plates, unconnected with each other, stand opposite to each other in the dilute acid, equally passive. 174 ELECTRICITY CH. XI. Probably the action of the mercury with which we have amal- gamated the surface of the commercial zinc is to soften that surface, and to render the zinc superficially free from all strain ; to give the zinc a mechanical homogeneity that apparently makes it behave like chemically pure zinc in being unacted upon by dilute sulphuric acid. // In the dosed circuit. If we repeat the above simple ex- periments, but now connect the copper and zinc plates, we get a current. In the figures, A is the zinc plate, B is the copper plate, C and C' are copper wires. The first figure represents the fact that the terminals C and C' in the open cell are at different V . s ; the FIG. i. FIG ii. second figure represents the direction of the current that ensues when the terminals are connected. Now we shall find that, in the case of common zinc, when the circuit is completed immediately there is an increase in the evolu- tion of hydrogen, the additional gas being given off the copper plate from the side that is turned towards the zinc plate. In the case of pure, or amalgamated, zinc, we shall find that on completing the circuit chemical action at once commences ; hydrogen is given off from the copper plate, while the zinc is dis- solved. In both cases we find, on testing the liquid, that it is the zinc that is dissolved, the copper remaining unacted upon by the acid. In the case of pure, or amalgamated, zinc, it would seem from this experiment that chemical action only occurs when a current passes, and that then, while it is the zinc that is acted upon, yet CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS I?5 the hydrogen is set free from the surface of the copper plate. Thus the hydrogen seems to travel in some invisible manner with the current, and to be set free off that surface by which the current leaves the liquid. In the case of impure zinc it seems likely that the solution, which proceeded while the cell was open, still goes on when the circuit is closed, but that now there is additional solution of zinc, the hydrogen corresponding to this portion being set free off the copper plate. Further investigations have made it almost certain that in no case will the zinc dissolve, saving step by step with the passage of a current ; but that in the case of impure zinc there are in- numerable small currents circulating between portions of the impure zinc, that differ from one another in chemical or in mechanical properties. These currents are called local currents, and the chemical action that occurs in the open cell is called local action. Amalgamation then is to prevent the occurrence of local action, which latter contributes no energy to the main current. 10. Polarisation. In the simple Volta's cell in which we use a copper and a zinc plate immersed in dilute sulphuric acid, it is found that the current falls off rapidly from its initial strength. We may show this by introducing a galvanometer into the circuit, when a gradual decrease in the deflexion of the needle will be observed (see 8). Note. Mercury cups. In all experiments where it is necessary to make and to unmake electrical connection between different conducting wires, it is very convenient to use merctcry cups. These may be simply small round wooden boxes filled with mercury, in the lids of which are bored holes. The wire terminals of the battery, or of any piece of apparatus such as the galvano- meter, may be rapidly dipped into or removed from these cups, and, if the wires be well amalgamated, the connections thus made are good. On examining into the cause of the falling off in current, we find it to be due to the film of hydrogen that adheres to the copper plate at which, as we have seen, it is set free. Experiment shows that this hydrogen acts to lessen the current in two very different ways. Firstly, it covers the copper plate with a layer that conducts the current very badly as compared with the dilute acid that ELECTRICITY initially was in complete contact with the copper, thus lessening the current by passive resistance. Secondly, owing to the presence of the hydrogen on. the copper plate there is a tendency to a back current opposing the main current. This back tendency is called polarisation^ and is of a very different nature from resistance. Indeed, if we now replace the zinc plate of the cell by a clean copper plate, and complete the circuit, we find that we actually get a reverse current ; this continues until the hydrogen has combined with all the oxygen that is mechanically dissolved in the liquid, as we shall see later on (see Chapter XII. n). It is pretty clear even at this stage that * polarisation ' can never actually reverse the original current, for this latter causes the polarisation. But the current can be nearly or quite brought to a standstill. Experiment. The figure indicates in a simple diagrammatic manner one mode of showing this 'back tendency' or 'polarisation.' G is a somewhat delicate galvanometer ; m m' are mercury cups ; Cu and Zn are the plates of the cell, in which we here use common water instead of dilute acid. C'u is a spare copper ^ * - that may be readily substituted for the plate Zn. (If we find the current too strong for the galvanometer, we may protect this latter, to a greater or less extent, by bridging over the cups m m 1 with a wire of less or greater resistance, as will be explained in Chapter XIII. 10 ) We first complete the circuit, as shown in the diagram, and notice that the initial current soon almost, or entirely, ceases. We next replace the plate Zn by the 'passive plate' C'u, and a reverse current, almost as strong as the original current, will be observed. If we use dilute acid in the cell, the current only falls off to some extent, and the reverse current is proportionally weaker. ii. Constant Batteries. The efforts of scientific men were soon directed towards the contrivance of battery-cells in which polarisation did not occur. The hydrogen, which in the simple Volta's cell adheres to the copper plate, was to be got rid of. One method of doing this was CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS J// a mechanical method ; viz., to use as a negative plate some plate to which the hydrogen could not stick in the form of a film. The other method was to give the hydrogen, as it appeared in its nascent condition on the surface of the negative plate (see 7), some chemical reducing work to do. We will now describe several forms of so-called ' constant ' battery-cells under the heads of mechanical and chemical respectively. Note. Electromotive force. The poles of the open cell exhibit a certain A\ r ; the greater this is, the greater (cceteris paribus) will be the current when the circuit is closed. When the current is running, that which urges it is called electromotive force ; it can be measured in terms of the A V that would appear were the circuit cut. We shall hear more of electromotive force (usually called E.M.F.) when we come to Ohm's Law in Chapter XIII. In Chapter XV. 13 we shall see that when the hydrogen is got rid of by chemical means we not only obviate polarisation, but we gain a positive advantage in a greater E.M.F. (see preceding note). I. Mechanical. The earliest form of cell in ordinary use that comes under this head is the Smee's. In this the copper plate is replaced by a sheet of thick silver foil that is covered with very finely divided platinum (or ' platinum-black '). Off these points the hydrogen rises to the top, and does not remain as a film covering the negative plate. The substitution of silver for copper gives, moreover, a slightly greater E.M.F. II. Chemical. Single-fluid cells. The most important single- fluid cell in which the hydrogen is used up in doing chemical reduction is the (i.) Bichromate cell. Here the negative plate is of carbon (a special kind) instead of copper ; and the liquid is composed of a solution of potassium-bichromate in water, mixed with sulphuric acid. The nascent hydrogen is employed in reducing the chromic acid ; so that chromic sulphate is formed. This battery has a high E.M.F., yields no fumes, is simple in its arrangements, and there- fore is adapted for general laboratory use. The zinc plates must be removed from the liquid when not in use. (ii.) Latimer Claris standard cell. For purposes of com- parison rather than for practical use, Latimer Clark proposed the following cell. As a negative piate, pure mercury. Over this a paste obtained N iyS ELECTRICITY CH. xi. by boiling sulphate of mercury with a saturated solution of zinc sulphate. As a positive plate, zinc ; this resting on the sulphate paste. Insulated wires are connected with both plates. This cell can be easily prepared. Its value consists in the fact that, when left with the circuit open, its E.M.F. is very con- stant, and can be used as a standard of comparison. When, how- ever, the circuit is closed, and a current flows, the E.M.F. does not remain constant. III. Chemical. Two-fluid cells. There are many elements in which the negative plate is separated from the positive plate by a porous pot or' partition. The positive plate is surrounded by a liquid that acts upon it when the circuit is closed, and the negative plate by a liquid containing something that the nascent hydrogen can reduce. (i.) Bunseris cell. In this battery the + plate is zinc, sur- rounded by dilute sulphuric acid. The plate is gas-carbon ; it FIG. i. stands in a porous pot, and is surrounded by nitric acid, the porous vessel thus preventing the two acids from mixing, while yet allowing the chain of chemical changes to pass with the current through its pores. In the figure, P represents the entire cell ; while F Z V C represent the outer vessel, zinc, porous pot, and carbon re- spectively. The zinc dissolves in the dilute acid, forming zinc sulphate ; while the corresponding hydrogen, set free against the carbon, reduces the nitric acid to lower oxides of nitrogen. CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 1/9 (ii.) Grove's cell. This is an earlier form than the preceding. It differs from it only in having platinum instead of carbon. The shape of the porous pot is flat, to suit the flat plates of platinum- foil. (iii.) Daniell's cell. In this the only essential difference from the above is that we have copper in a saturated solution of copper sulphate, instead of platinum (or carbon) in nitric acid. But it is usually constructed having the zinc and acid in the porous pot, while very often the copper itself forms the outside vessel. The zinc may be surrounded by a semi-saturated solution of zinc sulphate, or of common salt, instead of by dilute sulphuric acid. Here the hydrogen reduces the copper sulphate ; sulphuric acid is formed, while copper is deposited on the copper plate. It is therefore necessary to keep up the strength of the copper sulphate solution by a supply of crystals of that salt. In one very portable form we have but a single vessel ; at the bottom is a plate of copper on which is a layer of crystals of sulphate of copper ; over this is a layer of sawdust ; and, resting on this, a plate of zinc immersed in dilute sulphuric acid or in a solution of sulphate of zinc. In- sulated wires form connections with the plates. In ' gravity batteries ' the liquids are kept apart simply by the fact that the less dense liquid forms a layer above the more dense. (iv.) LedanchPs cell. In this the + plate is a zinc rod immersed in a strong solution of ammo- nium chloride. In the porous pot is a carbon rod, round which is tightly packed a mixture of manganese dioxide and of pow- FIG .. dered carbon. The porous pot is closed at the top with pitch, a hole being left for the escape of gases. N 2 180 ELECTRICITY CH. xi. The ammonium chloride solution soon soaks through and moistens the powder in the porous pot, and then the cell is ready for use. Zinc chloride is formed in the outer vessel, the zinc displacing ammonium. In the inner vessel is set free ammonia, while the re- maining hydrogen of the ammonium reduces the manganese dioxide. 12. Remarks on Cells and on Batteries. We have now men- tioned the chief cells that are of interest to the general student. A few remarks will be made in conclusion. Chemical action in the cell, For a discussion of the manner in which the chemical action takes place, in what way the hydrogen displaced by the zinc * travels with the current ' and appears at the negative plate, and what relation the amount of chemical action bears to the strength of the current, the student is referred to Chapter XII. , Efficiency of different cells. We have stated that different cells give a different A V at their terminals when the circuit is open, and a different E.M.F. (which, when the other conditions are the same, will cause different currents) when the circuit is completed. For a further discussion of this matter we refer the reader to the sections on ' Electromotive force ' in Chapters XIII. and XV. Again, cells differ from one another in offering more or less re- sistance to the passage of the current through them. This matter will be discussed in Chapter XIII. Coupling cells together. If we connect n cells as we connected the elements in the Couronne des tasses we get a A V in the open circuit, or an E.M.F. in the closed circuit, which is n times that of a single cell. Again, if we connect all the positive plates together and all the negative plates together respectively, we have what amounts to one large cell whose plates are n times the size of those of a single cell. Such an arrangement has the E.M.F. of only one cell. The advantages of the above two methods of coupling in dif- ferent circumstances respectively will be discussed in Chapter XIII., and the proper terms will be there given. Amalgamating zinc. If the zinc be wetted with dilute acid it is readily amalgamated with mercury. Or, again, if a little sodium be added to the mercury it enables one to amalgamate zinc and other metals with greater ease. CH xi. THE PHENOMENA OF ELECTRIC CURRENTS l8[ Use of carbons. In order to obviate the tendency of the car- bons to 'soak up' the liquids in which they are immersed a result very unpleasant and very injurious to the binding screws attached io the carbons it is usual to soak the upper part of the carbons in melted paraffin wax. The surface must then be scraped at the places where the binding screws make contact. 1 82 ELECTRICITY CH. XII. CHAPTER XII. THE CHEMICAL PHENOMENA ACCOMPANYING THE PASSAGE OF THE CURRENT. i. Introductory. In this chapter we shall consider the chemical phenomena that accompany the passage of an electric current. We shall assume that the student is acquainted with the elements of theoretical chemistry. Of the other classes of phenomena, the heat effects will be only briefly mentioned here, a fuller treatment being found in Chapters XV. and XVI. ; the magnetic effects will be discussed in Chapters XVII. and XVIII., &c.; and the induction phenomena form a large portion of uur subject and will occupy our attention in Chapters XXI.-XXIV. 2. Heating Effects; a Brief Account. When a current passes through a conductor it is found that the said conductor is heated. The stronger the current, and the greater the resistance of the wire, the greater is the quantity of heat evolved. The exact law relating to this matter will be given in Chapter XV. A battery of from three to six large bichromate cells will render white-hot, or even melt, fine platinum wire, and will illumine small ' glow ' lamps of ten candle-power. Now, whence does this heat-energy come? Our source of energy is the battery-cell. If in this a certain amount of zinc be dissolved, there will be a certain amount of heat evolved. If the circuit of action be contained within the cell, then all the heat due to the quantity of chemical action will appear in the cell. But if the cell drive a current and this current heat a wire external to the cell, so much the less heat will appear in the cell. The total amount of heat evolved during the solution of a given mass of zinc will be constant ; but, by making the external circuit of CH. xii. CHEMICAL PHENOMENA 183 some substance that offers a very high resistance to the current, we may cause nearly all the heat, due to the amount of chemical action in the cell, to be evolved outside the cell. We lose chemical-potential-energy, and we gain equivalent heat-energy (see Chapter X. 4). 3. Chemical Effects ; General View. Bodies may be roughly- divided into three classes with regard to their behaviour as to allowing a current to pass through them. (i.) Conductors. All those solids and liquids that allow a cur- rent to pass, while themselves undergoing no change saving a rise in temperature, are called conductors. Such are metals whether solid or molten, carbon, the bodies of animals, &c. (ii.) Insulators. Bodies that do not allow any appreciable current to pass are termed insulators. A few examples of insu- lators are glass, ebonite, paraffin oil, dry vapours, &c. These two classes merge the one into the other. The reader should refer to Chapter XIV. 14, for tables of resistances. (iii.) Electrolytes. A great many bodies allow a current to pass, but themselves suffer a chemical decomposition that pro- ceeds step by step with the current ; not allowing any appreciable current to pass without this chemical decomposition occurring. Such bodies are termed electrolytes. This class consists almost entirely of compound liquids or solutions of salts, or of molten salts; the word 'salt 'being understood in its widest sense. A few examples are aqueous solutions of adds, of metallic salts, or of alkalis ; also such liquids as melted potash or soda, c. The action, as we shall see hereafter, requires that the molecules of the bodies shall move with freedom ; hence all electrolytes are liquids or pastes. Explanation of terms used. Referring to fig. (ii.), 4, we see that usually the electrolyte is introduced into a glass vessel, while the current enters by a metal plate A and leaves by another plate B. This cell is called an electrolytic cell ; A is called the anode, B is called the kathode. These plates are usually of platinum, this being unacted upon by most liquids. In what follows it is understood that they are of this metal, unless the contrary is stated. It is found that the electrolyte is invariably split up into two molecular (or atomic) groups, the metallic radicle and non-metallic 1 84 ELECTRICITY ,CH. XII. radicle respectively. Thus H. 2 SO 4 splits into H 2 and SO 4 , not into H 2 O and SO 3 ; NH,C1 into NH 4 and Ci, not into NH 3 and HC1 ; CuSO 4 into Cu and SO 4 ; and so on. These groups are set free at the two electrodes ; the metallic groups (as H 2 , Cu, (NH 4 ) 2 , &c.) at the kathode ; the non-metallic groups (as O, SO 4 , C1 2 , &c.) at the anode. These groups are, from their 'travelling,' called ions ; the metallic groups are called kations or electropositive, from being set free at the kathode or negative plate ; the non-metallic groups are called anions or electronegative, from a similar reason. The process of chemical decomposition through the agency of a current is called electrolysis. Note. A slight acquaintance with Greek will enable the reader to discover for himself the derivation of the terms anode, kathode, electrode, ions, &c. Experiments in electrolysis. Experiments to illustrate electrolysis may be devised almost ad infinitum. We here describe a few typical cases. (i.) Decomposition of -water. It is not certain that pure water can be electrolysed. Certainly as we approach purity the water becomes almost an ' insulator ' ; it being remarkable that mere traces of acids or salts in solution have a very great influence in destroying the insulating power and in rendering the water an electrolyte. When we wish to decompose water we usually mix it with about one-tenth its volume of strong ILSO 4 . What part the FLSO 4 plays is not known for certain. Here we will assume that it only renders the water capable of electrolysis. The figure represents the usual arrangement, or one form of it. The vessel contains the acidulated water ; A and B are the anode and kathode respec- tively ; C and D are glass vessels at first filled with the liquid and inverted over the electrodes in order to collect the gases. When the current passes, the H. 2 and the O of the FLO are set free in chemical equivalents at the kathode and anode respectively. Errors in the volumes collected occur from the greater solubility of oxygen, from part of the oxygen being set free in the form of ozone, and from hydrogen being * occluded ' by the platinum electrode to a greater extent than is oxygen. The first and third errors can be nearly eliminated by allowing the action to proceed for some time before collecting the gases ; the second error by heating the tube, or by having electrodes of such area that the current is not too dense. CH. xii. CHEMICAL PHENOMENA 185 In several of the following experiments it is very convenient to have a V-shaped tube ; the electrodes occupying the two arms of the tube respectively. Or we may have an ordinary cell in which the two electrodes are separated by a porous earthen diaphragm, or two vessels connected by wet cotton wick. Each of these methods enables us to examine at leisure the condition of the liquid about the two electrodes after the action has proceeded for some time ; the mixing of these two portions of the liquid being to a greater or less extent prevented. (ii. ) Electrolysis of CziSO r If we employ a solution of CuSO 4 , we find Cu set free at the platinum kathode ; while from the platinum anode is set free O, the liquid about this electrode at the same time losing colour and showing the presence of free H 2 SO 4 . If we employ copper electrodes we find fresh Cu coating the kathode, while the anode is dissolved with the formation of CuSO 4 . We shall in 5 argue that in both cases there is primarily set free Cu at the kathode and SO 4 at the anode. (iii. ) Electrolysis of Na. 2 SO v In this case (using platinum electrodes) we find H 2 and 2NaHO appearing at the kathode ; from the anode is set freeO, while the liquid about it shows signs of free H 2 SO 4 . We shall in 5 show that this is equivalent to a setting free of Na 2 at the kathode, and of SO 4 at the anode. If some extract of red cabbage be mixed with the solution, and a drop or so of dilute acid be added (if necessary) until the whole is of a dull purple colour, the alkali and acid will be indicated by green and red colourations about the kathode and anode respectively. (iv. ) Electrolysis of JVff 4 . Cl. In this case we get Cl at the anode, and NH 3 together with H at the kathode (NH 4 C1 = NH 4 + C1). If, however, the kathode be of mercury, this latter swells up and forms what is by some considered to be an amalgam of mercury and the metal NH 4 . The solution of NH 4 .C1 must be weak and cold ; otherwise we may get the very unstable and dangerous ' chloride of nitrogen ' formed. (v.) Electrolysis of KHO. Some potassium hydrate is fused and is placed on a piece of platinum foil, which forms the anode. In a cavity on the upper surface of the salt is placed a drop of mercury which is made to form the kathode. The salt will have absorbed from the air, when it cooled after fusion, enough water to render it an electrolyte ; it will, in fact, be a very stiff ' paste.' After passing a current from a battery (say of four bichromate cells) for a few moments, we drop the globule of mercury into water. It is seen to give off hydrogen, while KHO is found in solution. ,, , This indicates that the KHO (2KHO = K,O + HX>) was decomposed, K._, being set free at the kathode and there forming an amalgam with the mer- cury, while O was set free at the anode. 1 86 ELECTRICITY CII. XII. (vi. ) Electrolysis of Pb. A.,. We here use a fairly strong solution of plumbic acetate, and a pair of lead electrodes. The Pb is set free at the kathode, there forming a beautiful ' lead tree.' With a small cell this can be projected on the screen by means of a lantern. A battery of four bichromates causes a very rapid growth of the tree ; any battery-cell will give the result in time. The anode will at the same time be dissolved, giving Pb.A 2 . 4. Grothiiss's Hypothesis. Nature of Electrolysis. In a later section we shall see that, as we should have expected, the ions set free at the two electrodes are always chemical equivalents of one another. Thus (we assume the reader to be acquainted with the exact meaning of chemical symbols) for H 2 and 2NaHO at the kathode we get O and H 2 SO 4 at the anode ; we could not get O 2 , since that is equivalent to 2H 2 or to 4NaHO. It may be noticed further that no signs of decomposition can be detected saving at the surfaces of the electrodes. We give here a view of electrolysis that' is consistent with the above facts ; it is the view of Grothiiss, or Grothiiss's hypothesis, slightly modified to suit the modem theory of compound liquids or of salts in solution. It is believed that, in a compound liquid, not only are the molecules in continual 'slipping' motion, but the atoms them- selves are being continually dissociated and recombined. Thus, in water, neighbouring molecules of H 2 O are continually ex- changing partners, the H 2 of one molecule taking the O of the other, and reciprocally. This cannot be observed, because the interchange is molecular only, and the average constitution of the liquid remains the same ; for much the same reason, indeed, we cannot observe the molecular motion called ' Heat.' It is considered that the presence of the electrodes, connected with the + and poles of the battery respectively, has the effect of directing this interchange ; or, when one electrode is made + and the other , then the interchange of partners is such that on the whole the H .,..$ in their changes move towards the electrode, and the O.s towards the + electrode. Probably the ' current ' is thus, and thus only, conveyed ; passing by convection in an electrolyte. In a metallic conductor it passes by conduction. CHEMICAL PHENOMENA I8 7 We thus have the H 2 .j- arriving at the kathode and the O.s at the anode ; while the liquid between, being in the same condi- tion as to average constitution as it was before we tried to pass a current, appears to be unaffected. This directed travelling of the groups has received the name of the migration of the ions. Experiment illustrating Grothiiss's hypothesis. The following experiment illustrates how the electro + and electro groups (or kathions and anions) travel from one electrode to the other, while the inter- vening liquid shows no signs of change. A, B, and C are three ves- sels containing a solution of /^ Na_,SO 4 coloured to a dull purple with extract of red cabbage. They are con- "^ nected by moistened lamp- wick, and in A is the kathode, and in C is the anode. If two Leclanche cells be employed, and the whole be left for a day or so, it will be found that A has become alkaline, and C is acid ; while B through which have passed both the alkaline and acid groups is still unaltered. We will now represent in chemical symbols the interchanges that are continually taking place in an electrolyte when the current is passing, choosing a few typical cases. The upper brackets represent the condition of the chain of molecules before inter- change ; the lower brackets give the new grouping after inter- change, with the ions set free at ^ ^_^ the two electrodes respectively. I. The case of water. Here we see that initially we have while Pt FiG. ii. groups of OH 2 on^y finally we have an odd H 2 at the kathode, an odd O at the anode, with groups of OH 2 between, as before. II. The DanieWs cell. Here we will consider the case of a battery-cell itself, since all we have said of electrolysis applies to the battery-cell equally with the electrolytic-cell, to chemical combination equally with chemical decomposition. In the battery- i88 ELECTRICITY FIG. iii. cell the copper plate is the kathode, and the zinc plate is the anode. The 'Zn' against the zinc plate represents an atom of the originally undissolved zinc. The arrangement of brackets repre- sents that when the circuit is closed this Zn takes SO 4 from the nearest H 2 SO 4 ; this leaves H 2 to take SO 4 from the next H 2 SO 4 , and so on ; Ho then passing through the porous cell takes SO 4 from the next CuSO 4 , and so on, until finally Cu is deposited on the copper plate (the kathode). As explained earlier, no change due to this interchange will be observable saving at the surfaces of the battery plates. 5. Primary and Secondary Decompositions. We will now complete the theory of electrolysis, explaining the results of de- composition noticed in 3, (ii.), (iii.), (v.), &c. There is every reason for, and no reason against, the view that in electrolysis we have primarily the metallic, and the non- metallic, groups set free against the kathode and anode respec- tively. But for reasons of ' chemical affinity ' such a condition of things may not be stable. Thus if Na. 2 SO 4 be decomposed into Na 2 and SO 4 , the Na 2 will decompose water and give 2NaHO-f H 2 , while the SO 4 with water will give HoSO 4 + O. Thus we have by primary decomposition, at anode SO 4 Na 2 at kathode ; and then by secondary decomposition, / tf;zy 3 ?2*< w J^J O'^T.6 Lead '. " ' .' V ^ j w 207-0 O J 103-5 oo 1-072 This table belongs to 9. Sometimes it is desired to coat the body (as a baser metal) with a permanent adherent layer (as of gold or silver) ; sometimes with a non-adherent layer that, when detached, gives us a 're- verse ' impression of the surface on which it was deposited. The processes of electro -gilding and electro- silvering come under the first head ; those of reproduction of engravings, casts of coins, c., come under the second head. Sometimes also we employ the process to etch designs on the anode. In ii we shall see that under the above conditions, i.e. when the solution remains unaltered in strength while there is a transfer- ence of metal from the anode to the kathode, any single cell however weak can effect the electrolysis (see also Chapter XV.). As a general rule a single battery-cell with large electrodes gives a compact film of metallic appearance, while a powerful battery, with electrodes too small for the current, gives a spongy non-metallic-looking coating. In a word, for good results the current must not be too dense. CH. xn. CHEMICAL PHENOMENA 193 I. Copper-plating. In this case we wish to coat metals with copper. In more technical books (such as Hospitaller's ' Formu- laire Pratique de PElectricien ' and Sprague's * Electricity ') the student will find recipes for copper solutions suitable for different purposes, and directions as to current-density, &c., such as will insure the best results. In the present Course we shall give little but the principles of the different methods, and shall therefore give but one recipe for a copper solution. ' A quantity of water is mixed slowly with from one-tenth to one-twelfth its volume of strong H 2 SO 4 , and the mixture when cold is saturated with CuSO 4 .' As a battery we may use one Dani ell's cell of sufficiently large size. This is connected up with two metal rods B and D as shown, so that conducting bodies hung from B form the kathode, while a copper plate hung from D forms the anode. All wires, &c., which it is desired to protect from action must be coated with wax, or. with caoutchouc, or other varnish. If the body is a non- conductor it should be covered with some conducting substance, and care must be taken, by encircling it with a copper wire or by some other means, to make the action uniform over the surface. The bodies to be coated may, more simply, be made to form the negative plate in a Daniell's cell. In this case the substitution of salt and water for battery acid, as the solution acting on the zinc, will make the action slower and the copper-film more com- pact. If the action is sufficiently slow, the deposit of a film of f\ mm. should take from twelve to twenty-four hours (Hospitalier), though a quicker rate of deposit may give good results. II. Reproduction of engravings^ &c. If the object to be re- produced be a delicate one, we may proceed as follows. We o 194 ELECTRICITY CH. xn. will suppose, e.g., that it is required to get a facsimile of one side of a rare coin. First, we take a wax cast (in reverse) of the coin. This wax reverse is then coated with plumbago, surrounded with a thin copper wire so as to distribute the current more evenly, and made to form the kathode in an electrolytic cell. The copper coating will, when removed, reproduce the surface of the coin in question. Or we may make complete casts of bodies and obtain electro- copper copies or reverses, according as the cast is a reverse or not. III. Electro- etching. By covering a copper plate with wax,, tracing characters through the wax, and using the plate as an anode, we may etch out designs to any required depth. It is, however, not very easy to obtain any delicate results in this way. IV. Obtaining designs in relief. If we use the plate, prepared as in III., as the kathode in the cell, we shall obtain the same designs in relief. V. Silver-plating. For this purpose we must employ a silver solution, and a silver plate as anode. One recipe for the silver solution is as follows (Hospitalier). * Dissolve 150 grammes of AgNO 3 in ten litres of distilled water ; add 250 grammes of pure KCN ; stir until there is complete solution ; then filter.' In silver-plating good results are more difficult to obtain than in copper- plating, and more technical books than the present Course should be consulted if the student intends more than mere illustrative experiment. ii. Polarisation of the Electrodes. In simple electrolysis,' i.e. where no simultaneous decompositions (see 6) occur, there are two cases to be considered. I. Where there is no polarisation. In the cases of electro- plating considered above, the condition of the plates and of the solution remains constant. The fact that the kathode increases, and the anode decreases, in thickness during the action obviously does not affect the question of whether there will be a ' back tendency.' Hence, there is no more tendency for the chemical changes to take place in a reverse order after electrolysis has proceeded than there was at first. There is, in fact, no chemical-potential-energy stored up by the electrolytic action ; and no tendency, therefore, for the electrolytic cell to drive a reverse current (see Chapter XL 10 and Chapter XV. 8 and 9). CH. xii, CHEMICAL PHENOMENA 195 Note. There may be a very slight amount of polarisation owing to the fact that the deposited metal is not of exactly the same quality as the anode plate. Experiment. We may pass a current through a Cu | CuSO 4 | Cu cell for some time, and then prove the absence of polarisation by connecting the terminals of the cell with a galvanometer, when no current will be observed. Care must be taken that the original current be not so dense as to give simul- taneous decomposition of water. There being no back E.M.F,, we can perform such electrolyses with any cell, however weak. (For further on this matter, see Chapter XV. n.) II. Where there is polarisation. But in all such cases as that of the decomposition of water, i.e. cases where the condition of the cell changes after electrolysis, we have a decomposition effected by the current against chemical affinities. We have, set free at the electrodes, ions that have a chemical tendency to reunite ; and this recomposition can take place most readily by means of a reverse current and a reverse chain of molecular interchanges. We have, in such cases, done chemical work ; we have gained chemical- potential-energy. The cell is now like a battery-cell ; and, like a battery-cell, it will tend to drive a current. The cell will have a certain ' electromotive force ' depending on the energy of the chemical affinities of the ions ; and this E.M.F. must obviously be opposed to that of the original current by which the electrolysis was effected. (For E.M.F. see Chapter XI. u, note.) Such cells are in fact battery-cells ; but, as a rule, they will not act as such for a long time, as the supply of ions ready to re- unite is limited. Experiments. (i. ) Grovels gas-battery. The figure represents several ' ceils ' for the decomposition of water. It will be noticed that the electrolysed gases are collected separately, and that the platinum electrodes are so long that their upper extremities are always immersed in the gas, however little of this latter there may be. First, a current is passed through each cell separately until a considerable amount of gas has been set free in each ; this amount must be the same in all. Then these cells are in reality battery-cells. They can be coupled up ' for small internal resistance,' all the kathodes together and all the anodes together; or ' for E.M.F'.,' anode to kathode and so on. The hydrogen in the one tube and the oxygen in the other tube of each tend to reunite ; and it is found that when the external circuit is completed this reunion will take place step by step with the passage of a current, the 02 196 ELECTRICITY CH. xn. action being merely the molecular interchange, explained in 4, reversed in direction. The current will be in the reverse direction to that by which the electrolysis was in the first instance effected. Here each cell answers to a Volta's cell in which the zinc plate has been replaced by hydrogen in contact with platinum ; the copper plate by oxygen in contact with platinum. Notes. (i.) The recombination can occur only when the circuit is closed ; and, further, only when there is a platinum plate that is in contact both with the gas and the liquid. This action of the platinum will be familiar to those who have studied elementary chemistry. (ii.) There will be a current if we replace the platinum -oxygen plate by plain platinum. This is due to the fact that there is always some oxygen dis- solved in the dilute acid. 12. Secondary, or 'Storage,' Cells. Any arrangement by means of which chemical-potential-energy may be used up in such a manner as to give us the equivalent energy of an electric current is called a ' battery-cell] or often more shortly 'a cell? Where the cell is put together out of certain materials and then does not require the passage of an electric current before it is fit for use, it is called a ' primary -cell? All the cells described in Chapter XL n, were of this class. But where the cell requires the passage of a current before it is ready for use, i.e. where the chemical-potential-energy is the result of electrolysis, in such cases the term ^ secondary-ceir or ' storage-cell] is employed. Such a cell, e.g., is the Grove's gas-cell. The latter term was intended to imply that we * stored up ' elec- CH. xii. CHEMICAL PHENOMENA 197 tricity in the cell, since we could use a primary current to effect the original electrolysis, and then could recover a current after a greater or less interval of time. So understood, the term is a mis- nomer. We use up some of the electrical energy of the primary current, and get equivalent chemical- potential-energy stored up in the secondary-cell ; later, when we complete the circuit of this cell, we lose this chemical-potential-energy and get again the elec- trical energy of a current. But we did not store up electricity ; a condenser does this, not a secondary-cell. Analogy. We might use up some wind-energy to turn a fan, and this might wind up a weight from a lower to a higher level. We might, later on, let the weight again descend, and so obtain a wind driven in the reverse direction by the fan. W T e should have stored up mechanical- potential-energy, not 'wind. The problem of inventing a good secondary-cell is of a two- fold nature. (i.) How can we retain a large amount of the ions in contact with the electrodes, ready to drive a reverse current ? (ii.) How can we prevent action occurring while the cell is lying idle with open circuit ? In our next two sections we shall describe one form of cell devised to satisfy these two conditions. 13. Plante's Secondary-Cell. In the original form of Planters cell there are initially two lead plates immersed in dilute H 2 SO 4 . I. Formation of the cell. A somewhat lengthy process is needed in the first instance in order to get the cell ready to act as a battery. (i.) A current is passed through it ; this resulting in hydrogen being given off at the kathode, while the surface of the anode is oxidised into the condition of PbO 2 . (ii.) The current is reversed, and is continued until the PbO 2 is all reduced to spongy lead, while the other plate is in its turn per-oxidised. (iii.) This process of sending currents in alternate directions is repeated until the lead has been acted upon to some depth. Thus, the plate that served last as anode is left coated deeply with PbO 2 , that which served last as kathode is deeply coated with spongy lead. This process is called ''Formation of the cell j and it 198' ELECTRICITY CH. XII. is said to be left charged' ; this is represented in fig, i. Subse- quent chargings will need only a single passage of the current. II. Discharging the cell. In our discussion of the cell we will at present consider only the degree of oxidisation of the lead, and will only touch on the part played by the dilute acid. It is clear that if we have PbO, the form- ation of PbSO 4 from this (PbO + H 2 SO 4 = PbSO 4 + H,O) can be re- garded as a secondary action, and has no direct bearing on the electrolytical theory of the cell. If we connect the terminals of the cell, a current is set up in a reverse direction to the current that was last passed through the cell in its formation. One atom of oxygen from each molecule of PbO 2 passes back to the other plate, giving PbO on each plate, instead of PbO 2 on the one and Pb on the other. This action from the one plate across the liquid to the other plate takes place only step by step with the current ; and the current will continue as long as there is PbO 2 left. When the cell has 'run down,' each plate will be coated with PbO (or, by action with the dilute acid, with Pb.SO 4 , which is equivalent to PbO as regards degree of oxidisation). III. For the cell to be recharged, a current must be again passed through the cell. If we consider that there is PbO on both sides, the action is simple ; by an electrolytic passage of H 2 to CH. XII. CHEMICAL PHENOMENA 199 the kathode, and of O to the anode, we get again Pb on the former and PbO 2 on the latter. If there be Pb.SO 4 on both plates we may represent the action as follows. [See fig. (iii.)] For convenience we have written Pb.SO 4 +H 2 O (there is plenty of available water in the dilute acid) in the two equivalent forms H 2 SO 4 .PbO and OPb.H 2 SO 4 . Lead Sulphate Pb H^SO^ Lead Free Acid FIG. iii. This indicates two main facts. (i.) That the electrolytic action of ' charging ' the cells is the same, whether there be PbO or Pb.SO 4 on the two plates of the exhausted cell. (ii.) That the recharging of the cell is practically a transfer- ence of O from the kathode to the anode. IV. Local action, and waste of energy, &c. It may naturally occur to the reader that there seems no reason why the whole action should not take place between the PbO 2 and the lead plate upon which it rests, instead of between the PbO 2 and the other lead plate that is separated from it by the dilute acid ; so that after a short period of lying idle with open circuit, the cell would have ' run down ' and be useless. No doubt there is some such action at first. But it seems that the insoluble layer of PbSO 4 , which is thus formed between the PbO 2 and the lead plate on which it rests, hinders further local action ; while the surface turned towards the other lead plate, being a free surface, remains always more open and porous. This appears to 'be a very important part played by the insoluble PbSO,. As regards the energy wasted in the charging of the cell, this will be more fully discussed in Chapter XV. We will here only 200 ELECTRICITY CH. xn. remark that we waste more energy in heat when the back E.M.F. of the secondary-cell is small as compared with that of the primary battery, or when the charging current is large. " It is of interest to remark that by charging cells arranged * in parallel,' and then coupling them ' end-on,' we can obtain a second- ary battery of as great an E.M.F. as we please ; one that could drive a current back through the primary battery. 14. Fame's Accumulator. In order to obviate the necessity for the lengthy and energy-wasting process of 'formation] Faure devised the following important modifications in Plante's cell. The two lead plates were coated with minium, this being Pb 2 O 3 , or PbO.PbO 2 . One passage of the current then sufficed, by the electrolytic setting free of H 2 at the one plate and of O at the other, to convert the one layer into spongy lead, and the other into lead peroxide. Since, however, Pb 2 O 3 requires but O in order to per-oxidise it, while it requires 3H 2 to reduce it, it is clear that we must have three times the amount of minium on the anode (where the elec- trolytic oxygen is set free) as on the kathode (where the hydrogen is set free). Thus the electrolysis of 3H 2 O will reduce Pb 2 O 3 on the kathode, and will per-oxidise 3(Pb 2 O 3 ) on the anode. In all other respects the theory of the * FaurJs accumulator] as it is called, is the same as that of the Plante's cell. The ques- tion of energy will be discussed more fully later on. CH. XIII. 2O I CHAPTER XIII. OHM'S LAW. i. General Ideas as to the Scope of Ohm's Law, Up to the present point we have, in our treatment of the battery-cell, of the current driven by it, and of the circuit through which the current flows, used terms in a vague and qualitative, rather than in an exact and quantitative sense. But in the present chapter we propose to discuss at some length the conditions which determine the magnitude of an electric cur- rent in any particular case ; and to state and explain the law which makes the calculation of the current a matter of simple arithmetic. The law referred to is that known as Ohm's law ; it was enunciated by Dr. G. S. Ohm in the year 1827. This law which must be accepted as confirmed by countless direct and indirect experiments and refuted by none states in the first place that when the circuit of a cell is completed we are concerned with three quantities only; and in the second place that these three quantities are connected by a very simple relation, viz., that given in 2. These three quantities are I. Electromotive force. Each battery-cell possesses a certain power of driving a current, which is directly proportional to, and can be measured by, the A V that is found to exist between the poles when the circuit is broken. Since that which moves matter is called force, so by analogy that which moves ' electricity ' was called l electromotive force? The reader must remember that this term is an inexact one, as the * electromoti 've force ' is not a force at all in the scientific sense defined in Chapter IL 4. To prevent confusion we shall henceforth use the letters E.M.F. instead; so that the actual word force will never be used except in the strict sense of Chapter II. 4. 2O2 ELECTRICITY CH. XIII. We may give an analogy from hydrostatics to make the relation between E.M.F. and A V somewhat clearer. Let us imagine two reservoirs, the levels of water in the two being Li and L 2 respectively, connected by a pipe filled with coarse sand. A current of water will be urged through the pipe, and its magnitude will depend, cczteris paribus, on the pressure due to the difference of level. The pressure can be measured by the difference of level (L, L 2 ) ; we might, indeed, if our system of units is suit- ably fixed, say that the pressure ' is ' the difference of level ; this would be scientifically inaccurate, but with our fixed system of units would lead to no error in calculations. So we can measure the E.M.F. of a cell by the A V that appears at the poles when the circuit is broken ; or by the greatest A V that we discover between two points in the circuit when the current is flowing. And we may sometimes, somewhat inaccurately per- haps, use A V as synonymous with E.M.F. This E.M.F. of a cell depends solely upon the nature of the cell. As we include the coatings of gases on the plates, &c., in the term ' nature of the cell] the reader will see that we have taken into account the phenomenon of ' Polarisation ' (see Chapter XL 10), which diminishes the E.M.F. of the cell from its initial value. However large a current be flowing, the E.M.F. remains unaltered save by polarisation. II. Resistance. Somewhat as, in the hydrostatic analogy given above, the pipe will offer more or less resistance to the current of water according to its length, its section, and the closeness with which it is packed with gravel or sand, so a conductor in the elec- tric circuit will offer more or less resistance according to its length, section, material, and temperature. Resistance is that which hinders the passage of a current and limits its magnitude without producing any tendency to a back- current ; it is purely passive. III. Current. This term has been explained earlier. It simply means quantity of electricity passing across any section of the circuit in one second of time. We may measure this quantity as in Chapter V. i ; but in dealing with electric currents another system of units (explained in Chapters XIV. i, *and XVIII. 3 and 4) will be employed. CH. XIII. OHM'S LAW 203 2. Exact Statement of Ohm's Law. Let AB be a con- ductor including no source of difference of potential. And let there be means of maintaining the points >^. >. A and B at various A V . s t and of measuring these A V.s by electro- meter or other methods ; and also of measuring by electrolysis or galvanometer methods the current that then flows in the conductor AB. Then the experimental law known as ' Ohm's Law ' is that * So long as the conductor A B remains unaltered in temperature and in all other physical respects, the current flowing in it is directly proportional to the difference of potential maintained be- tween its extremities A and B? We may in symbols express this by saying that C ~ (VA-V B ). On trying how the current varies when the conductor is varied, the AV between the extremities being maintained con- stant, our observations lead us to follow the hydrostatic analogy and to say that the conductor * offers a greater or less resistance ' to the current according to the dimensions, material, temperature, &c. Symbolising this resistance by Rg, we define R A by the relation p A R B It occurs to us to inquire next whether ' Ohm's Law ' applies to the entire circuit, including both connecting wires and battery. The difficulty now is to know what replaces the ' V A V B ' of the experimental law given 1 above. Let E represent the electrostatic A V between the terminals of the battery when the circuit is broken. Let B and r represent respectively the resistances of battery cell and of connecting wires (see 3 and 4, and Chapter XIV.) Then a great mass of evidence has convinced investigators that when the current is not dense enough to cause polarisation or other alteration in the battery, we may state that E 204 ELECTRICITY CH. xm. And if there are many cells in the circuit we have c _ where the E.M.F.s are to be reckoned as negative if they oppose the resulting current. Note. It is in fact believed that Ohm's law always holds, but that if the current is too dense then E and B have some new values not known from our previous measurements. 3. Resistance further Discussed, We will now discuss further the nature of resistance, and the physical meaning of Ohm's law. Ohm's law connects the three quantities C, E, and R ; of which, up to the present point, only the two quantities C and E have received an exact meaning. We might therefore be inclined to think that the law simply p defines R as such that 'C=~ ;' and therefore is no discovery at s\. all, but a mere truism. If this were the case the law would be of little use to us, since R might not be a fixed property of the conductor, one to be found once for all, but might depend upon E, so that the law would not then lend itself to the most important problem, viz., that of determining the current from a knowledge of the E.M.F.s and of the nature and dimensions of the conductor. Thus it might have been the case that when once a current was started with a certain E.M.F. E, the resistance of the conductor was once for all broken down ; so that an E.M.F. of 2E with the same circuit might give more than twice the current. But direct experiment gives us two important results. (i.) That R is a definite quantity for each conductor, and cap- able of measurement as such. Hence we can predict the current which any given E.M.F. will drive through any given conductor. (ii.) That R depends in a very simple way on the length, cross section, temperature, and material of the conductor. 4. The Exact Conditions on which Resistance Depends, If the resistance of such bodies as wires be examined experimentally, it is found that R is directly proportional to the length / of the CH. xui. OHM'S LAW 205 wire, inversely proportional to its sectional area A, and depends also on its ' specific resistance ' c, and on its temperature. We will define later the exact meaning of * specific resistance,' and we will for the present omit the factor expressing the depend- ence of the resistance on temperature. Thus we may state that R-*.i/. A Here k is a constant depending on the unit of resistance employed. The nature of the experimental methods of proving the above is indicated in what follows. Experiments. (i. ) External resistance. If we \vish to examine most simply the law of resistance r of such external conductors as wires, we must get rid of the internal resistance B of the battery (see 2, equation (ii. )). We can practically make B = o by employing, instead of an ordinary battery, a thermo-celL This can be constructed of metallic bars of such thick- ness as to make their resistance relatively negligible ; and a constant E. M.F. can be insured by keeping the two sets of junctions at two fixed temperatures (^Chapter XVI. ). With such a battery the formula of Ohm's law becomes C = We can measure C by means of a galvanometer, as will be explained fully in Chapter XVII. Then, by varying the length, material, and cross section of the wire, we are led to the result that r = k 2\. It is to be noticed that this result proves that the current flows through the whole body of a wire equally, and not along its sur- face. Provided that the temperature of the wire is constant, it makes no difference whether a cross section of (e.g.) i sq. cm. be circular, square, or of the form of a flat rectangle. Here is a notable difference between conductors as used for electrostatic purposes, and as used for conveying a current, re- spectively. (ii. ) Internal resistance of a battery. The internal resistance of a battery is a far more difficult matter to investigate. Indeed the conditions are so complex, that it is invariably the rule \ofind experimentally the resistance of a cell or battery, instead of to calculate it as we do in the case of a wire from a knowledge of the materials and their dimensions. But still it is possible to show that in all probability the same general laws hold ; or that B = k . . A. To investigate internal resistance we get rid of the external resistance r by employing very thick copper conductors, of no appreciable resistance, outside 206 ELECTRICITY CH. xnf. the cell. The cell itself is constructed with plates of various sizes (so as to vary A), whose distance apart can be altered (so as to vary /). The same general results are found to hold ; though, since we cannot confine the current to that portion of the liquid which lies directly between the opposed plates, we cannot arrive at very exact results. It is important to note that (i.) n cells in series offer to the current n times the resistance of one cell, since, in the former, the length / of liquid traversed is n times that in the latter. (ii.) If n cells be coupled zincs to zincs and coppers to coppers, or ' in parallel,' we practically make one large cell of plates n times the area of those of one cell. Hence the area A of the column of liquid traversed is increased n-fo\d ; and the resistance is -th of that of one cell, and of that offered by n cells in series. 5. Conductivity. Since C = E._*, we may call -- by the _K Ix name 'conductivity,' and say that the current is directly propor- tional to the conductivity, instead of inversely proportional to the resistance, of the circuit. In 9, fig. iii., we represent two points A and B in a circuit joined by several wires of resistances r lt r 2 , r 3 , r 4 . . . respectively. We may call their conductivities c^ c^ c^ and Whence - = _L + + _i + & c . ; R' r, r^ r 3 or the (equivalent conductivity^ (sum of the several conductivities]. 6. Application of Ohm's Law in a Simple Case. Suppose that we have a battery of n cells, and wish to know the best way of coupling them up in the two extreme cases of (i.) Practically zero internal resistance. (ii.) Practically zero external resistance. Let B be the resistance of each cell, r the external resistance, E the E.M.F. of each cell, and C the current. (Read again 4, end.) (I.) Let B, and even n B> be negligible as compared with r. (i.) The current from a single cell will be C = B ~- =? (approx.). 2O8 ELECTRICITY CH. XIII. (ii.) The current from the n cells in parallel (see 4) will be . p -p Ci = -g -- = - = C (approx.). + r n (iii.) The current from the n cells in series (see 4) will be ^ n E n E ^ 2 ^ ^B~+> = ~V '' = H Hence we see that with arrangement (ii.) we have no advantage over a single cell ; while with arrangement (iii.) we get n times the current that we get with a single cell. (II.) Let r be negligible with respect to J3, and even with respect (i.) With a single cell ...,,, Co=. = (approx.). (ii.) With n cells in parallel E (iii.) With n cells in series ... 17 *? "C* "FT The reader should compare this result with that in (I.), and should try to understand the physical meaning of these results ; it is not enough to follow merely the algebraic reasoning. Experiments. The figure represents diagrammatically three Leclanche cells whose + poles are connected with three mercury cups, +(i), +(2), and +(3), respectively, the poles being connected with the mercury cups (i), (2), and (3), respectively. By means of thick copper pieces we can connect these cells either in parallel or in series. By means of other mercury cups, and of other thick copper pieces, we can throw into the circuit of one or more cells the resistance-box r (see Chapter XIV.), and the galvanometer G. This latter may either be a common tangent galvanometer of no appreciable resistance, suitable for measurements of large currents ; or may be a more delicate instrument by means of which much smaller currents may be measured (see Chapter XVII.). The mercury cups and thick copper pieces give us connections of practically zero resistance. CH. XIIT. OHM'S LAW 2O9 (i. ) Let us put into the circuit a very large external resistance r of (say) l,ooo ohms ; and let us also put in our more sensitive galvanometer, as the current will be small. We can then, by coupling up our three cells in the different ways indicated in (I.) above, verify the results there deduced from Ohm's law. (ii. ) Next we may remove r, and have in the circuit only a tangent gal- vanometer of practically no resistance. The results of (II.) above may thus be verified. 7. Graphic Representation of Ohm's Law. By a graphic method, familiar to those who are acquainted with the elements of co-ordinate geometry, and capable of being understood also by those of less mathematical knowledge, we may represent to the eye in one view (i.) the difference of potential between two points A and B in a circuit ; (ii.) the resistance between A and B ; (iii.) the current which flows through this resistance on account of this difference of potential. Let distances measured along the line O r represent by their length the magni- tude of the resistances. If the wire be uniform, then equal lengths of wire will be represented by equal lengths along O r. If the conductor be not uniform, this will not be the case. A metre of a very thick good conductor may be practically a p 210 ELECTRICITY CH. xni. point on our resistance-line O r, while a millimetre of bad con- ductor may be represented by a considerable length on O r. Let distances along O v, drawn at right angles to O r, repre- sent potentials. If the reader will bear in mind that we are con- cerned only with differences of potential, he will see that it does not matter what potential we choose as our zero-potential, i.e. as the potential of our starting-point or ' origin ' O. We can, in fact, take the point O to represent any fixed point in the circuit that we choose, and can measure off the potentials of other points as so much above or below the potential of this point taken as our arbitrary zero ; this will make no difference in the magnitude of the line representing the A V of two points in the circuit. Thus if A and B represent two points in the circuit, and if lines A P and B Q be drawn perpendicular to O r, and if B M be drawn parallel to O r, then the line A P represents by its magni- tude the potential of the point A above our arbitrary zero, the line B Q that of the point B, and AM the A V between A and B ; while O P and O Q represent the resistances between our starting- point and the points A and B respectively, and P Q represents the resistance between A and B. T? A Now Ohm's law gives us that C = %-* ( see 2 ("*) ) '> wnence J^B C = ~T, if we have drawn our figure to proper scale ; or C = k . A ^, where k is some constant, if we have taken our scale AM of drawing at random. Let us suppose that C = But if < be the angle that the line A B makes with the axis O r^ we have = tan < ; whence we have C = tan <. Thus the ' slope ' of the line A B, as measured by tan <, in- dicates to us the current strength. Of course, since the current C is the same throughout the circuit, the line A B has the same slope whatever points in this circuit A and B may be. If we take in the whole circuit we have ordinate representing total E.M.F. in circuit tan = : -- = r . .- abscissa representing total resistance in circuit CH. XIII. OHM'S LAW 211 This last result, if known, obviously enables us to find the position of the point A in the diagram when we know that of B, and when we know also the resistance Q P between B and A ; just as we can algebraically find the potential of A when we know the potential of B, the resistance between A and B, and the current flowing in the circuit (see n). 8. Applications of the Graphic Method. We can thus represent to the eye each result arrived at algebraic- ally from Ohm's law. We will indicate in a few cases how this may be done. (1) Referring to 6 (I.). To represent case (i.) the reader should mark off from O v a distance O A representing the E.M.F. of one cell \ and from O r a distance O a representing the external resistance r, B being negligible. The current is then represented by tan ty^ where < is the angle which A a makes with O r. To represent case (ii.) we do not (appreciably) alter the resistance O #, and the E.M.F. O A is unaltered. We have, therefore, the same line A a making an angle c^ = < ; so that the current is unaltered. In case (iii.) the resistance O a is practically unaltered, but we mark off on O v a distance O N = n x O A, and we join N a. This line N a makes with O r an angle $ 2 such that tan < 2 = ;z . tan , or the current is n times greater than in case (i.). (2) The reader can easily represent the three cases of (II.). (3) It will be a useful exercise for the reader to draw the rises and falls of potential through a circuit in which there are abrupt changes at different points (due, let us suppose, to contacts of different metals at those points), as well as a general fall following Ohm's law. The method will be as follows. From a knowledge of the total E.M.F. and the total resistance in the circuit we can find C, or can find tan < ; thus we know the slope of the line between points separated by homogeneous conductors. The abrupt changes of potential must be given. It is easy then to trace the slope of, and abrupt rises and falls of, potential from any point in the circuit round to that point again. (4) Note. For ordinary purposes it is sufficient to mark off from O v a length representing the total algebraic sum of the E.M.F.s in the circuit, as if this E.M.F. were an abrupt rise of potential occurring at one point in the circuit and nowhere else. The slope (i.e. tan <>) ofHhe line is not affected by this convenient assumption. 9. Divided Circuits, We will now discuss more fully than in 5 the case of a divided circuit. The notation employed is UNJVER p 2 212 ELECTRICITY CH. XIIT. sufficiently explained by the diagrams and by what has preceded in 5. (Read 5 again.) We will take several cases. I. Two equal branches. Here we have for the equivalent resistance .... ............. . . Hence, if R be the resistance in the rest of the circuit and in the battery together, and if E be the E.M.F. of the battery, we have. ..... . .. E FIG i. Total current C = R' + R And since E B = C } r C 2 r, it follows that C l = C 2 = ; as is otherwise evi- dent. II. N equal branches. Here in a similar manner we can show that . . .. r = C, = Co = C, = &C. = III. Two unequal branches. Here we have or Also, C = And since E B = C^ = C 2 r 2 = CR', we get the results that .......... OHMS LAW 213 r - ' 2 c i - --- . V^ = -- . V^ = . C = L_. . C . . . (iv.) IV. N unequal branches. In this most general case we have ^-L + ^-K .&c. + l .) or, R' = ^ + fj ^3 . . . r n + &C. And by a similar reasoning to above we find that . . . R' + R R' , R/ r \^xO \^ r> 2 R/ Example in divided circuits. 1 Two wires, A D B and A E C B, of uniform section, &c. , and of equal resistance, connect the points A and B. A third wire, AFC, of equal resistance, connects the point A with the middle point C of one wire. When a current flows from A to B, find what fraction of D it passes through the wire AFC.' In the figure we represent the l""^- ^_ 1 arrangement intended, and we have called the resistances of ADB and AFC each I, while those of AEC and of C B are each _. We will first find the equivalent resistance of the paths from A through C to B, and so find the fraction of the current that passes through ADB; and next we will find what fraction of the remainder takes the route AFC. Now the resistance of A D B = I, or -. And the equivalent resistance FIG. iv. 214 ELECTRICITY CH. xm. the R' between A and C is given by -L = - + I = 3 ; whence R' = . Hence total resistance of the route by C is - + = >. Therefore the resistances by the 326 routes D and C are in the ratio of 6 : 5 respectively. Therefore Ji of the current passes by D, and passes from A through C. Again, this will be distributed between the branches A E C and A C in the inverse ratio of their resistances ; that is, in the direct ratio of 2 : i. Hence I of will pass by the route AFC; or the answer required is - 2 . 10. 'Shunts.' The case of two unequal branches has a very important application. It is often necessary (i.) to protect a sen- sitive galvanometer against a strong current by allowing only a part of the current to pass through the galvanometer, and yet (ii.) to measure the current. This is done by connecting the terminals of the galvanometer by a wire of greater or less resistance, and thus leaving a greater or smaller fraction of the current to pass through the galvanometer. If this fraction be known, and its mag- nitude be measured, we can easily calculate the total current. We shall see, however, that the introduction of this 'short cut,' or 'shunt] has the effect of increasing the total current by decreasing the total resistance ; and hence the fraction measured is a certain fraction of the new total current, not of the original current. Let R be the resistance of the battery and rest of the circuit combined; let G be the resistance of the galvanometer which here takes the place of r\ in 9, case (III.) ; let S be the resistance of the shunt which here takes the place of r. 2 ; and let R' be. the equivalent re- sistance of G and S. This R' is of course less than either G or S. Let C be the total current ; C G the current passing through the galvanometer ; and C s the current through the shunt. Let C be the original current passing before the shunt was used. Then we have ................. . . . . ' _E . r E i i i GTS J R' + R ' R' G "*" S " ' We will now proceed to consider several points. CH. xni. OHM'S LAW 215 (i.) If we wish to allow -1th, th, ? th, &c., of the total 10 100 1000 current to pass through the galvanometer, then - must be - G + b 10 (~* C* (~* - 1 , or - 1 respectively ; or S must be , , , &c., re- 100 1000 9 99 999 spectively. In general, if S = . G, then will -thof C pass through n i n the galvanometer. (ii.) If G be negligible as compared with R, then will C = C , whatever shunt we use. This follows from the above formulae, since we there may neglect G, and therefore a fortiori may neglect R 7 , which is less than G. (iii.) If G, and -- G, be so great that R may be neglected, then we findC = /zC ; and as a consequence we should have passing through the galvanometer a current = -lc = . TZ C = C : thus n n exposing the galvanometer to a current as strong as the whole original current. By having in the circuit an adjustable resistance, we can intro- duce a compensating resistance r, and so maintain C equal to C . Note. To keep C = C , we must have the total resistance of circuit the same ; or the compensating resistance r must be such that G + R = -^ 5- + R + r ; or r = G ~ - = -. Thus if S = G, so as togive C G = - . C, we have G + SG + S 11 i n n\ G. In this case C = C ; and C G = -- C , as was desired. n n ii. Fall of Potential through the Circuit. The following view of a circuit, a view indicated by the figure to Chapter XI. 7, is convenient, and, though not accurate in detail, it can for general purposes be substituted for a more accurate view, and can lead to no error in such considerations as follow. In the figure referred to, it is supposed that in the cell the E.M.F. occurs as an abrupt rise in potential ; in fact, that the energy of the cell is spent in, as it were, pumping the electricity straight up from the lower to the higher level. Thus, the line C Z 2l6 ELECTRICITY CH. xm. represents, in the graphic method, the total E.M.F. E of the circuit. The potential falls round the circuit from the higher to the lower level again ; the total fall being E. Ohm's law, as expressed by the formulae given in 2, means that this fall proceeds proportionally to the resistance. Hence, if R' be the resistance of any portion of the circuit, and if R be the total resistance of the whole circuit, then the fall of potential between the beginning and end of R' will be Tr-E. We deduce the following. (a) In the case of negligible external resistance the whole fall of potential will (approx.) take place in the cell itself. We find (approx.) no AV between the beginning and the end of the external circuit ; or the poles of the battery will have (approx.) the same potential. (b) In the case of negligible internal resistance the whole fall of potential will (approx.) take place externally to the cell. We shall find (approx.) between the beginning and the end of the external circuit a AV equal to the whole E.M.F. of the battery ; or the poles, of the battery will have (approx.) the same AV as if the circuit were broken. (c) Where internal resistance equals external, we shall have the fall of potential equally divided. The poles of the battery will exhibit a AV equal to - 1 - E. (d) Where there is a multiple arc between two points A and B, and we wish to find the AV Eg between A and B, we have merely to calculate the equivalent resistance R'. We then have E A = R; K Total resistance ' Note. All the above cases are readily exhibited graphically (see 7). 12. Kirchhoff's Two Laws. By a careful application of Ohm's law, and of the principle that when the V of a point is constant there must be as much electricity flowing away from it as flows to it in each second, it is possible to investigate the distribu- tion of current and potential in very complicated cases; cases where we have any number of cells connected by a net-work of conductors in any way whatever. CH. xin. OHM'S LAW 217 Kirchhoff has enunciated in the form of two ' Laws ' the principles that must guide us in such an investigation. Law I. If any number of conductors meet at a point, and if all currents flowing to the point be considered + , and ail currents flowing from the point be considered , and if the condition of things be steady, or the potential at the point be not altering, then the algebraic sum of the currents meeting at the point must be zero. Or 2 . C = o. Law IL Let us suppose there to be such a net- work of con- ductors as that imagined above ; there being cells of various E.M.F.S, and turned various ways, in this net- work. If we imagine ourselves to start from any point in this net- work and to make a circuit through the conductors back to our starting- point, we shall have passed through conductors of various resist- ances, shall have passed through various cells whose E.M.F.s are directed so as to drive a current against or with us. and shall have found various currents, some with us and some against us. As long as we are passing along a single conductor r }t to which there are no outlets, the current has some fixed value Cj ; but on passing a point where two or more conductors meet we may find a different current C 2 , which will remain constant over the next piece of resistance r< up to the next place where two or more branches meet. We can thus divide our circuit into por- tions of resistances r lt r 2 , r 3 , &c., respectively ; along each of which will be a current C )5 C 2 , C 3 , &c., respectively. If we call these currents + or , according as they flow with us or against us respectively ; and if we call the E.M.F.s that we encounter + or according as they tend to drive a current with us or against us respectively ; then it can be shown, by successive applications of Ohm's law to these different portions of our circuit, that in the complete circuit we must have Ci r \ + C 2 r 2 + C 3 r 3 + . . . = e + e' + e" + . . . where *, e', &c., are the various E.M.F.s that we pass ; attention being paid to signs both of the C . s and of the e . s. These E.M.F.s e, e', e", may occur in any way in the circuit, and do not necessarily (or indeed usually) occur in the pieces of circuit r^ r. ta &c., respectively. 218 ELECTRICITY CH. XIII. This law is usually expressed thus. ' In any complete circuit, . C R = 2 . e. 1 Proof of Kirchhoffs second law. We will here indicate the manner in which Kirchhoff's second law may be proved. The figure repre- sents part of a net-work of conductors, in which are introduced any number of battery-cells, e^ e.,, e y &c. Between any two consecutive junctions, as A and B, the current has some con- stant value CAB '> and between B and the next junction C there will be a con- stant current C BC , which will in general be different from C AB . The total resistance of conductor and cells between A and B is represented by r AB . The symbols Then R -f x = -v- coto 2 <'... . (ii.) K Next remove x, and introduce some measured resistance /, and let the new deflexion be a 3 . Then we have R + / = -?. cot a 8 . . . . . . (iii.) By subtracting (i.) from (ii.) and from (iii.) respectively, and by Q 226 ELECTRICITY CH. XIV. dividing the one result by the other, we finally eliminate both R -p> and ; the final result being k X COt a 2 COt n l / COt U 3 COt a/ which gives us x in terms of known quantities. Both the above methods assume that E is constant. Now E varies with time, rendering method (I.) inaccurate ; and in method (II.) E varies not only with time, but with the strength of the current flowing. The Wheatstone's bridge method, next to be described, is free from the above and other objections. 4. Wheatstone's Bridge, General Principle. Fig. i. repre- sents diagrammatically the principle of Wheatstone's bridge. FIG. i FIG. u. Let the current from A to C divide into two branches ABC and A B' C ; and let the resistance A B be measured by r, B C by r 1 , A B' by s, and B' C by s 1 . Then A is at one potential V A , and C at a lower potential V c ; and along both branches the potential falls from V A to V c propor- tionally with the resistance. If (r + r 1 ) be great as compared with (s + s'\ then the fall through A B C is gradual as compared With the 'steeper' fall through AB' C. But whatever proportion CH. XIY. MEASUREMENT OF RESISTANCES 22^ (r + /) bear to (s + s'), there will be on the branch A B' C points of the same potentials as those of any points on ABC re- spectively. Now if we connect two points B and B' by a * bridge ' B G B' containing a galvanometer G, then according as we see indicated a current from B to B', no current, or a current from B' to B, we shall conclude that B is above B' in potential, B and B' are at the same potential, or B' is above B in potential, respectively. The case of no current through G is the most important ; in this case B and B' must be at the same potential. Now in fig. ii. M P represents the potential V A ; and we have taken V c as our zero potential, so that the point C is repre- sented by O or by O'. The fall of potential down the one circuit is given by the line P Q O, the fall down the other circuit by P Q' O. The points Q and Q' represent B and B' respectively ; when these latter are at the same potential, as proved by the absence of current in G. The rest of the figure is clear after what we have said in Chapter XIII. 7. Then when B and B' are at the same potential, or when QN = Q' N', we have, by Euclid, Book VI r+7' = MP j and T+T? = M F ' r 1 s' and therefore r' s + s' r s or = Hence one of the resistances is determined by the other three. Algebraic proof 'of the same. We will now use a notation that will be readily understood with reference to fig. i. of 4, the notation being similar to that employed in Chapter XIII. 9 and 10. Since there is no current through G, we have C r C,/ ; and C s = C s >. Applying Kirchhoff 's second law to the circuit A B G B' A, we have . . C r r - C s . s = o ; or C r . r = C s . s. Applying the same law to the circuit B C B' G B, and remembering that C r = C r ', and C s = C s >, we have also in like manner ........ Cr . r 1 - C s . s' = o ; or C r . r 1 = C, . /. Whence we readily obtain the result that Q2 228 ELECTRICITY CH. XIV. It is clear that this formula remains true, however the E.M.F. of the battery may alter. In this fact lies the main superiority of this method, over the previous methods, of measuring re- sistance. 5. Slide-form of Wheatstone's Bridge. The figure here given represents roughly the simplest and least expensive form of Wheatstone's bridge. On a board are fixed three bands, A, B, and C, of stout copper ; the pieces A and C being elbow-shaped. Between A and B, and B and C, are gaps ; and here can be inserted, most conveniently by means of mercury cups, any resist- ances we please. The other ends of the pieces A and C are joined by a thin uniform wire of high specific resistance; and parallel to this runs a scale that should have its initial and final points exactly op- posite the points where the wire is soldered to the thick copper pieces. This scale is graduated to read both ways ; and, on the sup- / position that the wire is of uniform resistance, and that the scale corresponds properly to the beginning and end of the wire, we can read off the ratio of the two resistances s and s' into which any point such as B' divides the wire. Over the wire slides a spring key B'; by pressing the button of this key we make contact with the wire by means of a metallic edge or fine wire. The point where contact is thus made should be exactly opposite to the point on the scale at which the little index carried by the key is then pointing. The ' bridge ' B G B' connects the piece B and the key B'; the battery terminals are connected with A and C. Referring to fig. i. of 4, we see that we have complete cor- respondence; only the points A, B, and C of that figure have been replaced by the metal pieces A, B, and C, which are, from a resist- ance point of view, equivalent to points. Further, we see that CH. xiv. MEASUREMENT OF RESISTANCES 22Q the point B' is moveable, so that we can make the ratio s : s 1 anything that we please. In using the instrument we insert our unknown resistance at r, and a resistance box at r'. We then adjust the position of the key B' until the making of contact with the wire at B' gives no current through the galvanometer in the bridge. When this is the case, B and B 7 must be at the potential; and we then have which determines r. Note. There is often some want of exact correspondence between the ends of the scale and the ends of the wire. Such a defect produces less error in the result if we have s and s' not very unequal. We can contrive this by having a resistance box at /, and by removing the plugs (i.e. by throwing in resist- ance) until we get our zero deflexion of G when B' is not far from the centre of the wire ; in other words we make r 1 not very unequal to r. 6. Wheatstone's Bridge; Resistance Box Form. The above described instrument is liable to error. The wire, by much use, loses its uniformity; so that equal lengfhs no longer correspond to equal resistances. Moreover, there is, even in the best instru- ments, some uncertainty as to the exact correspondence between the point of contact B' and the point on the scale against which the index rests. (This second defect can, however, be elimi- nated.) In the instrument used for technical purposes B' is fixed, and s and s' are resistance boxes containing coils of (e.g.) i, 10, 100, and 1,000 ohms. For the known resistance / we have another resistance box. We can thus, by means of the two boxes s and /, measure from I th'to 1,000 times any resistance in the box r' ; TOOO this is simply a matter of removing the suitable plugs. We can thus with certainty measure to th of i ohm, which is better. 1000 than measuring to smaller fractions with uncertainty. 7. Resistance of a Galvanometer. This can be measured as any other resistance ; but a very simple method is as follows. Sir Wm. Thomson's method. Here we put the galvanometer G, whose resistance is required, in the place of one of the resist- ances r ; and, in the place of the bridge with its galvanometer, we 233 ELECTRICITY CH. XIV. have a key K, by means of which we can make or break con- nection between B and B' at will. The galvanometer G will have a deflexion due to the current through A B ; the battery power and resistance in the external circuit A P C must be so adjusted as to give a reason- able deflexion of G. We adjust the resist- ances s and s' until the making of contact with the key K produces no alter- ation in the deflexion of G. When this is the case, it can be proved that no current then flows through the bridge ; and that as before, ....... G = ^ . r\ where G is the galvanometer resistance. 8. Resistance of a Battery-Cell. The measurement of the resistance of a battery-cell is not at all a simple matter. For whereas in the above measurements we had only one con- stant unknown, viz. the constant resistance x that we desired to measure, in the case of the battery-cell we have two unknowns, one of which is variable from moment to moment ; we have, in fact, the resistance x, and also the E.M.F. y, which, owing to vary- ing degrees of polarisation, is not constant. We here give two methods for measuring x. I. The E.M.F. reduced to zero. If we have any even number 2 m of similar cells, and if we set m against m, the total E.M.F. will be zero. When the current, employed in measuring the. resistance, passes through these 2 m cells, the polarisation pro- duced should still be zero in total amount. With cells thus matched m against m, we can measure the resistance of the whole 2 m cells as a * dead ' resistance ; that of one cell is found by dividing by 2 m. Against this method it may be stated that, as a rule, we require the resistance of the cell or battery as it is at any given time ; and CH. XIT. MEASUREMENT OF RESISTANCES 231 we cannot generally provide another cell or battery, equal to this in E.M.F.. to oppose to it. We may call this method the ' Method of opposition? II. Mance's method. Let us arrange a Wheatstone's bridge as in 5, but with the cell or battery, whose resistance is to be de- termined, in the place of the resistance r, and with a simple contact key in the place of the usual battery. The cell at r will drive a current ; and no arrange- ment of resistances can pre- vent there being a current through the bridge and the galvanometer G. Hence, G will be deflected. It can be shown that if the resistances be so adjusted that G maintains the same deflexion whether the key K be opened or closed, then we have the old relation V The proof is not very simple. By application of KirchhofTs laws we obtain equations by means of which the unknowns can be determined. The current through G will be found to involve the external resistance of A K C in such a way that it is only independent of it when = . Whence it follows that, if this current is the r s' same whether the key is open (giving an infinite resistance) or closed (giving a finite resistance), this relation must hold. (For further discussion of this method see * Phil. Mag.,' Series V., Vol. 3, 1877, Professor O. J. Lodge ; and also in Glazebrook and Shaw's 'Practical Physics.') 9. Measurement of E.M.F. Let us take our standard cell, whose E.M.F. we suppose (for simplicity) to be just i volt, and let us measure the AV between its terminals when the circuit is 232 ELECTRICITY , en. xiv. open (see Chapter X. 34). Let us measure in the same absolute potential measure the AV between the terminals of any other cell or battery, the circuit being open. Using the AV of the standard cell as unit, we thiis can express the AV of the latter in volts. We thus get a clear idea of what is the initial E.M.F. of any battery, expressed in volts. But when the circuit is closed, polarisation sets in ; and we find that the same cell has a different E.M. F. for each strength of current passing through it. 'The E.M.F. of a cell' is, therefore, a term whose meaning is undefined unless it be stated what current is running. As a rule, therefore, the initial E.M.F., or the AV between the poles of the open cell, is given ; and it is then stated with more or less exactness to what extent this falls off with currents of various magnitude. The E.M.F. when the current is zero, or the statical AV (ex- pressed in volts) between the poles of the open circuit, is therefore what we record as ' the E.M.F. of a cell.' To find the E.M.F. with any given current is a very difficult matter. For, if we try to determine the E.M.F. and the internal resistance of a battery by means of two equations derived from Ohm's law, we must use two different currents ; and in this case neither the E.M.F. nor the resistance of the battery will be the same in the two equations. 10. Electrometer Methods; Open Circuit. (i.) If we have an absolute quadrant electrometer we can measure the E.M.F. of the open cell by measuring the AV between its poles. Dividing this absolute value by that obtained for i volt, we express the E.M.F. of the open circuit in volts. (li.) If we have an ungraduated electrometer we may proceed as follows. We take m of our cells, and n standard cells, arranged end-on. We put one terminal (say the terminal) of each battery to earth ; and connect the -f terminals of each battery with one pair of quadrants respectively. We arrange m and n until either we have no deflexion, or until with n standard cells we get the deflexion of the electrometer needle one way, and with n + i cells we get the deflexion the other way. Then, if e be the E.M.F. of the cell required, and if e be that of the standard cell, we have ".*.. CH. xiv. MEASUREMENT OF RESISTANCES 233 (either e = e , m or e lies between e and - e ; thus obtaining a measure- ment to within th of e . m u. Volt-meter Galvanometers. Let us suppose that we have two points A and B in a circuit ; and that it is required to measure the AV in volts between them, or the E.M.F. E B . We have above indicated the electrostatic method of doing this. A current method is also possible. For if we have a galvanometer G of a very high resistance as compared with the resistance between A and B, the current that passes through G when its terminals are connected with A and B will be, by Ohm's law, directly propor- tional to E B ; and at the same time, owing to the relatively very high resistance of G, the A V E^ will not be appreciably altered by connecting the points through G. Thus (i.) If we connect the terminals of an open cell through G, we do not appreciably alter the E.M.F. of the open cell, the resistance of G being relatively so great (see Chapter XIII. n ()). (ii.) If we connect G with two points in the circuit, we do not, by introducing a branch circuit of so high a resistance, practically alter the A V between A and B. Hence, if we have a galvanometer of a high known resist- ance G, and which measures currents accurately, we can from it deduce the AV between the points A and B with which its TT A terminals are in contact, by means of the equation C = -* ; and G this will be practically the same AV as existed before we made connection with the galvanometer. Sir W. Thomson's graded volt-meter galvanometer is such an instrument. In Chapter XVII. we shall give some detailed description of it, and shall indicate how it can be used to measure both large and small E.M.F.s. Here we shall say little more about it From the formula C = L the instrument can be experimen G \ 234 ELECTRTCITY CH. XIV. ally or theoretically graduated, so that we can read off at once in volts the A V E between the terminals A and B of the instru- B ment, answering to any particular current C. 12. Method of Opposition. Here, as in 10, we have zero current ; but we use a galvanometer instead of an electrometer. If m of the cells, whose E.M.F. we require, be opposed to n standard cells, and if /;/ and n be so adjusted as to give if possible zero-deflexion of a galvanometer in the circuit, then, as in 10, we can measure to within th of the E.M.F. of a single standard m cell 13. Latimer-Clark's Potentiometer. In this instrument the method of opposition is employed. P is a battery ; R is a Wheatstone's rheostat ; A B is a uniform graduated wire of high resistance, fixed between two terminals A and B ; e is a standard cell ; e is a cell of a smaller E.M.F. that we desire to measure ; G and G' are delicate galvanometers to indicate when there are zero currents. Contact keys are not repre- sented. Note. As an example we may state that in a certain instrument the wire A B has a resistance of about 53 ohms. It is about 53 metres long, and is coiled in a screw-groove cut round an ebonite cylinder. Now the total fall of V due to the battery P will be distributed between the portions A R B and A Q B of the circuit, according to their respective resistances. Hence, by manipulating R, we can arrange the AV be- tween A and B until it equals the E.M.F. e of the standard cell ; this will be indicated by the fact that when E* = ? , then no current will pass through G, since the two equal and opposite E.M.F.s are opposed. Since there is no current through this cell, and so no polarisation, therefore e remains constant, and therefore E is also constant. As the CH. XIT. MEASUREMENT OF RESISTANCES 235 E.M.F. of P gradually alters by polarisation, we can so diminish the resistance of R as to maintain G undeflected, or E B constant All the above may be regarded as a contrivance for keeping the two points A and B at a constant AV equal to the E.M.F. e of the standard cell. And the uniform wire A B gives us by its graduations any fractional parts of e that we desire ; since, by Ohm's law, the AV, e m falls uniformly from A to B. Now, if we arrange the cell e as shown, and if we get zero- deflexion of G' when we make contact at Q, where A Q = a and B Q = b, it follows that then e = E Q , or that since evidently E A = . E A . * Q AB B Thus we can measure e accurately when less than e . Note. It may make matters clearer if we point but that the positive poles of the battery P, and of the cells e and e, are all connected with the same end A of the wire A B ; and their negative poles, are all connected with the other end B. If e be greater than e we have merely to interchange the posi- tions of e and e, and the same experimental method gives us that a 4- b e = e . a 14. Table of Resistances in Ohms (or w), We shall now give some information as to the actual resistances of various sub- stances ; and, in order that the statements may be scientifically exact, we shall explain certain terms that will be used. Microhms. A microhm is the T FOWTTo tn of an ohm ; or i microhm = wX io~ 6 . Megohms. A megohm = 1,000,000 times an ohm ; or i megohm Copper, annealed ..... 1*616 ,, Copper, hard drawn . . . . 1*642 ,, Gold, hard drawn .... . . 2*154 ,, Aluminium, annealed . . . . 2*946 ,, Zinc, pressed . . . . . 5*690 ,, . Platinum, annealed . . . . .9*158 ,, Iron, soft . . . . . . 9*827 ,, Tin, pressed . . . . . 13*360 ,, Lead, pressed ...... 19*847 ,, Mercury, liquid . . . .- . 96*146) to 99740) German silver. * .21*170 ,, Brass , \ - _--- . 5 '800 ,, For further facts as to resistances, such as resistances of different frnasses of wire of known gauge, we refer the reader to Hospitalier's 'Formulaire Pratique de l'Electricien'(Masson, Paris), and to Lupton's ' Numerical Tables and Constants ' (Macmillan). II. Specific resistances of liquids in ohms. (Taken from Lupton's book.) /Water at -75 C. . . '."'. . .. . ... i*i88x io s (a*) 4 Water at 4 C. . . ' * .'*<.'.- 9*100 x 10" (a) (Water at 11 C. .'...:.. . . 3*400 x io 5 (w) (Dilute hydrogen sulphate at 18 C. ; 5 per cent, acid . ,4*88 () I Dilute hydrogen sulphate at 18 C. ; 30 per cent, acid . .1*38 (w) Hydrogen nitrate at 1 8 C., density 1*32 1*61 () Saturated solution of copper sulphate at i o C. .... 29*3 (w) Saturated solution of zincic sulphate at 14 C. . . . . 21*5 (o>) Hydrogen chloride, 20 per cent, acid, at 18 C 1*34 () Ammonium chloride, 25 per cent, salt . . . . . . 2*53 (o>) Sodium chloride saturated, at 13 C. . . . . . . 5*30 (w) The above show the very great difference that exists between pure water and solutions of salts, with respect to the conduction of a current. CH. xiv. MEASUREMENT OF RESISTANCES 237 III. Specific resistances of bad conductors in megohms. Ice at - 12-4 C 2240 ( x io fi ). Glass; (soda-lime, density 2-54) at 20 C. . 9-1 x io 7 ,, (Glass ; (crystal, density 2-94) below o C. . practically infinite. (Glass; (crystal, density 2 -94) at 105 C. . I'i6 x io 7 (w x IO H ). Shellac; at 20 C 9-00 x io 9 ,, Paraffin; at 46 C. . . . . . 3-4 x io 10 ,, Ebonite ; at 46 C 2-8 x ro 10 Air ; usual pressure . . . . practically infinite. True vacuum ...... practically infinite. Very dense gases, it would seem, do conduct (or ' convect ') to some extent. In very rare gases, or ' imperfect vacua,' there appears to be con- duction ; but it seems certain that in this latter case, at all events, the discharge is disruptive, and that there is no true conduction. 15. Table of E.M.F.s in Volts. We here give some results as to the E.M.F.s of certain cells. The reader must, however, bear in mind that the E.M.F. of a cell depends (i.) upon its general make ; (ii.) upon details which are different in different cells pro- fessedly of the same E.M.F. ; (iii.) upon the current. The E.M.F.s here are therefore only approximately correct for cells in general. They are the initial E.M.F.s ; and so we have some anomalous results, as, e.g., that the E.M.F. of Volta's cell is greater than that of a Smee's a result certainly not true when there is much current running. /Volta's cell ; Zn \ dilute If. 1 SO^ \ Cu ; about i-oo volt. Smee's ; Zn \ diL H.,SO 4 \ platinised silver ; from 0-50 to I'OO volt. Darnell's; Zn \ dil. H.SO^ \ CztSO 4 solution \ Cu\ about 1-12 volt. Grove; Zn \ dil. H. i S^^ \ HNO 3 \ Pt ; nearly 2-00 volt. Bunsen ; Zn \ dil. H.,SO^ \ HN0 3 \ C ; 175 to I -94 volt. Clark's standard cell ; ~Hg \ Hg,.S0 4 \ Zn ; at I5C. 1-438 Leclanche ; Zn \ NH^Cl sohition and C\ ' ' I<42 " De la Rue ; Zn \ NH.Cl \ AgCl \ Ag . . . 1-04 ,, Marie-Davy ; Zn \ dil. ff,SO 4 \ HgSO \ C . . 1-52 ,, Bichromate ; initially ; solution new . . . 2-00 ,, Plante secondary cell , , . about i'8o to 2-50 ,, These results are for the most part those given in Lupton's book above referred to. 238 ELECTRICITY CH. xv. CHAPTER XV. JOULE'S LAW, AND CONSERVATION OF ENERGY. i. General Survey. We have already pointed out how (i.) all forms of energy may be measured in work-units (i.e. in * r &Y> ; . (ii.) energy is indestructible, or in any self-contained system we have transformation of energy from one form to another, but no loss or gain of energy. In addition to these statements we may now add the law, which seems to apply to all inanimate nature at any rate, 'that all energy tends to run down to the form of .uniformly diffused heat.' This law is called the law of ' Degradation of energy? Now in the battery-cell our store of energy is in that form which is called chemical potential -energy. In any particular cell the total energy at our disposal is proportional to the total mass of zinc (or other metal) to be dissolved. This energy also depends upon the other chemical changes that take place ; and thus it is that in different cells, in which the rest of the chemical action is different, we have different amounts of energy answering to the same mass of zinc dissolved. This energy may pass (apparently) direct into the form of heat energy, as when we simply dissolve unamalgamated zinc in acid. Or we may obtain electrical energy as an intermediate form ; and part of this electrical energy may be converted again into chemical-potential-energy if it be caused to decompose an elec- trolyte, or it may be transformed into mechanical energy if it be caused to work an electro-motor (jw Chapter XXIV.), or it maybe allowed to be entirely converted into heat-energy if it be .left simply to heat the circuit. But in every case we have simply the energy answering to the CH. xv. JOULE'S LAW 239 nature of, and amount of, the chemical changes that have taken place in our sole source of energy, the cell or battery. 2. Units of Heat, Work, and Activity. I. Unit of work. In the absolute system this is the erg, already discussed in Chapter X. II. Unit of heat. The calorimetric unit of heat is that amount ef heat which will raise one gramme of water from o C. to i C. ; or we may with sufficient accuracy measure the number of calorimetric units of heat by the product of the (number of grammes of water] into the (number of degrees Centigrade through which they rise or fait in temperature]. This unit is called the calorie. The calorie is not a C.G.S. unit, as the Centigrade scale of temperature is independent of the C.G.S. fundamental units. We have, therefore, to connect this unit with the C.G.S. system by direct experiment, measuring the heat, as we can measure all forms of energy, in ergs. Now it has been shown experimentally that i calorie 4*175 x /o 7 ergs = 42 x io 6 ergs [nearly], or that one gramme-degree unit of heat would, if totally used up in doing mechanical work, do 41,750.000 ergs of work ; or the same would raise 424 grammes through one metre against gravity (where g is taken as 981). It was Joule who first established the definite measurement of heat in units of work. The factor which reduces calories to ergs will be called J, so that J = 4*175 x io 7 . We may, for ordinary purposes, take J to be equal to 42 x IO G . Note. We use the true calorie, the gramme-degree-Centigrade unit. Another calorie is sometimes used, viz. the kilogramme-degree-Centigrade calorie ; but its use will probably soon be entirely discontinued. III. Unity of activity, or rate of work, or power. It is often very important to consider the rate at which work is done, or the work per second. It is convenient to give the name of 'activity ' to rate of work. The C.G.S. unit of activity is the activity of one erg per second. (In English engineering practice the horse-power is much used as a convenient unit of activity when work is being done on a large scale.) All energy, then, can be measured in *rgs or units of work, and all energy per second in units of activity. 240 ELECTRICITY CH. xv. v 3. Energy of the Electric Current. Consider a current C running steadily between the points A and B of a circuit, where A is at the potential V A , and B is at some lower potential V B . Each unit of electricity at A is at a higher potential than when it comes to B ; in fact, there is a continual discharge from the constant potential V A to the / ^ n X- > *"*o! constant potential V B . Each / \ unit, therefore, loses potential / \ energy to the amount repre- sented by V A V B (see Chap- ter X. 28) in its passage from A to B ; and, if the current be C, or if C units pass per second, there will be a loss of electrical- potential-energy between A and B to the amount of C x (V A V B ) per second. We may write this as C x E A . If C and V be B measured in suitably chosen units, the above product will give us the electrical energy lost in ergs per second, or in absolute C.G.S. units of activity. Now the absolute electro-magnetic system of units, referred to in Chapter XIV. i, and explained in Chapter XVIII. , is such that (current] x (E.M.F.) does give us this activity in absolute C.G.S. units. If, however, we use amperes to measure C, and volts to measure E M.F. or ^V, then the product C x E gives us the activity in a new unit which we call a watt. Since i ampere = T ^th absolute unit, and i volt io 8 absolute unit, it follows that - r"-5 i watt = io 7 (ergs per second}. To repeat ; when a current of C amperes runs between two points whose difference of potential is measured by E volts, then electrical energy disappears between these points at the rate of. .... Cx^ watts (i.) or C x E x i o 7 ergs -per- second. We may add that One English horse-power = 746 watts [nearly]. If there be no source of E.M.F. between the points A and B, A then by Ohm's law we have C = . 1, where R is the resistance between A and B. In this case, which is the case where A B is a CH. XV. JOULE'S LAW 241 simple conductor, and includes no form of pile or electrolytic-cell or thermo-cell, we have that Activity = C 2 R watts (ii.) But if we simply know E A as the AV between A and B, and if B we do not know whether or no there be any source of E.M.F. be- tween A and B, then we cannot write C R instead of E B . For if there be such an E.M.F., e, between A and B, then by Kirchhoffs extension of Ohm's law, we have that C R does not equal E B) but equals (E B db e). Hence the expression (i.) given above is of the more general application. In the case of a battery of E.M.F. E, through which is passing a current C, the above considerations will show us that the rate at which chemical-potential-energy is used up in giving electrical energy must be represented by an activity of C E, since electricity to the amount of C per second is being raised through a AV represented by E. In fact, the rate at which chemical energy is used up in ' pumping up ' electricity in the cell from the lower level to the higher (the difference of level being E volts) is repre- sented by the activity of CE watts. Inside the cell, then, we are using up chemical or other energy in raising electricity from a lower to a higher level, i.e. in gaining electrical energy ; while partly inside and partly outside the cell this efectrical energy is being again lost giving us, as we shall see later on, other forms of energy. 4. Joule's Law. The same Joule who first clearly showed that heat was a form of energy, and could therefore be measured in work-units, was also the first to point out the exact connec- tion between the electrical energy lost in any portion of a circuit and the heat evolved in that portion of the circuit. He showed experimentally that if a current C be flowing through a conductor of resistance R, then the heat evolved per second is proportional to the product C 2 R ; R 242 ELECTRICITY CH. xv. and further investigation has shown the exact relation that the amount of heat evolved bears to the electrical energy lost in this portion of the circuit. The nature of his experiments is indicated by the accompany- ing figure. A wire of known resistance is immersed in a known mass of water, a current measured by a galvanometer is passed through the wire for a measured time, and the rise in temperature of the water is noted. We thus measure C, R, and the heat evolved. From a comparison of many experiments in which C and R were varied, the law above given was deduced. Further experiment showed that the same law held with respect to the battery-cell itself. It is then true that in the whole circuit or in any part of it, the ; . (Heat evolved per second} is proportional to (the product C 2 R). If we express our heat, current, and resistance, in suitable units, then this proportionality becomes equality, and we have . . (H per second) = C 2 R. We shall now for convenience introduce a new symbol IS having a very simple meaning. Let IS be the algebraic sum of the E.M.F.s in any portion of the circuit considered ; so that, by Ohm's law or by KirchhofFs extension of it, we have the relation C = ' R or C R = IE.- (Thus if we consider the portion of the circuit A B, and if no source of E.M.F. lie between A and B, we have } the same as Eg or (V A V B ) ; but if there be a direct E.M.F. e, or an opposed E.M.F. e, between A and B, then IB = EB e). We can then write C IS instead of C 2 R ; and we have, when suitable units are employed .....' H (per second) = C 2 R = C IE. If C, R, and IS be measured in amperes, ohms, and volts, then the product C 2 R or C 2 gives us the heat-activity in watts. If watts and calories per second be compared, by referring both to ergs per second, it can be shown by the data given earlier that . i watt = '24 calories per second (approx.). This relation may be verified directly by experiment, CH. xv. JOULE'S LAW 243 We may then express Joule's law as follows. If in any circuit or portion of a circuit C be the current in amperes, R the resistance in ohms, and 15 be the algebraic sum of the E.M.F.s (in such a sense that by Ohm's law C R = !S) measured in volts, then there will be evolved in that circuit or portion of circuit heat to the amount of C-R x -24, or C3&X '24, calories per second. We have thus two formulae for the heat evolved in calories per second ; the formula C 2 R x '24 is always true ; and C IB X '24 is 3S true if IS be such that C =^ . 5. The Heating of Uniform Wires. The most important practical application of Joule's law is the application to the case of conducting wires. We will use formula (i.) of the last section. (I.) If a wire have resistance R, and there be flowing through it a current C, then there is evolved C 2 R x '24 calories per second. With a given current we can thus calculate the calories per second evolved if we know the length, section, and specific resistance p, of the wire. For, if p be given in ohms, or be reduced to ohms, we have that " . R = p : ohms. cross section If r be the radius of a wire of circular section (the most usual make of wire) then / _.; R = p . _. ohms. TT r- (II.) The temperature to which a wire is raised. If the heat were not dissipated by convection, conduction, and radiation, no limit (saving that imposed by the fusion of the wire) could be fixed to the temperature to which a wire would rise. As a matter of fact, however, there soon obtains a state of equilibrium between the heat evolved per second and the heat dissipated per second. Hence the temperature will be, with more or less exactness, proportional to C 2 and to R ; or is higher as the current is larger, and as the wire offers a greater resistance. Experiment. Put in the same circuit lengths of Pt and of Ag wire of the same radius. The platinum is the worse conductor of the two ; and therefore R 2 244 ELECTRICITY CH. xv. .for it the resistance is greater, while the current is the same for both. It there- fore is seen to attain a white-heat, while the silver remains much cooler ; in the final condition, the greater dissipation due to the higher temperature com- pensates for the greater amount of heat evolved per second. Note. If in the same circuit we have (were it possible to find such) two wires of same radius, same conductivity, same surface power of dissipating heat, but of different masses and specific heats, then it is sometimes stated that the temperatures will depend upon the masses and on the specific heats, varying inversely as both. This, however, is not the case. That wire whose water- equivalent (or mass x specific -heat] is the least, will first attain the final tempera- ture. But, when this is once attained, only the resistance and the dissipating- power come into the question ; and, sines by hypothesis this is the same for both, it follows that the final temperatures will be the same. III. Temperature as dependent on radius. Let us now consider wires of different radii, but of the same material, included in the same circuit, so that the same current necessarily passes through each. And let represent the number of degrees by which the temperature of the wire exceeds the so-called ' temperature of the air,' when a steady condition of temperature has been arrived at. We have then, for each unit length of the wire t Heat produced by current per~\ _ VHeat lost by radiation, &>c., per~\ second J L second J If we assume [but it is not at all accurate to do so] that the resistance of the wire does not depend upon its temperature, we may under the conditions assumed say that . {Heat produced per second} oc _L where r is the radius of the wire. Again, if we assume [but at such high temperatures this is not accurate] that the rate of loss of heat is simply proportional to 6 and to the area of surface exposed, we have (Heat lost per second] oc 2 TT r . 6. With these assumptions, then, we have that for wires of same material in the same circuit (The rise in temperature, &} oc - _ _ . 6. Distribution of Heat in the Circuit, Let the current in a circuit be C ; the resistances of battery ^and of various parts of the CH. xv. JOULE'S LAW 245 circuit be B, R 1} R 2 , R 3 , &c., respectively ; the total resistance be R ; the total heat evolved per second be H calories ; the several portions evolved in the above different portions of the circuit be H B , HJ, H 2 , &c., respectively ; then we evidently have the follow- ing relations holding. ( H = C 2 R x -24 = C 2 (B + Rj + R 2 + &c.) x -24. J H B = C 2 B x -24. [Hi = C 2 R! x -24. &c. Whence also HB ' Hj : H 2 : . . . = B : HB = |.H. ; ! H, =|l H. &c. 7. Heat Evolved with various Arrangements of n Cells, Let there be n cells, each of resistance B and E.M.F. E. Let R be the total resistance of the whole circuit including the battery. Let H be the total heat per second evolved ; and let H } and H e be the portions evolved internally to the battery, and in the external circuit, respectively. () Let the cells be arranged ;;/ end-on and / in parallel ; and let the external resistance be r. Then we have H = C 2 R x "24 calories. We have also C R = m E by Ohm's law ; and R = r + "~ (see Chapter XIII. 13, &c.). We can thus substitute for C and R in terms of known quantities ; and we have . H = C 2 R x -24 = x '24 calories-per-second. r+m i m B ^ H = &c., calories-per-second. V ^ H = &c., calories-per-second. 246 ELECTRICITY CH. XV. (/?) If the cells be so arranged that the internal resistance v ~ = r, then we have H = C 2 R x -24 = n ~ x -24 calories. 2 r [, = H e = H = I H. 2 r 2 8. Case of no Back-E.M.F. in the Circuit. - Where there is in the circuit only the E.M.F. E of the battery, and no other E.M.F., then by Joule's law : t \ ; H = C 2 R = C E x "24 calories-per-second. .or H = CE watts. And by 3 we know that the total electric activity is W = C E waits. In this case, therefore, all the electric activity runs down into the form of heat-per-second. Experiment and theory, moreover, concur in establishing it to be a fact that in a cell where there is no local action, i.e. in which no chemical action occurs until the circuit is closed, all the chemical- potential-energy lost per second appears as electrical activity. Hence in this case the total chemical action gives first the equiva- lent electrical energy, and then, finally, the same amount of heat that would have been given had the chemical action taken place without the intermediency of a current. 9. Case of a Back-E.M.F. e in the Circuit. Now let us have in the circuit an electrolytic cell, or some other arrangement giving a reverse E.M.F. -e the total resistance being still R. As argued in 3, the battery is expending energy at the rate of W = C x E watts. By Joule's law we have heat evolved at the rate of H = C 2 R watts. And since by Ohm's law C = g-~j this heat activity H may R be written as C (E e) or C E - C e. Hence of the activity C E expended by the battery, we have accounted for part in heat ; but we have not yet accounted for the remainder C e. CH. xv. JOULE'S LAW 247 ' Conservation of energy ' alone would, therefore, lead us to conclude that activity measured by C e watts is being expended on the electrolytic cell. And, in complete accordance with this result, we have the argument that in order to drive a current C against an E.M.F. e we must expend an activity of C e watts. In an electrolytic cell it is fairly evident that this activity can only be expended in the storing up of chemical-potentiai-energy, since no other work is being done. Then we have expended by the battery an activity of C 2 R or C E watts ; this amount of chemical-potential-energy being lost each second in the chemical changes there taking place. Of this C (E e) watts reappear as heat evolved per second ; while C e watts are stored up each second in the electrolytic cell in the form of chemical-potential-energy, chemical decomposition being there effected. Case of charging a secondary battery. In the charging of a secondary battery whose back-E.M.F. during charging is e, we store up the activity C e, and waste in heat the activity C (E - e). Hence it is more economical to charge the cell with a battery whose E.M.F. E is only a little greater than e ; for then we store up nearly the whole activity. But if E nearly equals e, the current C will be small, and the process of charging will be a long one. As a rule we compromise matters, and do waste a good deal of activity in heat in order to get the cell charged within a reason- able time. 10. Numerical Examples. (i.) A battery has E.M.F. = 50 volts ; the total resistance is 20 ohms. Find the current in amperes, and the activity (or work per second) in ergs per second, in watts, and in horse-power. Here, C = = ^ = 2 1. amperes. R 20 2 Activity = CE = lx5o=i25 watts. 2 Activity = CExio 7 = i25xio 7 ergs per second. Activity - C E -f- 746 = -?| horse-power. 746 248 ELECTRICITY CH. xv. (ii.) In the same case the external resistance R e is 15 ohms, and the internal resistance R ; is 5 ohms. Find the external and internal activities both in watts and calories per second. Here, 'External activity = x 125 =-? x 125 watts. K 4 External activity = 3 - x 125 x -24 calories-per-second. 4 Internal activity =- ixi25 = -xi25 watts. K 4 \Internal activity = - 1 . x 125 x -24 calories-per-second. 4 (iii.) A battery has E.M.F. 20 z/0//.y and current 10 amperes. Find the total heat per second in calories. Here H = C x E x -24 -= 10 x 20 x -24 calories-per-second. (iv.) A battery has E.M.F. = 10 volts ; current is observed to be 6 amperes ; and in the circuit are a set of electrolytic cells of total back-E.M.F. equal to 4 volts. What is the chemical activity (or ejiergy-per- second) stored up in the electrolytic cells ? And how many calories per second are given out? Here, - J Chemical activity stored up = ^=4x6 = 24 watts. {Heat activity = C (E ^) =6x6 = 36 watts. ii. Failure of a Smee's Cell to Decompose Water. If the back-E.M.F. e of an electrolytic cell would be greater than the E.M.F. E of the battery, then such a battery will fail to drive a current through, and decompose, such a cell. For if the action be supposed to take place, we get . * : . . (i.) C = - = a negative quantity; which would mean that _K the electrolytic cell would be driving a reverse current through the battery-cell ; and that the latter would not be decomposing the former. (ii.) Again, we should have the activity C E expended by the battery less than the activity C e expended on the electrolytic cell, which would be a breach of conservation of energy. It is for this reason that a Smee's cell cannot decompose water. CH. xv. JOULE'S LAW 249 If E = (a) The nature of the metals. () The difference of temperature of the two junctions. (c) The absolute temperature of the junctions. Thus we have (a) a different E.M.F. if we make our cell (cceteris paribus] of Bi and Fe from what we have if we make it of Bi and Sb. Again, we have in general (b) a different E.M.F. if the differ- ence of temperatures of the junctions be 20 C., from what we have if (cateris paribus) it be 50 C. And, finally, we have in general (c) a different E.M.F. if the junctions be at, e.g., 20 C. and 22C. from what we have if (cceteris paribus] they be at 50 C. and 52 C. Thermo-electric series at 20 C. Hence we cannot give a list of metals showing their thermo-electric relation to one another for all temperatures, but only for some stated temperature. In the following are given in micro-volts the E.M.F.s of various cells, one metal in each case being lead, and the other the metal opposite to which the number stands. The junctions in each case are at 20^ C. and 19^ C. respectively ; or there is a temperature-difference of i C., while the mean temperature is 20 C. For the metals that occur higher in the list than lead, the current passes from that metal through the hotter junction to lead ; for those lower in the list than lead, the current passes from lead to that metal through the hotter junction. This is indicated by the sign of the E M.F. Experiment shows also that the E.M.F. of a cell composed of any two metals is simply the algebraic difference of their respective E.M.F.s with reference to lead. (The list is taken from Lupton's Numerical Tables.) E.M.F.s, in micro-volts, of cells whose junctions are at 20^ C. and K)\ C. respectively ; lead being one of the metals in each case. Bismuth, pressed ; comm. + 97-0 Antimony, pressed . ' y - 2-8 Bismuth, pressed ; pure . 89-0 Silver, pure; hard. . * -3-0 Cobalt . . . . 22 -O Zinc, pressed ; pure . - 37 German silver " . H75 Copper, electrolytic ; . -3-8 Mercury . 418 Antimony, pressed ; Lead . - , . - . O'O Commercial ... ^ - 6-0 Tin -I Iron wire, soft . .. - I7-5 Copper, comm. . -I Antimony crystal, axial . -22-6 Platinum - -9 Antimony crystal, equatorial - 26-4 Gold . - 1-2 Selenium . * . - 807-0 UNIVERSITY OF CALIFORNIA 256 ELECTRICITY CH. xvi. Explanation of table, The following two or three cases may explain further what the table means. (i.) If we make a cell of cobalt and lead, the junctions being at 20^ C. and 19^ C.,the E.M.F. will be 22 x io- b volts ; and the cur- rent will pass from cobalt to lead through the hotter junction. (ii.) With lead and silver the E.M.F. is 3 x io~ 6 volts ; and the current passes from lead to silver through the hotter junction. (iii.) With pure bismuth and cobalt \ht E.M.F. is (89 22) x io~ 6 = 67 x 10 ~ 6 volts ; and the current passes from former to latter through the hotter junction. (iv.) With cobalt and pure silver the E.M.F. is [22 - ( - 3)] x io~ 6 = 25 x io- 6 volts ; and the current passes from former to latter through the hotter junction. Such a series is called a * thermo-electric series at the tempera- ture (i.e. 20 C.) in question.' 5. Thermo -Electric Powers, The effectiveness of any par- ticular cell (i.e. a cell composed of any two specified metals), at any specified mean temperature f C., may be reasonably measured by the E.M.F. of the cell when the junctions have a temperature difference of i, the mean temperature being t C. To express this ' effectiveness ' we use the term thermo-electric power. The meaning of this term may be seen from the following statement. The thermo-electric power of any pair of metals at t C. is measured by the E.M.F. of a cell composed of these metals when there is a temperature difference of i C. between the junctions, the mean temperature being t C. The E.M.F.s are usually expressed in micro-volts. The more exact meaning of thermo-electric power may perhaps be further explained with some advantage. It is found experi- mentally that if the junctions are at any two temperatures / 2 and /! of which the mean is *, and if E^ be the E.M.F. of the cell under these conditions, then .-, E' J The thermo-electric power at t C. = ^ where t = CH. xvi. THERMO-ELECTRICITY 257 Example. Thus, if 30 C. were taken as / C. , we have the thermo-electric < < E S : power at 30 C. given by any such fraction as ^-, , , &c. Or we may say that if we know the thermo-electric power at t C. in micro-volts, then we get the E.M.F. in micro-volts when the junctions are at temperatures (/ + 0) C. and (t 6) C. by multiplying the thermo-electric power at / C. by the temperature difference 2 0. Note.\ide be the infinitesimal E.M.F. due to the infinitesimal difference dQ in temperature of the junctions, we may in infinitesimal notation say that . The thermo-electric power at t C. = f If the thermo-electric power of a single metal be spoken of, it is understood that lead is the other metal. A metal A is said to have a greater thermo electric power than a metal B at t C. when it stands higher in the list for /C.; in this case the current passes from A to B through the hotter junction. 6. The Neutral Point It is found experimentally that for every pair of metals there is a particular temperature possessing properties to be described in the present section. This tempera- ture is, for reasons to be given later on, called ' the neutral point ' for that pair of metals, and is usually designated by the letter T. This temperature, T C., is different for each pair of metals. Sup- posing that we are dealing with two metals A and B, whose neutral point is T, the facts observed are as follows. (I.) If one junction, say A | B, be at T while the other junc- tion be at some higher temperature t^ there will be a certain E.M.F. which we may call E^, and the current will pass from one metal to the other (say from A to B) through the hotter junction, i.e. through that at /,. We should for these temperatures say that A is above B in the thermo-electric series. The higher /j be- comes, the greater is the E.M.F. (II.) If the temperature of the other junction be lowered from T to some lower temperature / 2 > step by step, the E.M.F. (which we may now call E ') decreases step by step ; and when t^ and t. 2 s ELECTRICITY CH. XVI. are equidistant from T, or when /, T = T / 2 , the E.M.R is zero. (III.) If the temperature of the junction that was at T be lowered still further, so that T / 2 is greater than ^ T, an E.M.F. acting in a reverse direction to the former is observed. This in- creases step by step as the temperature / 2 is still further lowered step by step. (IV.) If now the upper temperature t be brought down to T, this reverse E.M.F. is still greater. We find, in fact, that the current flows from A to B, or from B to A, across the hotter junction, according as /, T is greater or less than T / 2 respectively ; or that A and B change their thermo-electric order when the mean temperature of the junctions crosses the neutral temperature T. (V.) As we should perhaps have anticipated from the fact that when /! T = T / 2 we get no E.M.F., we find that, e.g., if one junction be 20 above the neutral point T while the other is 10 below it, we have the same E.M.F. as if the former were 20 above T, and the latter were 10 above T. It is usually said that A and B have zero thermo-electric power at T ; and that above T and below T their thermo-electric powers are of opposite sign respectively. Note. In more advanced treatises it will be found that certain pairs of metals have more than one neutral point. But such cases will not be considered here. Experiment. The following experiment will illustrate the above statement. Round the ends of a piece of iron wire are twisted two pieces of copper wire. The ends of these latter are con- nected, by means of mercury cups m m', with a galvanometer G. This latter must be delicate, and, on account of the higher resistance due to the bad contact of the Cu and Fe, it may be of higher resistance than the form commonly used with a simple thermo-cell. The end A is left at the temperature of the air, say about 20 C, while the end B is heated in a Bunsen's flame, For iron and copper the neutral point T is about 260 C. Te wire B CH. XVI. THERMO-ELECTRICITY 259 As B rises in temperature an increasing current is observed ; its direction can be shown to be from Cti to Fe across the hotter junction. This current increases until B is at 260 \ It then decreases, and becomes zero when Bis at 500 C. , i.e. is zero when /, - T = T / 2 . On raising B above 500 C. a current in the reverse direction is observed ; and this increases as the temperature rises still further. It is clear, from what has been said in this section and in 4, that two metals A and B have zero thermo-electric power at .their neutral point T. 7. Ther mo-Diagrams. From the experimental and theoreti- cal investigations of Professor Tait, Sir W. Thomson, and others, it is found that the thermo-relations of most metals for all temper- atures (at least for a wide range of temperature) can be simply exhibited in one diagram. We shall give and explain this graphic method ; the reader understanding that the construction of the diagram follows from the experimental lesults given in 5 and 6. In fig. i. O v is the axis of ordinates, along which measurements represent micro-volts ; and O t is the axis of abscissae along which measurements repre- sent degrees Centigrade. The diagram is drawn to scale suitably, so that the lengths of or- dinates and abscissae give micro- volts and de- grees respectively ; but the ordinates and ab- scissae need not be measured in the same units of length respec- tively. The three straight lines given represent the metals lead, metal A, and metal B, respectively ; in what sense they ' repre- sent ' these metals will soon be seen. Let us erect ordinates from the points / 2 > A and t^ on the line \? /, * 3 we shall in general have A | B different from A | B. Hence, as regards the E.M.F.s that occur at the junctions only, we have a resultant E.M.F. in the circuit given by ; . / , t l E*~A|B-A|B, The reader must observe that we do not state that the E.M.F.s at the junctions are the only E.M.F.s in the circuit. We shall see in (IV.) that in general such is not the case. These junction- E.M.F.s occur abruptly, and may be measured by the abrupt change of potential that occurs at the junctions, the circuit being broken at some point where it is homogeneous (i.e. not at a junction). (III.) One metal ; its extremities at different temperatures. When any metal (the one known exception being lead) has its extremities at different temperatures / 2 and f l9 there is in general an E.M.F. either from hot ter cold or from cold to hot, according to the nature of the metal. Thus, in the two cases respectively, it will be found that the hotter end has the lower or the higher potential ; the end towards which the E.M.F. acts being raised to the higher potential. This rise or fall of potential is 1 'gradual. If a complete circuit of one unequally heated metal be made, there will be no resultant E.M.F. if the temperatures change gradually from point to point ; for in that case the E.M.F. due- to the rise of tempera- ture through one part of the circuit will be balanced by an exactly equal but opposite E.M.F. due to the fall of temperature through the rest of the circuit. This will be true even if the rise and fall be very different in * steepness ' ; provided that there is no abrupt change of temperature at any section. If, however either owing to two very unequal heated ends touching, or owing to a very abrupt alteration in the diameter of the conductor at any point there be some abrupt discontinuity in the variation of temperature from point to point, then there will be in general a resultant E.M.F. in the circuit. It is generally said that this is the case when there CH. xvt. THERMO-ELECTRICITY 26? is a finite change of temperature within a distance so small that it may be considered to be within the limits of molecular action. In such a case there is a A V occurring abruptly, as in (II.). (IV.) Two metals ; the junctions at different temperatures ; the E.M.F.s all round the circuit being considered ; compare with (//.). If there be two metals A and B with junctions at different tem- peratures, the total E.M.F. in the circuit will be the algebraic sum of (the E.M.F.s at the junctions) + (the E.M.F.s in the metals A and B). The former were considered in (II.). As regards the latter, these w r ould have a zero sum were the metals the same, as stated in (III.). But in general a metal A w r ith its ends at to and /j does not give the same E.M.F. as another metal B with its ends at / 2 and /, ; and so in general the sum of these ' gradual E.M.F.s' (as we may call them to distinguish them from the abrupt junctioh-E.M.F.s) is not zero. In the case of lead there is no E.M.F. due to its unequal heating. Hence, in a cell of which one metal is lead w r e have a somewhat simpler sum of E.M.F.s in the circuit. (V.) Thus when the current is running in a thermo-cell of junctions at /._> and /j respectively, the distribution of potential in the circuit is a resultant distribution comprising three com- ponents. (i.) The abrupt changes of potential at the junctions. These measure w r hat we may call the juncticn- E.M.F.s or Peltier- E.M.F.s. (ii.) The rises and falls of potential occurring gradually along the conductors, due to the unequal heating of these conductors. These measure what we may call the Thomson-E.M.F.s. (iii.) The regular fall of potential that follows Ohm's law, discussed in Chapter XIII. The total E.M.F. in the circuit could be measured by break- ing the circuit at some place where it is homogeneous (i.e. not at a junction) and measuring the A V between the broken ends. (VI.) Experiment tends to show that at the neutral point T C. of two metals A and B there is no junction E.M.F. between them, though this is hardly certain ; and most certainly shows that in a cell where the junctions have T C. as their mean tempera- ture, the total E.M.F. in the circuit is zero. 268 ELECTRICITY CH. XVI. The above statements represent, in a necessarily very imper- fect form, the present views as to the sources of the E.M.F. in a thermo-cell. For practical purposes it is simpler to depend upon the diagram and formula of 7 ; since these (when fully given) embody all the known results of experiment. ii. Theory of the Peltier and Thomson Effects. (I.) Let there be a cell of two metals A and B whose junctions are at / 2 C. and t\ C. respectively. We will suppose, for the sake of more clearly defining the particular case that we at first con- sider, that / 2 ^ s higher than /j, and that both temperatures are above T ; 'further we will suppose that, for temperatures above T, the metal B is of higher thermo-electric power than A. This is the case in which the total E.M.F. in the circuit is represented by the area B 3 B 4 A 4 A 3 in the diagram, fig. i., of 7. (II.) When a current is running we have manifested electrical energy. If this energy be not converted (in part) into chemical- potential-energy or into mechanical or other energy, it will all be converted finally into heat energy. Now this electrical energy must have been derived from some original form of energy of which an equivalent amount must have disappeared. In the voltaic-cell it was chemical-potential-energy that was so used up. But in the thermo-cell the only available source of energy is the heat energy supplied to the cell by a flame or other source of heat. (III.) When, therefore, the current does no other work it must be that (i.)heat energy is absorbed, and disappears, some where in the circuit ; (ii.) this is transformed into an equivalent amount of electrical energy ; (iii.) and this again is finally transformed into an equivalent of heat energy, distributed over the circuit according to the resistances of the several portions of the circuit. The question arises, from what part of the circuit is the original heat energy derived ? (IV.) In attempting to answer this question theoretically, we must be guided mainly by two considerations. Firstly, we may feel sure that the well-established law of ' Degradation of energy ' will hold here as in all the phenomena of inanimate nature. That CH. xvi. THERMO-ELECTRICITY 269 is, the source of heat must be on the whole the hotter part of the cell j and the general result of the action must be that the tem- peratures of the cell tend to become equalised. Secondly, we must remember that when a current runs against an E.M.F. work is done ; and, if the work takes no other form, it will appear as heat evolved. Further, it seems almost certain that this heat will be evolved in that portion of the conductor in which the work is done, i.e. in which the E.M.F. lies. Conversely, when a current runs with an E.M.F., this E.M.F. does work on the current ; and, when no heat is supplied, this work will be done at the expense of heat absorbed from the conductor ; further, it seems almost certain that this heat will be absorbed from that portion of the conductor in which the E.M.F. lies. (V.) The consideration of (IV.) will lead us to predict that, on the whole, the hotter portion of the cell will be cooled and the cooler portion will be heated ; while some of the heat derived from the hotter portion of the cell will, after passing through the intermediate form of electrical energy, reappear as heat distributed round the circuit in accordance with Joule's law. (VI.) Where the current crosses a section (e.g. a junction of two different metals) at which occurs an abrupt E.M.F., or [see 10 (V.)] a 'Peltier E.M.F.,' there we should expect to have heat absorbed or given out according as the current flows with the E.M.F. or against it respectively. Such an absorption or disengagement of heat, occurring at a mere section and not over any finite length of the conductor, is the true Peltier effect referred to in 8. There is probably no Peltier effect when the junction of the two metals is at their neutral temperature T ; for it seems probable that at that temperature the two metals are as one. (VII.) Where the current flows through an unequally heated metal, it flows (in all metals excepting lead) with or against the Thomson E.M.F. spoken of in (III.) and (V.) of 10. We should then expect heat to be absorbed or given out re- spectively, over a finite length of the conductor in question. This is, in fact, observed in the Thomson effect of 9. Since the unequally heated ends of each metal are at the two junctions, the result of the Thomson effect will be to alter the temperatures of the two junctions. Unless, therefore, special ex- 2/0 ELECTRICITY CH. xvi. periments are performed from which we can calculate each effect separately, we shall in general observe the sum of these two effects (i.e. of the Peltier and Thomson effects) at the junctions, and shall not be able to ascribe the heating or cooling of the junctions to the Peltier effect only. (VIII.) We may then say that the transformations of energy re- ferred to in (IV.) and (V.) take place through the intermediency of the Peltier and Thomson effects together. When the hotter junction is at the neutral temperature T, then if the Peltier effect be zero, heat energy must be supplied through the intermediency of the Thomson effect only. (IX.) Now let us suppose that we cease to supply heat from external sources. The junctions will arrive at the same tempera- ture, by the cooling of the hotter and the heating of the cooler junction. If now the current be continued from some external source in the same direction as before, there is no reason for supposing that the above Peltier effects would cease, though there would now be no Thomson effects, since the metals are at one temperature. We should predict then that the junction which was the hotter would now become the cooler, and conversely. This would raise up an E.M.F. opposed to the former E.M.F., and therefore opposed to the current that is flowing. Some such reasoning as the foregoing would therefore lead us to predict the ordinary case of the Peltier effect ' ; the case, namely, where a current is sent across a junction of two metals A and B, and where it is found that the junction is so heated or cooled as always to raise up an E.M.F. opposing the current, provided that the junction is not at the neutral temperature. In other words, there will be heating or cooling according as the current flows from the metal of lower, to the metal of higher, thermo-electric power, or vice versa. The whole of the above reasoning is necessarily rather of the nature of guessing at probable results than of strict argument The fact is that the theory of thermo-cells is beyond the scope of an elementary book. CH. XVII. 2/1 CHAPTER XVII. GALVANOMETERS ; WITH A PRELIMINARY ACCOUNT OF THE MAGNETIC ACTIONS OF CURRENTS. i. Magnetic Field about a Simple Rectilinear Current. We will now turn our attention to the very important class of pheno- mena coming under the head of ' the magnetic actions of currents} On these magnetic phenomena depend the construction and use of that important class of instruments called galvanometers, whose use, as current detectois and current measurers, is so essen- tial in the modern science of electricity. In the present Chapter we propose describing various forms of galvanometers. But, in order the better to understand their theory, we shall give some preliminary account of the magnetic fields due to electric currents ; leaving, however, the main part of this impor- tant subject to be pursued further in Chapters XVIII.-XX. When a conductor is charged statically with electricity, we have about it what we call an electrostatic field. This field acts on a -f unit of electricity in lines of force that run to or from the conductor, as explained in Chapter X. The case of a conductor carrying a current is very different. It is true that still a + unit of electricity would in general find a field of force about the con- ductor, for this conductor will be in general at potentials different from that of the earth, and different from point to point of the conductor. But this electrostatic field is quite unimportant and negligible compared with the new field of force that springs into existence directly the 'electricity' moves, or directly there is a current. This new field is a magnetic field ; and we shall therefore consider its action, not on a -f unit of electricity, but on a + unit magnetic pole. 272 ELECTRICITY It is easily shown by direct experiment that the lines of force in which this field urges our unit pole form circles round the wire that carries the current, so that a pole is urged, not to or from the wire, but continually round and round it (the wire being assumed straight). Each line of force is a closed circle (not a spiral), lying in a plane perpendicular to the rectilineal wire, and having its centre in this wire. The field is weaker further from the wire, and stronger nearer to the wire. As with the electrostatic field in Chapter X. 13 and 14, so with a magnetic field we can mark out a certain number of lines of force and leave in the field such a selection that the number piercing i sq. cm. held perpendicularly to the lines at any place in the field measures the strength of the field at that place, i.e. gives the number of dynes with which a unit magnetic pole would be urged at that place. Experiments. (i. ) A hole is bored (with a bow drill and turpentine) in a sheet of glass, and a wire carrying a current is passed through this hole per- pendicularly to the glass plate. On passing a strong current, and scattering iron filings on the plate, these latter will (on tapping the plate) be observed to arrange themselves in con- FIG. i, centric circles about the hole as centre. If the wire be not perpendicular to the plate, the lines will be ellipse-shaped. This indicates that the lines of force are circles lying in planes perpendicular to the wire. (ii. ) If the wire lie on the plate, we find the filings arranged in straight lines perpendicular to the wire. These straight lines are the sections of the concentric circles, made by the plate. CH. xvn. GALVANOMETERS 273 Now let a magnetic needle, so balanced as to turn any way, be placed near such a current ; and let us for the present consider only the field due to the current, leaving ,the earth's field out of the question. It is clear that the needle can be in equilibrium only when it lies in the plane passing through its point of sus- pension and perpendicular to the wire carrying the current ; and when, further, its poles are equidistant from the wire. If it is very small it may be said to come to rest when it lies in (or is tangent to) one of the above circular lines of force. (Compare Chapter II. ii.) In such 'a position its poles are urged by equal forces tending in opposite directions round the wife ; these equal forces being inclined at equal angles with the needle, since this forms a chord of the circular line of force. This pair of forces will direct the needle as stated, and will also give a resultant urging the needle broadside-on towards the wire. The reader should make the above clear to himself by means of a figure. (iii.) A magnetic needle is fixed on a cork, and so floats on the surface of water. To one side of the needle is placed a wire perpendicularly to the surface of the water, and a strong current is passed through the wire. It is better so to place the wire, and to pass the current in such a direc- tion that the action of the current is not opposed to that of the earth, but acts with it. It will then be observed that the needle will be dragged into a position in which its poles lie on the circumference of a circle which has for its centre the point where the wire cuts the water, i.e. lie on the same circular line of force ; and that then the needle will be urged broadside-on towards the (iv.) In the same way are steel or iron filings urged broadside-on towards a wire carrying a current, and caused to adhere to it, the filings having become ' magnetic needles ' by the inductive action of the field. They are thus arranged in very close circles round the wire. (In the case of a magnetised steel wire the filings adhere end-on, pointing along the lines of force that in this case radiate from the wire.) 2. The -f and Directions of the Lines of Force. It is easily shown, by experiment with a magnetic needle, that the + and directions of the lines of force are given by either of the following rules. (I.) Amperes rule. 'If one swims with the current (i.e. so that it flows from feet to head) and looks at a n-seeking (or +) pole, this will be urged to one's left hand. A s-seeking (or ) pole will be urged to one's right hand.' 274 ELECTRICITY CH. xvn. Oersted's experiment. The his- toric experiment of Oersted illus- trates well this rule of Ampere. The figure here given sufficiently explains this simple experiment. (II.) Another form of the rule is often very useful. No one who has driven in an ordi- nary (or right-handed} screw can forget how the hand turns as tne screw advances away from the person driving it. As one drives it from one, the hand turns as do the hands of a clock that faces one. Now the above experimental rule of Ampere means that if the current advances with the screw, a + pole is urged round it in the direction in which the screw turns. And since the direction in which a + pole is urged gives us the + direction of the lines of force, we may give the following rule. ' The -f- direction of the lines of force about a wire carrying a current is associated with the direction of the current in just the same way as the direction of rotation of an ordinary (or right-handed) screw is associated with the direction of the onward movement of the screw? (III.) Field due to a circle of ivire carrying a current. If the wire be bent into a circle it will easily be understood that at the centre of the circle all the lines of force there combine to give a line of force running perpendicularly to the plane of the circle. To any side of this line the lines of force bend away ; they ulti- mately curve completely round the wire and run into themselves again. Thus there is, if we wish to be very exact, only one straight line of force, viz. that which runs through the centre of the circle perpendicularly to its plane. [Mathematicians would say that the two ends of this line join at infinity, so that it also forms a -'closed curve.'] But practically, if we take a portion of the field that is near the centre of the circular current and is small as compared with the diameter of the circle, we may consider this portion to be uniform, and to have its lines all running perpendicular to the plane of the circle. A little consideration will show us that the same ' right-handed screw relation ' holds here in this somewhat converse case. If we CH. xvii. GALVANOMETERS 2/5 follow this central line of force in its + direction (i.e. if we travel as a + pole is urged), then the current round the circle is asso- ciated with our direction of movement as the direction of rotation of a screw is associated with its onward movement. .g., if we face a 'circle in which the current goes dock-wise (i.e. as do the hands of a clock when we face it) our + pole is urged towards this circle. 3. Simple Form of Galvanometer. If we balance a mag- netic needle so that it moves in a horizontal plane, it will come to rest when it is in the plane of the magnetic meridian. If we de- flect it from this position of rest, there will be a couple due to the horizontal component of the earth's field tending to restore it to its original position of rest. Now let a loop of wire be passed round the needle, above and below it, so that its plane coincides with the plane of the magnetic meridian. The figure here given exhibits this simple arrangement. If a current be passed round this wire loop, it is easy to see that the upper and the lower wires give fields about the needle in the same direction, tending to set it at right angles to the plane of the loop, i.e. at right angles to the plane of # the magnetic meridian. Now the earth's couple is greatest when the needle stands E. and W., and is zero when the needle stands N. and S. ; while the converse is the case with the current's couple. Hence the needle will settle at some angle of deflexion from the magnetic meridian ; the magnitude of Q (cceteris paribus] de- pending upon the current-strength. Such is the principle of the galvanometer. In 5, 6, 12, 13, 14, 15, we shall describe instruments designed for measurement of currents. In 7, 8, 9, 10, u, the instruments described are mainly for detection of very small currents. (For further general remarks on galvanometers, see 16.) 4. Relation of Strength of Field to Current-Strength, It can be shown that strength of field is (cater is paribus] directly 276 ELECTRICITY CH. xvn. proportional to strength of current ; the strength of field being measured, e.g., by the vibration method of Chapter III. 3 and 6. We make a vertical frame, round which insulated wire may be wound, and place it in the plane of the magnetic meridian. In the centre of this frame the needle vibrates under the field due to earth and current together. We then try the effect of having one, two, three, &c., turns of wire round the frame. By including in the circuit a rheostat, by means of which more or less resistance may be introduced, and an auxiliary galvanometer whose deflexion we thus keep constant, we can insure the constancy of the current that we employ. We then compare the fields, as in Chapter III. 3, and show that the above law holds. It is easy to show that the field depends strictly upon current-strength and not upon current- density ; that is, upon the quantity per second passing over any section of the conductor, not upon whether this current forms (so to speak) a broad slow stream or a quick narrow one. Experiment. One way of proving the above is to show that when a current flows through a thick wire, and returns through a thin wire wound round the former, the total resultant magnetic field is zero. This is another example of proving equality by means of the 'zero method.' The reader will observe that from this it follows, as assumed above, that with the same current n turns of wire act as one turn carrying n times this current. Hence, the law stated at the begin- ning of this section is proved by the method given, in which we employ one, two, three, &c., turns of wire. 5. The Tangent Galvanometer. The simplest form of true galvanowgte/' (or current- measurer] is that called the tangent- galvanometer. Referring to 2 (III.), we shall see that about th centre of a circular wire carrying a current the field may be considered uniform. Or, if a needle whose length is small as compared with the diameter of the circle (see note (iii.) at end of this section) be suspended at the centre, then, however much it be deflected it will remain practically in the same strength of field as long as the current remains constant, If then we can measure the strength of this field by the amount of deflexion, we measure at the same time the current that gives rise to the field ; whereas, if the needle were too long or situated unsymmetrically with respect to the circle, the needle would, for GALVANOMETERS 277 FIG. i. the same current, pass into different strengths of field according to the extent of its deflexion ; in which latter case we could measure the fields but not the current. Fig. i. gives a general view of the tangent galvano- meter. There is usually a short needle provided with a long and light index that moves round a graduated circle. Very often arrange- ments are made by means of which we can either use one thick circle of copper for very powerful currents, or two or more turns of wire for cur- rents not so strong (see 7). This circle or coil is placed in the plane of the magnetic meridian. In fig. ii. we are supposed to be looking down from above, and so to be viewing the top of the circle of copper and the needle, projected together on to a horizontal plane. G G' is part of the pro- jection of the copper circle ; it lies, as mentioned in 3, in the magnetic meridian. n s represents the needle ; this lies in reality below GG', at the centre of the circle of which GG' is the top. The dotted lines run- ning parallel to GG' repre- sent the direction of the earth's horizontal field, H ; those perpendicular to G G' represent the field I due to the current. 2/8 ELECTRICITY CH. xvn. Using the notation of Chapter III. we have that (i.) Earth's couple acting on needle is H /u I sin a. (ii.) Current's couple acting on needle is I fj. I cos . When there is equilibrium we must have these couples equal ; or H n I sin a = I p. I cos a. Whence I = H . tan o. Now, in the present case we have said that whatever be the deflexion of the needle we may consider I to be constant while the current is so, and to be directly proportional to the current C. Hence we may write C = k I, where k is some constant depending on the dimensions of the coil, the number of turns of wire, and the unit of current that we employ. We have then C = k H . tan a. Hence with the same instrument, and with H the same, the current is proportional to the tangent of the angle of deflexion ; whence the name of this instrument. It is clear that it can be used for currents up to any strength, since tan a can have any value. But it fails to give accurate measurement if a exceed (say) about 70, since then large changes in current give small changes in a. The best angle of deflexion is about 45, as may be shown mathematically. Notes. (i.) Other forms of the instrument. In Gaugain's form we have the needle placed in the axis of the coil, at a distance from the centre of the coil about one-fourth of its mean diameter. In Helmholtz's modification of Gaugain's instrument there are two coils lying symmetrically on either side of the needle, and the wires are wound on small portions of cones, of which, if completed, the centre of the needle would be the common apex. Such an arrangement gives a more uniform field, and permits one to use a somewhat larger needle. (ii. ) To set the coil in the magnetic meridian. When the coil is in the magnetic meridian, the same current, passed in opposite directions, should give opposite deflexions of equal magnitude. The coil should be adjusted until this is the case. (iii.) As regards the relative dimensions of needle and diameter of coil, giving results accurate enough for practical purposes, we quote here the follow- ing authorities. Wiedemann ('Die Lehre von der Electricitat,' iii. 247) gives length of needle at most one-eighth diameter of coil. Kempe (' Handbook of Electrical Testing,' p. 18) gives needle about f -inch for a 6-inch or 7-inch ring, as accurate enough for most purposes. S. P. Thompson gives length of needle about one-tenth of diameter of coil. CH. XVII. GALVANOMETERS 279 B rough gives needle one-sixth of diameter of coil. Andrew Gray gives needle I centimetre for coil of diameter 15 centimetres , in a standard instrument. Hence, needle one-tenth of diameter of coil seems a good relative dimension. 6. The Sine Galvanometer. Fig. i. shows us a somewhat different form of galvanometer. Here the coil turns round on a vertical axis, the movement being measured over a horizontal graduated circle. As before, the coil is initially in the plane of the magnetic meridian. We may here turn the coil until the current does not affect the needle at all. In this position it must be that the lines of force due to current coincide with v those of the earth, or the coil is due magnetic east and west. We then turn the coil back through 90 over its graduated scale, and it will lie in the plane of the magnetic meridian. When the needle is deflected the coil is turned in the same di- rection ; and finally, if the current be not too strong, can be brought again directly over the needle. When this is the case we note the angle through which the coils have been turned. The current will be proportional to sin a. Fig. ii. shows us the theory of this instrument. In it is re- presented the final position of equilibrium, in which the coil G G' lies again over the needle at an angle a from the magnetic meri- dian N S. Since now the lines of force due to the current are perpendicular to the needle, we have j (i.) Current's couple = I /u /. 1 (ii.) Earth's couple = H p I . sin a. FIG. 280 ELECTRICITY CH. XV] T. With similar reasoning to that given in the last section we have finally . ." .-^ . . C = k H . sin a whence the name of the instrument. It is evident that we cannot, without using shunts, measure any current exceeding k H, since the greatest value of sin u is i, which it has when a = 90, or when the needle stands E. and W. With a larger current we should chase the needle completely round. Since the coil is always over the needle we may have this of any length we please, for it will always be in the same part of the cur- rent's field. 7. The Multiplying ,.f Galvanometer. The extent of deflexion of the needle depends upon the strength of the magnetic field due to the current. With the same current, this can be multiplied many-fold by passing the wire many times round the needle. In fact, we wind the wire in a coil and suspend the needle inside it. The advantage of this can be demonstrated as in 4. A figure of a multiplier is shown in the next section. 8. Astatic Galvanometer; Two Needles. In the last section we showed how to increase the current's action on the needle by increasing the field- strength due to the current. There is another very effective manner in which the deflexion for a given current may be almost indefinitely increased. This other method is based upon the device of making the earths re- storing couple as weak as we please, while we leave the current's deflecting couple unaltered. We cannot do this by weakening the magnetic moment of a CH. XVII. GALVANOMETERS 28l single needle ; since, as we see in the formulas of 5 and 6, we should then weaken equally the current's action on the needle. But we can so combine two needles that, they form with re- spect to the earth a single needle of as small a magnetic moment as we please ; while, by coiling the wire round one needle only, we can cause the current to act on a single needle of considerable magnetic moment. A system of needles on which the earth has little (or, strictly, no) action, is called an astatic system ; and a galvanometer in which such a system is employed is called an astatic galvanometer. There are two com- binations that give us convenient astatic sys- tems. (i.) Let us connect rigidly two needles ns and ri s' of nearly equal strength, so that, when they are suspended, they are in the same vertical plane ; the poles being directed opposite ways. If ns be some- what the stronger, the two needles will act with respect to the earth as one needle of the same length /, but of pole- strength measured by the difference (u /*') ; FIG. i. so that the magnetic moment with respect to the earth is l(pp'\ and may be very small indeed by having the pole- strengths p and // nearly equal. The coils of the galvanometer, however, as seen in the figure, are so arranged that one needle of moment /* / is in the strong field within the coil ; while the other needle, //, inasmuch as it is reversed in direction and is also in the reverse field above the 282 ELECTRICITY CH. xvn. coil, is acted upon by a weaker couple in the same direction as is the lower needle. Thus we can, by making the pole-strengths ^ and p' very nearly equal, cause the earth's restoring couple to become very small, without diminishing the deflecting couple ; and so can cause the galvanometer to become very sensitive. (ii.) Next let two needles be fixed and suspended as before, but let the vertical planes through them make some small angle with each other. In fig. ii. we give a projection of the two needles as viewed from above. Let them further be of equal strength, so that p = //. Then it is pretty clear that with respect to the earth they will act as a short needle n" s" whose axis is at O, standing at right angles to the bisector of the angla ; its pole-strength being 2 p, f\ its length being n" s" or I sin -. It follows that its magnetic mo- ment will be 2 p. /sin. This can be made as small as possible by making as small as we please. It is to be observed that the needles will stand so that. ".$" is in the magnetic meridian ; or the general direction of the two needles themselves will be E and W. The action of the current is as before. 9. The ' Controlling Magnet ' Method . A needle balanced so as to move in a horizontal field is influenced by the horizontal component of the earth's field only, as we have said before. Now we may superimpose on this horizontal field another due to a magnet ; and, if the lines of force of this field run parallel to those of the earth's, but in an opposite direction, we may weaken, to any desired extent, the resultant field in which the needle moves, by suitably adjusting the strength of the magnet's field. The most convenient arrangement is to have a magnet directly above the needle, capable of being slided up or down a metal stem so as to make the field about the needle weaker or stronger ; and also capable of revolving in a horizontal plane about the stem,. so that it may either stand in the magnetic meridian or may make any angle with the meridian. en, xvii. GALVANOMETERS 283 For our present purpose our object is to leave the needle directed in the magnetic meridian by a very weak resultant field. We thus should have the magnet's field a little weaker than the earth's, and opposite in direction. The needle is acted upon by the current as before ; and, as the restoring couple is very small owing to the weakness of the restoring field, the instrument may be very sensitive. We may add that in the most sensitive instruments the methods of 8 and 9 are combined. 10. Sir W. Thomson's Mirror Galvanometer. The trans- mission of signals by means of interruptions and reversals of currents will be described in a future Chapter. At present we will merely observe that galvanometers will indicate such reversals and interruptions, and so will serve as instruments to receive signals of this nature. We shall see in a later Chapter that when such signals are sent through long submarine cables, the currents are very small on account of the great resistance ; and further, that interruptions and reversals lose their abrupt character and become mere fluctuations in the weak current transmitted. To indicate such weak currents and such slight fluctuations, Sir W. Thomson invented a form of galvanometer without which signals by cable would hardly have been possible. In his mirror galvanometer we may notice the following main points of interest. (i.) As the instrument is usually intended to be used in circuits where the resistance is already very great, there are often many thousand turns of (necessarily) fine wire ; this ' multiplies ' the strength of the field due to the current without perceptibly diminishing this latter. For the total resistance will not relatively be much increased. (ii ) The ' needle ' is .a very small bit of magnetised watch- spring. Having little mass and inertia, its movements follow any fluctuations in the current almost instantaneously. (iii.) A controlling magnet enables one to vary the sensitive- ness of the instrument at will, as explained in 9. (iv.) The same controlling magnet enables one, if need be, to bring the needle to rest in any position ; and. aided by the method of suspension mentioned below, to work the instrument even on 284 ELECTRICITY CH. xvir. CH. XTII. GALVANOMETERS 285 board ship. In such cases we over-compensate the earth's field and direct the needle by the magnet alone. The fibres supporting the needle are fixed at top and bottom, and so prevent it from swinging against the sides of the coil in which it works. (v.) To the needle is attached a very light mirror. A beam from a lamp falls upon this, and the reflected ray gives as index a spot of light that moves over a graduated scale. We thus (as in the case of the quadrant electrometer of Chapter X. 33) can have an index as long as we please, possess- ing no mass or inertia. The figure on the opposite page indicates the coil, lamp, beam and its reflection, scale, and controlling magnet placed above the coil. The needle and mirror are inside the coil and are not seen. An additional controlling magnet T is sometimes of service in bringing back the spot of light to the zero mark. Notes. (i.) The mirror maybe concave or plane. In the latter case an auxiliary lens is used, in order to bring the reflected beam to a focus on the scale. (ii.) Either in the reading of, or in the construction of, the scale, we must make allowance for the fact (proved by elementary optics) that if the needle and mirror be displaced through an angle 0, the reflected beam is displaced through an angle 20. ii. The Differential Galvanometer. In the differential galvanometer the needle is suspended symmetrically between two coils. These coils must fulfil the following requirements. (i.) Their resistance must be exactly equal, so that equal E.M.F.S at the terminals give equal currents. (ii.) Equal currents in the same direction must give equal but opposite fields about the needle ; so that, when the same current in the same direction is passing through the two, the needle is un- affected. An exact test that these conditions are fulfilled is to send the same current through two coils coupled end-on, so that the current passes in the same direction through each coil. The needle should not be affected. Usually a small portion of one coil is left moveable, allowing exact adjustment to be made when the in- strument is used. It is clear that by coupling the coils suitably this instrument 286 ELECTRICITY CH. XVII. may be used as an ordinary galvanometer, the two coils giving fields acting on the needle in the same direction. The differential galvanometer has several uses, (i.) One use is to compare resistances ; the method is somewhat as follows. A current is divided into two branches, passing through the two coils respectively. Since the resistances of these are equal, the currents will also be equal and the needle will be unaffected. The unknown resistance is now introduced into one branch, thus causing an unequal distribution of the two currents. Then known resistances are introduced into the other branch until the needle returns again to zero. When this is the case the resistances must be equal. In this method we are indepen- dent of changes in the E.M.F. of the battery. (ii.) Another, less simple, use of this instrument is given in Fleeming Jehkin's 'Electricity,' p. 242, to which book we refer the student for details as to this method. We will only say here that by shunting one branch of the galvano- meter we may either measure a small resistance to a small fraction of an ohm, or may measure a resistance which is a large multiple of the greatest resistance contained in the resistance box. 12. The Ballistic Galvanometer. As we have seen, a current can be measured by the couple that it exerts (if we may use the expression) on a needle in an instrument of known con- struction ; this couple depending also in a known way on the angle that the needle makes with the plane of the coil. Now quantity of electricity is given by (current} x (time}. Hence we can, in the case of steady currents, measure the total quantity of electricity that has passed by means of the, tangent galvanometer, by observing both current and time of duration. But supposing that we wish, eg., to measure the quantity of electricity in a condenser by discharging it through a galvano- meter. In the ordinary case we have a current whose strength rises from zero to a maximum, and then falls again to zero. While it passes the needle is deflected, and part acts when the arm of the couple is the full length / of the needle, part when the arm is I cos (see 5), where rises from zero to a maximum and then decreases again. It is impossible, under these conditions, to calcu- late in any simple way the total quantity that has passed. In order to do this an instrument is devised analogous to the ballistic pendulum used in mechanics. In this latter instrument we can estimate what are usually called ' impulsive forces,' or ' impulses,' CH. XVII. GALVANOMETERS 28 7 by allowing them to act on such a massive and slowly-moving pendulum that there is no appreciable change of position until the impulses have ceased to act. The impulses thus act on the pendulum in one known position (viz. when it is hanging vertically); and the sum total of these impulses can be deduced from the extent to which the pendulum swings, its mass and dimensions being known. In the ballistic galvanometer we have a needle of such a mass that it does not move appreciably from zero until the discharge has entirely passed. Thus each portion of the discharge acts on the needle with coilples whose arms are all the length / of the needle. The extent of swing of the needle then depends solely upon the sum of (each cur rent- strength) x (the infinitesimal time during which it remains constant}. In spite of the fact that the current changes continuously, thus making this 'sum ' a matter for the ' infinitesimal calculus/ even a beginner can see that the sum of all the products above given measures the total quantity of elec- tricity that has passed. If Q be this quantity, and be the maximum angle of deflexion of the needle, it can be shown that . . . where k is a constant depending upon the construction of the in- strument and upon the strength H of the earth's horizontal field. This constant can be found or calculated. Note. The student can find the proof of the above formula in more advanced books. Use of ballistic galvanometer. We may use the foregoing in- strument to measure, or compare, the capacities of condensers. In the accompanying figure xy a b is a condenser. The plates ab are to earth, while the opposite plates xy can be connected, by means of a key M, either with the pole P of a battery, wwith the one terminal of a ballistic galvanometer G. The other pole C of the battery, and the other terminal of the galvanometer, are to earth. When the plates 288 ELECTRICITY CH. xvn. xy are connected with the pole P of the battery, the condenser is charged so that xy are at a potential V, while a b are at zero poten- tial. (It is clear that V measures the unpolarised E.M.F. of the battery, for the other pole C is to earth, and there is no permanent current.) Then if K be the capacity of the condenser, the quantity Q of the charge will be where Q x is in coulombs, if V is in volts and K t is in farads (see Chapter XVI 1 1. 4). Now let the condenser be discharged through the ballistic galvano- meter. The total impulse given to the needle is, we have seen, directly proportional to Q ; and this impulse is also, we have stated, propor- /\ tional to sin - 1 , for mechanical reasons that we have not given ; 1 being, as above, the ' angle of throw ' of the needle. J? A Hence we have K x = ^ . sin -* k So, for another condenser, we have K 2 = . sin -% ; whence sin ^ K K 2 sin - 2 2 We can thus compare any condensers with standards, or with each other. Note. Let / be the time in seconds of a complete to-and-fro oscillation of the needle when no current passes. Let 6 measure the angle of throw in some unit of angle such as degrees. Let r\ measure in ohms that resistance through which the E.M.F. measured by V (see above) would drive a steady current, giving a permanent deflexion of one unit of angle the same unit as is used in measuring 9. And let K be the capacity of the condenser \n farads. Then it can be shown that ...... ... ........ . . . t . sin - K-- -i 7T. fj This formula is given in Fleeming Jenkin's ' Electricity.' The more mas- sive the needle, and the less friction there is in its movement, the truer is this formula. r ui~ DEPARTMENT OF P'- CH. xvir. GALVANOMETERS 289 13. Sir W. Thomson's 'Graded' Potential Galvanometer To meet the present requirements of exact electrical measure- ment, Sir W. Thomson has invented two galvanometers of con- venient form and of wide range of sensibility, the one designed es- pecially for measurements of E.M.F.s or of potential differences, the other for measurements of currents, In both there are arrange- ments for altering to a known extent and through a wide range the sensibility of the instrument ; and both are so marked by the makers that, with simple corrections for the strength of the earth's horizontal field H at each place, results can be read off in volts and amperes respectively. For these reasons the instru- ments are called graded galvanometers. A very full account of them is given in Gray's ' Absolute Measurements in Electricity and Magnetism.' They have not, however, for reasons alluded to in 17, satisfied practical requirements. Sir W. Thomson has since , invented other instruments, alluded to at the end of Chapter XXV. The volt-meter galvanometer. The general principle of a volt- meter galvanometer has been already explained in Chapter XIV. n. In the figure C is the coil. This has a great number of turns of wire, and a large resistance, generally exceeding 5,000 ohms. M is the magnetometer, or system of needles. This consists of a system of short needles, parallel to one another, disposed symmetri- cally about a common pivot. We can consider this system to be equivalent to one short needle, and shall often speak of it as 'the needle.' To this is rigidly fixed a long and light aluminium index, standing at right angles to the direction of the needle. It is this index that is seen in the figure. It moves over a scale properly graduated. Over the magnetometer and resting on the magnetometer box, but removable at pleasure, is seen a semicircular magnet. By the use of U 2QO ELECTRICITY CH. xvn. this magnet the { restoring-field ' (which would otherwise be only the earth's horizontal field H) may be much increased in strength. Further, when the magnet is used, any error in the measurement of H at any particular place will be a smaller percentage of the whole field than it would be if H alone were the whole field, and so will cause a smaller error in the measurements made with the instrument, providing that the field due to the magnet is known. (This latter has, in practice, to be re-determined from time to time.) But it must be remembered that, according to the principle of 9, any strengthening of the restoring- field diminishes the sensibility of the instrument. The instrument is so constructed that the axis of the coil C, along which runs the central line of force (see (III.) of 2), passes through the point of suspension of the needle, and thus the current tends to direct the needle along the line of the axis of the coil. On the other hand, the magnet is so placed that its magnetic axis passes through the point of suspension of the needle, in a direction perpendicular to the last. When used, the instrument is levelled, and is so placed that the planes of the coil and of the magnet are in the magnetic meridian. When this is the case, the index should stand at zero when no current passes, and equal currents in opposite directions should give equal deflexions from zero. In this position we have as a restoring-field the earth's horizontal field H, to which may be added at pleasure the stronger horizontal field due to the magnet, coinciding in direction with H. The deflect- ing-field &OK. to the coil is horizontal, and is perpendicular to the last field. Thus the theory of the instrument^ since its needle is relatively short and moves in a uniform field, is that of the tangent galvanometer. The sensitiveness of the instrument can be varied, both abruptly, by means of the magnet, which may be used or removed, and also, step by step, by sliding the magnetometer box nearer to, or further from, the coil. The feet of the box slide in a groove in such a way that the whole moves parallel to itself, while the point of suspension of the needle moves along the axis of the coil. Instructions as to the conversion of the readings into volts under all conditions of sensitive- ness are sent with each instrument. The terminals are of an ingenious construction, designed partly to obviate accidents in measuring very large AV.s, and the ' leads ' are so arranged that the current passing through them has no action on the needle. 14. Sir W, Thomson's Graded Current Galvanometer. In the construction of the last instrument it was desired to measure CH. xvn. GALVANOMETERS 291 the AV between two points without appreciably diminishing this -W by lessening the resistance between them. Hence we saw that the coil was made of a very high resistance. In the present case it is desired to measure currents without appreciably diminishing them by the introduction of resistance. The coil, therefore, must be of as low a resistance as is consistent with the requisite sensitiveness. The current galvanometer, or am-meter, is in general construction almost identical with the last. But the coil is composed of a few turns of thick copper wire, or copper strip, of a resistance equal to about jJ^ ohm. There is also a special arrangement of the terminals, to obviate inconveniences or accidents in dealing with large currents, and the ' leads ' are so arranged that the current passing through them has no action on the needle. 15, Weber's Electro-Dynamometer. It is sometimes desired to measure currents by some instrument that does not depend upon the magnitude of the earth's field of force at any particular place in fact, to construct an instrument that can be used without reference to the variable quantity ' H,' that has appeared in the formulae of the preceding instruments. Now we shall see in later Chapters that a coil carrying a current acts as a magnet of the same shape whose poles answer to the two faces of the coil. It is found that the magnetic moment of such a coil depends solely upon its shape and the strength of the current flowing through it ; so that, as long as these remain constant, we have an absolutely invariable magnetic needle. Let the large coil of 5 (or one with more turns of wire, if suitable for the purpose) stand with its plane vertical and coinciding with the U 2 2Q2 ELECTRICITY CH. xvn. plane of the magnetic meridian. And let there be suspended at its centre by a * torsion wire ' a relatively small coil, whose faces are turned due magnetic N and S, and whose plane therefore is perpendicular to that of the larger coil. * If a current be passed in the right direction through the small moveable coil, it will be in equilibrium with respect to the earth's hori- zontal field, since its faces (or poles) are turned N and S. If another current be passed through the larger coil, the field due to this will act on the smaller coil, tending to set it parallel to the larger. We might have given the small coil free suspension, and have used the small coil as we used the needle in 5. But since we wish to get rid of H from the formula of calculation, we twist the torsion wire until we bring the small coil back to its zero position, in which the earth's field did not act on it. The angle of torsion then measures the couple with which (to speak intelligibly enough, even if not very accurately) the large coil acts on the smaller. We have then a balance between (i.) the torsion of the wire on the one hand, and (ii.) the electro-magnetic couple between the coils on the other. The former can be determined by ordinary mechanical methods, and the latter depends (in a manner known to those who have studied the theory of electro-magnetic actions) on the dimensions of the coils, and on the product of the strengths of the two currents. Thus, for each instrument a formula can be once for all constructed, giving the value of one current when the other is known. If we pass the same current through both coils- then we can measure this current by measuring only the above given angle of torsion. This last form of the instrument is adapted for the measurement of currents that pass in rapidly alternating directions, since in this case the reverse current in the large coil finds a needle of reverse polarity upon which to act. The angle of torsion will in this case be proportional to the square of the current-strength. 1 6. Some General Observations on Galvanometers. We will end this Chapter with a few general remarks on galvano- meters. (I.) Long-coil and short-foil galvanometers. Fleeming Jenkin uses these two very clear terms to indicate instruments in which the current passes many times, or few times, round the needle respectively. For obvious reasons we cannot increase the field acting an the needle by the device of passing the current many times round without at the same time unavoidably increasing the resistance of CH. xvn. GALVANOMETERS 293 the instrument, for we cannot make coils having many turns of any but fine and therefore highly-resisting wire. In using an instrument we must, then, consider whether the resistance R of the rest of the circuit be large or small as com- pared with the resistance G of the galvanometer. If G be negligible as compared with R, the field is multiplied n-fold by using n turns of wire, and the deflexion for a given current is thus made much greater. But if R is negligible as compared with the resistance of one turn of wire, then we get but th of the n current when we use n turns, and thus gain nothing but incon- venience in using a long-coil instrument. 17. Galvanometers for Practical, or Commercial, Use. Now that electrical measurements have to be made every day in places where the presence of large magnets and powerful electrical currents renders the magnetic field unknown and variable, it is neces- sary to have instruments whose action is independent of the earth's field of force, and whose accuracy is not perceptibly impaired by such disturbances as are likely to occur. At the end of Chapter XXV. will be found some explanation of the principles of several of the more recent types of instruments. 1 8. Calibration of Current-meters and of Voltmeters The system of units employed in modern electrical science is based upon the electro-magnetic actions of currents, as will be explained in Chapter XVI 1 1. And the practical unit of current employed, the ampere, can be determined by observing the attraction or repulsion between accurately constructed coils of wire carrying a steady cur- rent. [It was (e.g.} in some such way that Lord Rayleigh measured in amperes the currents he employed.] The ampere being thus measured electro-magnetically, it can be determined how much silver or copper is set free in i second by a current of i ampere. From the experiments t>f Lord Rayleigh and others it appears that about i 1 1 8 milligramme of silver is set free by I ampere in i second. In practical text-books the student will find it explained how this result renders it a comparatively easy matter to calibrate any given ammeter in amperes. As regards voltmeters, we may [see Chapter XIV. n] graduate them also by means of currents measured electro-chemically. Or we may make use of the known E.M.F.s of certain standard cells ; these E.M.F.s having been previously determined once for all, in absolute value, by long and careful laboratory experiment. For further on ' Calibration ' we refer the student to practical text books. 294 ELECTRICITY OH. xvur. CHAPTER XVIII. ACTIONS BETWEEN CURRENTS AND MAGNETIC POLES. MAGNETIC EQUIVALENT OF A CURRENT. - ACTION BETWEEN CURRENTS AND CURRENTS. i. Action of an Infinite Eectilinear Current on a Magnetic Pole. If we pass a current through a vertical rectilineal wire A B, and observe its action on a horizontally-balanced magnetic needle ns placed near it, we arrive experimentally at several important results. (i.) We find (as also was indicated by the experiments of Chapter XVII.) that the lines of force form circles whose centres lie on the wire and whose planes are per- pendicular to it. (ii.) We find that when once the wire is so long that it makes the angle A OB nearly 180, then, as far as its action on the needle is concerned, it is practically infinite. This suggests the idea that the current gives a field at O only so far as it gives a ' broadside-on ' projection when viewed from O ; and that a current running directly towards O would give no field (lii.) We find that the field strength at O is directly propor- tional to the current-strength, as has been otherwise shown in Chapter XVIT. 4. (iv.) We find that, for the same current, the field-strength is inversely proportional to the perpendicular distance from O to the wire. 2. Action of an Element of a Current on a Pole. If we wish to be in a position to calculate the action of a current of any shape on a pole having any position with respect to the current, CH. xvin. CURRENTS AND MAGNETIC POLES 295 we must know the action of each little bit (or element) of the cur- rent on the pole, and must sum up all these actions in order to get the total action of the entire current. Such summing-up belongs to the integral calculus in general ; but in one case it can be done very simply. For the benefit of those readers whose mathematical knowledge is elementary, we shall explain further the meaning of an element of a current, and shall then discuss in a simple manner the law of action that is found to hold, and the application of this law to determine the field at the centre of a circular current (see 6). Let ab represent a bit of a current so small that the following results may be considered to be true. (i.) a b may be considered to be straight, although it be part of a curved circuit. Its length will be designated by ds^ where in general ds is a very small fraction of a centimetre. (ii.) The length p. a (where p represents the position of the pole acted on, the pole- strength being \L units) may be considered as equal to /* b, in whatever posi- tion a b stand with respect to p.. Thus, in the figure, r centimetres is the length of either p. a or pb. This is because we suppose ds to be exceedingly small with respect to r. (iii.) If we draw b a' so as to make an isosceles triangle bp.a, we may consider b a to be at right angles to both /j.a' and ph. (iv.) And, following from (iii.), we may speak of If a as making 'the angle 6 ' with either /z a, or with /LI b produced ; we shall generally say ' with r? When the above conditions hold, ab is called an element. These conditions do not hold in the figure, since that is exaggerated in order to make details sufficiently plain. From the experiment of i it was not difficult for mathe- maticians to guess at the law of action of each element of current on the pole or on the short needle. The law guessed at was such as, applied to the case of i, would enable us to predict the results there experimentally arrived at. It was then applied to other cases. The total action of a circuit was predicted by means of the integral calculus from the assumed law of action of an element. In each case thus tested, experiment fulfilled the predic- tion. When any assumed law stands thus the test of experiment, 296 ELECTRICITY CH. XVIII. we must be satisfied to consider it to be true. We now give the law, thus established, of the action of an element of current on a pole. Let n be the pole-strength, r the distance in centimetres of the element d s from the pole, d s the length of the element, d s . sin the projection of ds perpendicular to r (or the length b a' in the figure), C the current-strength. Then it is found that the pole is urged with a force proportional to ^ ' 2 ' ' in a direction perpendicular to the plane containing the element ds and the pole ^, the direction of this force being further determined by Ampere's rule. If we analyse this law with the aid of the figure given, we see that istly. The force acting on /j. is directly proportional to p and to C. This fact is what we should expect. In the last Chapter it was partly proved, and partly assumed as following from the defi- nition of IJL and C. 2ndly, For the same length ds of the element, the force is pro- portional to sin 6 ; that is, the element acts solely in so far as it presents a ' broadside-on projection ' (which is a! b in the figure) to the pole. 3rdly. The force varies as - 1 - ; thus following the same law as we found to hold in gravitation, in electrostatics, and in magnetism. . 4thly. If the current have the direction of the arrow, and the pole be + or north-seeking, this latter is urged straight up from the plane of the diagram. 5thly. The action is not in the line joining the pole and the element, but is perpendicular to the plane containing the two. The action then is very different from any with which we had to do in electrostatics or in magnetism. Note on the experiment 'of I. In the case of the infinite rectilinear cur- rent, we found the total field at ns to vary as -, whereas the elemental law r has -4 This may perhaps perplex the learner. The reason is as follows. As we recede further from the current A B, we certainly increase the distance from each element. But at the same time the upper and lower parts of the wire begin to present an increasingly great ' broadside-on projection ' to the pole or needle. CH. xviir. CURRENTS AND MAGNETIC POLES 297 3. The Absolute System of Electro-Magnetic Units. The reader should at this point look again at Chapter XIV. T, and Chapter XV. 2. We intend now to explain what are the funda- mental units to which we there referred. (I.) Unit of current. \i a wire be bent into the arc of a circle, and a pole be placed at its centre, we see by the results of 2 .that the forces due to all the separate elements will act in the same direction on the pole ; and, further, since each element is the same distance from the pole, the total force on this will be simply proportional to the current- strength, the pole-strength, the length of the arc, and the inverse square of the radius. Now we have already denned the units of force, pole-strength, and length. It is, therefore, simplest to define the electro- magnetic unit of current to be such as makes the force unity when all the other above quantities are unity. Or ..... The elect?-o -magnetic unit of current is such that flowing through an arc of unit length (i.e. i centimetre), whose radius is unity (i.e. i centimetre), it acts with unit force (i.e. i dyne) on a pole of unit strength (see Chapter II. 7) placed at the centre. As has been already stated, the practical unit, called the ampere, is one-tenth of the above absolute unit. (I I.) The unit of E.M.F.If the E.M.F. betiveen two points in a circuit be such that unit current flowing for unit time does unit work between these two points, then this E.M.F. is called unit E.M.F. in the electro-magnetic system. It will be evident to the student that this definition of unit E.M.F. is, in theory at least, simple and clear; for the work can be conceived of as measured by observation of the number of calorimetric heat units given out between the points (as in Chapter XV. 4) ; while the current can be measured in the above given absolute units by means of a galvanometer of known constants. Such a definition of unit E.M.F. is far better adapted to the requirements of electro-dynamical measurements than is the electrostatical definition given earlier. As has been already stated, the above given unit is incon- veniently small. The practical unit, called the volt, is io 8 times this absolute unit. 29$ ELECTRICITY CH. XVIII. (III.) The unit of resistance. Ohm's law defines the unit of resistance as that through which unit E.M.F gives unit current. The practical unit of resistance, called the ohm, is io 9 times the absolute unit. Thus, as stated, we have ..... T O^ T The ampere = = th absolute unit of current. T/VJ T/-\ 4. Summary of Electro-Magnetic Units (see 3). ( Absolute unit of current as defined in (I.). i Practical unit of current, the ampere . . = JL absolute unit io (Absolute unit of E.M.F. or of A V v . . ' . . . . as defined in (II.). (Practical unit of E.M.F., the volt = io' s absolute units. f Absolute unit of resistance as defined in (III.). ( Practical unit of resistance, the ohm = 10 absolute units. /Absolute unit of activity is the rate of work when unit current runs under unit E.M.F., and is i erg per second. j Practical unit of activity, the watt, is the rate of work when one ( ampere works under one volt E.M.F. ; it = io 7 ergs per second. | We get watts by multiplying the number of volts by the number of j amperes ; or, if we are considering only the heat given out in a ( conductor, by (number of amperes} 2 x (number of ohms]. (about -24 calorie per second, I or about ^ English horse-power. (Absolute unit of quantity is that which crosses any section of a con- ductor in one second when absolute unit of current flows. Practical unit of quantity, or the coulomb, is the same when one ampere flows. Absolute unit of capacity is that of a condenser in which the bound charge is absolute unit of quantity when the AV of the plates is absolute unit of E.M.F. or AV. Practical unit of capacity, or the farad, is that of a condenser in which the bound charge is one coulomb when the AV of the plates is out volt. We may add that the more usual practical unit is the micro-farad (see below). The megohm, mega-volt, c., are respectively i,coo,ooo times the ohm or volt, &c. The microhm, micro-volt, &c., are respectively - of the ohrn i oooooo or volt, &c. CH xvm. CURRENTS AND MAGNETIC POLES 299 Note. Determination of the units. The three quantities, C, R, and E, are related to each other by Ohm's law. It may be well to indicate how the absolute units, or any convenient multiple of them, might be determined independently of one another. Current. When the dimensions and construction of a galvanometer are known, and when the magnetic field H in which it is situated has been deter- mined by the method indicated in Chapter III. 15 and 16, it is possible to measure current in absolute electro-magnetic units by observations of its action upon the galvanometer needle. For such a measurement no knowledge of E or of R is either directly or indirectly implied. Resistance. The absolute measurement of the resistance of a conductor is one of the most important experiments in physical science, and one of extreme difficulty. Of the several methods that have been employed to obtain a direct measure of the resistance of a wire, the simplest consists in passing a current through the wire which is contained in a calorimeter similar to that shown in the figure on page 241. Theoretical considerations give for the heat developed per second in the calorimeter the value JH = C-R. Here C is the current, the value of which can be calculated from the indications of a tangent galvano- meter placed in the circuit ; H is the heat developed per second, which can be found from the rise of temperature of the liquid by the principles of calorimetry ; and J is the mechanical equivalent of heat (p. 239), which is known from Joule's experiments to be very near to 41,750,000. The only quantity remain- ing is R, which can therefore be calculated from the other three. Other methods depend upon the laws of electro-magnetic induction given in p. 357 ; for the details of these the student must refer to' more advanced works upon the subject. Electromotive force. Having methods of measuring current and resistance absolutely, E.M.F. may be found by the arrangement shown on p. 234. e is the cell whose E. M. F. is required ; a tangent galvanometer is placed between R and B, and the current Ae (l GB is not required. The current given by the battery P is first adjusted to a convenient strength by the rheostat R, then the contact piece Q is moved along the wire A B until no current flows in G. When this is the case the reasoning of 13, p. 234, shows that e = aC ; a being the resistance of the wire A Q, and C the current in the circuit P R B Q A P, which is given by the tangent galvanometer : thus e is determined. 5. The Dimensions of the Derived Units. When any system of units has been constructed, each derived unit can be expressed in the fundamental units. The manner in which each derived unit involves the fundamental units can be exhibited in a simple form, giving what is called the dimensions of that physical quantity. Thus, in Chapter II., the dimensions of force were given by the relation ..... -" .............. In any one system of units physical quantities are of different natures if their dimensions (i.e. the way in which they involve the 3OO ELECTRICITY CH. XVIU. fundamental units) are different. This statement may be taken as axiomatic. But the same physical quantity may have different dimensions in two different systems of units respectively. This is not so easy a matter to understand, but the fact is clear enough, as will be seen from the table given on the next page. Thus the dimensions of current are different according as we regard it from the point of view of electro- static quantity (see Chapter V. I passing across a section in unit time, or from the point of view of magnetic actions. If two physical quantities have different dimensions in the same system, it follows that not only are they different in essential nature, but that in general any alteration of the fundamental units will alter in different ratios the numerical values of the two quantities. On p. 301 we give a table exhibiting the dimensions of the various physical quantities in the two systems, electro-magnetic and electrostatic, respectively. It would be out of place in this Course to show how the dimensions of each are found, but we give one example. Example. The dimensions of magnetic pole-strength. By defini- tion and experiment we have .:"; force between two equal poles = no e-s reng ^ (distance oetween tnemj- Using symbols to represent the units of pole-strength, force, and length, we may write -. " ^ = F x L 2 = ML x L o = M L 3 T _ 2> And therefore /u, = M . L . T" 1 gives the dimensions required. With respect to the fact that in the electro -magnetic system the dimensions of R are LT~', or are those of a velocity, the reader is referred to Chapter XXI. 7 (II.), note. The ratio-velocity v. We must now call attention to the very remarkable fact that when a quantity is expressed both in electrostatic and in electro-magnetic measure, the ratio between the two sets of dimensions always involves simply the dimensions of a velocity ; this we have designated by v in the last column of the table on the next page. Direct experiment shows (within the limits of experimental error) that this velocity is a constant. If C and C' represent the numerical magnitude of the same current measured electro-magnetically and electrostatically respec- tively, and if the same convention be assumed with respect to the other symbols, then by the table on the next page we have . . ., .^ v = 2: = c : = E = /IT = /K 7 Q C E' VR> So the formulae of the last section become, for an electric circuit, p C Q and ft C (Oj - O 2 ). Experiment. De la Rive's floating battery. A small battery-cell is con- structed, composed of two strips the one of zinc and the other copper immersed in dilute acid in a short wide test tube. The cir- cuit is completed by an insulated wire wound into a small circular coil that has its plane vertical when the test tube is vertical. This small cell and cir- cular current is floated on the surface of a large vessel of water, by means of a cork or other float through which the test tube passes. We thus have a vertical circular current capable both of rotation on its vertical axis and of horizontal translation. If we test it by means of a large bar magnet we shall find that it acts as would a thin circular magnetic shell. As could be deduced from Chapter CH. xvin. CURRENTS AND MAGNETIC POLES 3O/ XVII. 2, that face in which (as one stands opposite to it) the current appears to circulate clockwise will act as a south-seeking magnetised surface, while the face in which the current appears to run coimter-clockwise acts as a north-seeking surface. The coil will thus turn until the face of opposite polarity to that of the magnetic pole presented is turned towards this latter, and will then be ' attracted ' and move towards it. If the current be strong and the water be still enough, the coil will under the earth's action only ' set ' with its faces turned north and south respectively. The equivalence of a magnetic shell and an electric circuit was discovered and stated by Ampere. 10. This Equivalence is for the External Field only. There is one obvious difference between the circular current and its ' equivalent' magnetic shell. In the circuit, the field is continuous; and a pole is urged through the circuit as well as up to it. In the magnetic shell the external field stops at the surface of the shell. ii. Principle of Sinuous Currents. Referring to 2, we see it stated as the result of experiment that the element a b acts on the pole as does the element a' b. Extending this to the case of wires of finite length we should predict that a circuit of any form in which the wire has small sinuosities, loops, and other irregularities, will act on an external pole just as does a circuit of the same general form in which the wire is without such irregularities ; provided that the irregularities are of very small size as compared with their distance from the pole acted on. This equivalence of simple and sinuous wires is called the * Principle of sinuous currents. ' Experiment. Employing the 'zero method,' as before, it is easy to show that a circuit, in which the current goes one way round a simple wire, and returns by another wire wound over the former in small sinuosities, has zero action on a pole. This shows that simple and sinuous currents, which have the same general form, are equivalent as regards action on an external pole. Thus, a complete circuit in which the wire has sinuosities that are very small as compared with the size of the circuit, is equi- valent to a magnetic shell of the same average boundary as the circuit and same * strength.' Thus, we may regard the arrangement of the above experiment to be equivalent to two equal shells superimposed on one another, X 2 308 ELECTRICITY CH. XVIII. the faces being turned contrary ways ; such a magnetic arrange- ment giving neutrality. 12. Keaction of a Pole on an Element of Current. We have spoken of the force exerted on a pole by an element of current, assuming the pole to be moveable and the current to be fixed. But by Newton's laws, which may be regarded as so con- firmed by universal experience that they are axioms, there must be between the pole and the element a reciprocal action of which either half becomes manifest as we fix the pole or fix the current respectively. The pole and the element are urged round each other in a direction given by the 'screw' analogy of 2. If we fix the pole, then the current will be urged round it, the rotation following the same direction as before. We can thus give the rule that 1 If we stand in a + pole, so that we are exactly parallel to, and facing, a person swimming with the current in the element (see Ampere's rule, 2), then the element is urged off to our left in a direction perpendicular to the plane containing the element and the .pole ; the force being measured by ' - as in 2.' Of course a pole gives the force in exactly the opposite direction. 13. Action of a Pole upon a Closed Circuit. What fol- lows applies to any closed circuit ; but, since the mechanical reasoning is then simpler, we will take only the case of a plane closed circuit. Let N be the north pole of a magnet, and N.A' A"X a straight line through N ; we suppose the pole to be con- centrated at the point N and to be of strength /*. Let the circuit C be so fixed that it can only turn about the axis N X, swinging round upon the points A' A" as a door swings upon its hinges. We will consider whether or no the pole N has any tendency to make the circuit C revolve as a whole round the axis N X. CH. xvin. CURRENTS AND MAGNETIC POLES 309 Take two lines N a' a" and N b' b", so close together that a' b' and a" b" may be considered to be elements (see 2), and drop perpen- diculars on N X from a" and a'. Since the circuit is closed, each such pair of lines will intercept two elements a b' and a" b", in which the current flows opposite ways. As seen in 2, we may, as regards the action of N, replace a' b' and a" b' f by the intercepts made on circles through b' and b" respec- tively ; these intercepts are represented by dotted lines, and will be called s' and s" respectively, their distances from N being called r' and r". Now the pole N urges a" b" perpendicularly down into the page C s' f with a force measured by "- f ] ; while a' b' is urged upwards from the page with a force '-^- . (r ) But from plane geometry we have the proportion s" : s' = r" : r' ; and hence the force on a" b" : force on a' b' = _L : L. The effect these will have in turning the circuit as a whole about N X as axis depends upon the product of these forces into the arms, a" A" and a' A' respectively. But, again by simple geometry, it is clear that arm a" A" : arm a' A = r' f : r'. Combining this result with that obtained for the two forces, we have that the two moments are equal, being in the proportion of (r" x JL\ : (r' x M or oif I : i. They are also opposite in direc- tion. Hence they neutralise each other. In the same way the whole closed circuit may be divided into pairs. of elements that, together, give zero moment about N X. Hence the whole has zero moment about the axis N X. By similar, but more advanced, reasoning it can be shown that for any closed circuit [whether plane or not] there is zero moment about' any axis through a pole N. By the above method of investigation it is shown that a com- plete circuit has no tendency to revolve as a whole about any axis through the pole of a magnet ; of which axes one particular case is the magnetic axis that joins the N and S poles of the magnet. This means, when interpreted mechanically, that the two poles N and S act on the closed circuits in lines passing through the poles > ELECTRICITY CH. XVIII. since, if they did not act in such lines, we should have a 'moment' about axes through the poles. In other words, the action between the circuit considered as a whole and the poles is a direct action, and not such as that considered in 2. This result, arrived at from the consideration of the action of a pole on the elements of current forming the complete circuit, agrees with the view which regards the circuit as equivalent to a magnetic shell. For the action between a pole and a magnetic shell is also a direct action. The line of action between N and the circuit C will pass through some point in C ; and, if the circuit be moveable about some axis through which the line of action does not pass, the circuit will revolve into a position of equilibrium, or will ' set itself,' as would a magnetic shell. In this position it will then rest, being prevented, owing to the fixed axis, from moving up to the pole. 14. Action of a Pole on an Incomplete Circuit. If there be such an arrangement that the circuit external to the pole be not FIG. i. complete, then this 'incomplete circuit' will be urged round the pole. Every voltaic circuit must be complete ; but part of it may be completed through the magnet itself, leaving the portion external to the magnet incomplete. In fig. i. we see that, as regards rotation round the pole N, the portion P Q of the circuit C is uncompensated, and therefore the whole will be urged round an axis through N. CH. xvrn. CURRENTS AND MAGNETIC POLES 311 In practice this can be effected by arranging a circuit A C B A so that the part C is mobile about the axis A B ; the part A B of the circuit being the axis of the magnet N S. It can be shown that if the points A B lie both between N and S, or both outside N S, the resultant action of the two poles N and S will be such that C does not revolve about the axis. In the case drawn it will so revolve, 15. Action of the Earth's Field on Currents Completely or Partly Mobile. When currents are ai ranged so as to be completely or partly mobile about vertical or horizontal axes, there will be in general movements due to the action of the earth's field. If we resolve the earth's field into its vertical and horizontal com- ponents, the nature of the movements referred to can be predicted without much difficulty, by means of the principles explained in Chapter XIX. 4, 5, and 7. There is, however, no especial interest attaching to these actions, and we shall accordingly omit further discussion of them. 1 6. Actions Between Currents; Ampere's Laws. The first law arrived at by means of direct experiment is very simply ex- pressed. It is as follows. Law I. Parallel currents in the same direction attract, parallel currents in opposite directions repel, one another. There are many pieces of apparatus by means of which this fact can be shown ; the form that is perhaps of the most general use is the ' Ampere's stand. 5 It is not necessary to describe in detail the arrangements in this or in similar pieces of apparatus. It is sufficient to say that, by means of pivots and mercury con- nections, wires bearing currents are given, to a greater or less degree, freedom of movement ; and that thus the action of other currents on them can be observed. Experiment. The figure on the next page represents the portion M of a rectangular coil M N acting upon the portion B of the moveable piece B C. As drawn, the currents in M and in B are in opposite directions, and repulsion between M and B will be observed. When two wires bearing currents are not parallel, but are in- clined to one another, the following law is found to hold (see also Chapter XIX. 8). 312 ELECTRICITY CH. XVIII. Law II. When two currents make an angle with one another they attract one another if they run both towards or both from the vertex of the angle, and repel one another if they run the one toivards and the other from the vertex. FIG. i. Or, two currents crossing one another tend "to move into a posi- tion in which they are parallel and in the same direction. Experiment. This may be illustrated by means of the arrangement here indicated. The |jf[ p current PQ acts mainly on the portion CD of >-!J _ A the mobile current, and movements ensue which are in accordance with the above law. It may be observed that P.Y acts on the portions Y C and C B, and the portion Q Y acts on D Y and on A D, so as to give movement in one and the same direction. The action on the upper part BA will be in a contrary direction, but this action will be negligible if B is relatively very far from P Q. [Law ITT. In a rectilinear current there is repulsion between each two consecutive elements of the current. Ampere devised experiments CH. xviii. CURRENTS AND MAGNETIC POLES 313 in which it appeared that collinear elements of a current repelled one another, as if, indeed, this were an extreme case of Law II., in which the angle between the two currents was 180. It is, however, doubtful whether any experiment has shown directly this action between two elements of currents that are truly collinear. It is indeed true that calculations based upon the assumption of this action have been veri- fied. But the simple statement of this law, as given above, does not fit in well with the modern theory of electro-magnetic and electro- dynamic actions, and, as no direct application of it will be made in this Course, we think it best to pass it by without laying any stress upon it.] For the more modern view of Laws I. and II. we refer the reader to Chapter XIX. 8. 17. Continuous Rotations of Currents. If O A be a current mobile about O as centre, and if P Q be a rectilinear current, the above first two laws show us that there will be continuous rotation of _-- x O A through the posi- tions O A', O A", O A'", &c., round and round. So again, if A B C be a circular current and p m be a current mobile p about p as centre, then from the same laws we can again predict continuous rotation. Experiments have been devised that illustrate these actions. For the more modern view of the same we refer the reader to Chapter XIX. 9- 1 8. Ampere's Laws of the Actions Between Elements of Currents. By means of experiments on finite currents, and by deductions from these aided by arguments based upon the principle of ' symmetry,' Ampere investigated the laws governing the action of 'elements of current ' (see 2) upon one another. These laws once established, he employed them to calculate the actions of circuits upon each other in various cases, the calculations being as a rule long and laborious. In fact he worked out, in a manner which commands our admiration, a very complete system and method of attacking electro-dynamical problems a method based upon the * action at a distance ' between the current elements. Now, however, this method is not employed. Faraday introduced, 3H ELECTRICITY CH. xvm. and Clerk Maxwell further worked out and systematised, the electro- magnetic method of treating all electro-dynamical matters. In this method the currents are treated as magnetic systems giving magnetic fields and acted upon by magnetic fields. This view is far more fruit- ful than is Ampere's, and probably represents more accurately the physical nature of the actions. The views and methods of Ampere were rather those of a mathematician than of a physicist. With respect to his views we must, however, allow that for many purposes, e.g. for predicting the nature of the movements that will ensue in certain cases, the Laws I. and II. given in 16 are very convenient ; and they should certainly be committed to memory by the student, as expressing and summing up a group of experimental, results. CH. XIX, CHAPTER XIX. LAWS OF THE MOVEMENTS OF CURRENTS AS DEDUCED FROM THE CONSIDERATION OF MAGNETIC FIELDS AND POTENTIALS. i. Magnetic Fields and Potentials. In Chapter X. we dis- cussed, with reference to electrostatic phenomena, various matters of importance connected with fields of force, lines of force, poten- tials, equipotential surfaces, and the like. The reader must ob- serve that all the general propositions there given hold good, mutatis mutandis, for any field where the forces that give rise to the field vary as -^. They hold good, e.g., for magnetic fields. Thus we can test and investigate a magnetic field by the magnitude and direction of the force acting on + unit-magnetic-pole in the various parts of the field : and again, we can map out a magnetic field in just the same way as, in Chapter X. 13, 14, we mapped out the electrostatic field. So also magnetic-potential-differences can be measured by the work done on + unit-pole, the work being measured in ergs. And the 'tubes of force' property given in Chapter X. 16 applies also to magnetic tubes of force. 2. Movements are from Higher to Lower Potentials, When a pole, system of poles, or electric circuit equivalent to a magnetic system, is placed in a magnetic field, it will as a rule be urged by forces. By the very definition of ' potential ' // will be urged from a place or position of higher, to one of 'lower, potential ; or it moves from a place or position to which it could have been brought from an infinite distance with the expenditure of more work, into a place or position to which it could be brought from infinity with less work. This, in the case of electrostatics, amounts only to saying that a -f charge moves down, and a charge moves up, the lines of force ; for, in electrostatics, we deal always with ELECTRICITY CH. XTX. charges that are either + or . In the case of magnetic fields we may be dealing with simple + or poles, in theory at least ; and with regard to these the same may be said. But we may be dealing with magnetic shells or with electric circuits that are equi- valent to magnetic shells ; and in such cases the movements may be of both rotation and translation, or of either alone. The italic- ised statement given above is therefore the more general form of the law. To make matters clearer we will consider the case of a field due to a + magnetic pole N. A simple + pole placed in this field will move off to infinity from N, and a simple pole will move from infinity up to N ; in both cases the final position is that to which the + pole could be brought from infinity with least positive (or with most negative) work. A magnetic shell placed in the field will turn until its face is turned towards N, and will then move up to N ; the final position being that into which it could have been brought from infinity with the most negative work pos- sible. If the shell were compelled to slide on rails so as to keep its + face towards N, it would then move off to infinity ; this being, under the assumed conditions of constraint, the position of lowest potential possible. Whether the final position of the pole or magnetic system be one of low positive-, of zero-, or of negative-potential, depends upon the conditions of each particular case. But this final position will always be that from which the system could be restored to its initial position only with the expenditure of the greatest amount of work that was, under the existing conditions as to freedom of movement, &c., possible. 3. Potentials on Poles and on Circuits. Where we have magnetic /?. JU C ( + 4 <2 7T + Oj - Q 2 ). If the pole comes back to the same position, so that Qj = Q g , we have for the work the expression + 4 an p C. In the case of an indefinite rectilinear current, the same formula holds with respect to complete tours round the wire ; the case of in- complete tours will not be noticed here. We regard the indefinite rectilinear current as the edge of an indefinitely large circuit; and the work done in a complete tours of the pole /* will be given by the 1 formula + 4<37TfiC. [Here C is in absolute electro-magnetic units.] n. Potentials on Circuits (continued). So, again, the expression ; C ' /' \ *v *""*-* -*--*'"" / ' ^ FIG. i. . g ; of force are also given, the arrows representing their + direction. It will be seen that inside the solenoid the lines run in a contrary direction to that which they have outside. In fig. ii. is given in section a hollow cylindrical magnet, in like manner. Here, the lines along the inside have the same direction as outside. The outside field has the same general character. But while all the lines of the solenoid run continuously through and round the hollow tube, in the case of the hollow magnet, on the other hand, all the lines run into, and end in-, the solid steel that forms the side of the hollow cylinder. In fact, we must remember that the complete solenoid is equivalent to a magnet, as regards external field, but any longitudinal slip of it taken alone is not equivalent to a magnet ; while each such 328 ELECTRICITY CH. XX. longitudinal slip of the hollow magnet is itself a complete magnet, the hollow cylinder being not a simple whole, but being a com- pound arrangement formed of a system of magnets arranged as are the staves in a barrel. 6. Matter Placed in a Uniform Magnetic Field of Force. Let us consider a uniform field of magnetic force, and a cylinder of any material, whose two ends are plane faces standing perpen- dicular to its axis, so placed as to lie with its axis along the lines of force of the field. In the case of soft iron, or of steel previously unmagnetised, we find that the magnetisation is such that we have evident mag- netism at the two end-surfaces only ; if we neglect, as relatively unimportant, the irregularities that occur at the edges where the molecules are free towards the outside and in contact with other molecules towards the inside. Over these end-surfaces the density p of magnetisation (see Chapter XVIII. 7) will be approximately uniform. There is no experimental reason for supposing but that cylinders of any material are, if sensibly magnetised at all, mag- netised in a similar. manner to the above. The value of this density p depends (i.) upon the field-strength I, and (ii.) upon the nature of the material of which the cylinder is composed. , Now experiment indicates that, so long as the bar is far from saturation (see Chapter I.), then p is directly proportional to the field-strength I. Thus we may write , ...... where k is a quantity depending upon the nature of the material. When the field-strength I = unity, then k is numerically equal to p. CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 329 This quantity k is called the coefficient of magnetisation of that ma- terial, and is measured by the value which t> has when I = unity. Without at present discussing whether the following assump- tions are physically possible (we shall see later that they are), let us assume that k may be a + quantity, zero, or a quantity ; and let us consider what would be the observed condition of the cylinder in the three cases respectively. (I.) Let k = a + quantity. This would make p positive. That is, remembering what is the + direction of the lines of force, we should observe a north-seeking polarity at the end lying furthest down the lines of force, and a south-seeking polarity at the other end. Or this case is the usual one of soft iron or other magnetic matter placed in a field of force. (II.) Let k zero. Here we observe no polarity, since f>= o. That is, the material is one whose presence in the field makes no difference to it (III.) Let k = a quantity. In this case p will have a con- trary sign to that which it had in case (I.)., Or the material would be one in which induction takes place in a contrary direction to that observed in iron ; thus a north-seeking pole of a magnet would, at any rate apparently, induce in a bar of such material a north-seeking pole at the end nearest to the former. In order that we may now use convenient names for different classes of bodies, we shall to some extent forestall what will be discussed more fully later on in this Chapter. We may, therefore, state that there is in the first place a class of bodies for which k is + ; or for which induction takes place down the lines of force, as in the case of iron. Such bodies, of which iron (including steel) is by far the most important, are called magnetic, or more properly paramagnetic. In the case of very pure soft iron, k has a large value ; thus the presence of a bar of such iron in a magnetic field may increase the field-strength near the poles even fifty-fold. There is, secondly, another large class of bodies for which k is ; or for which induction takes place up the lines of force, or in the contrary direction to the above. For such bodies k is very small. For example, if a bar of bismuth be placed in a magnetic field, this field will appear to be slightly weakened near the poles of the bar, in virtue of the opposed induced polarity of the bar j 33O ELECTRICITY CH. xx. but, from the smallness of k, the field as a whole is but very slightly affected. Such bodies are called diamagnetic. (For further discussion see 14 and 15.) 7. Movements of Small Bodies in a Non-Uniform Magnetic Field. Let us now consider a field that is not uniform, and a small body placed in the field. By small body we here mean one so small with respect to the whole field that it can be considered to be all of it in a stronger or weaker part of the field at the same time ; and yet not so small but that one side of it is, in our non- uniform field, in a part of the field of somewhat different strength to that in which the other side finds itself. We will consider what will be its behaviour. (I.) Small magnetic bodies. It can be shown that a small magnetic body, such as an iron pellet, for example, is urged from weaker to stronger parts of the field. Thus, if a small iron pellet be presented to a pole of a magnet (the usual case of a non-uniform field) there will be induced an opposed polarity on the side next to the pole, and a similar polarity on the side more remote. The former will be attracted, and the latter will be repelled, by the pole ; these two polarities are equal in magnitude but opposite in sign. Now the former polarity is in a stronger field than is the latter, and hence attrac tion will predominate, and the pellet will move towards the pole. (II.) Small diamagnetic bodies. For similar reasons a small diamagnetic body is urged from a stronger into a weaker part of the field. 8. The ' Setting' of a Long Body in a Uniform Magnetic Field. Next let us consider the case of a long cylinder placed in a uniform magnetic field, at an angle with the lines of force of the field. We will represent our cylinder as composed of a series of small spheres placed near to one another. This is a convenient representation, and though not an accurate one, will not invali- date the very general results at which we shall arrive. (I.) A magnetic cylinder. Fig. i. represents a cylinder of iron. Each of the little spheres A B C D would, if it stood alone, be mag- netised in the direction of the lines of force of the field. Tnis is represented by the lettering n s in each. There is, however, inductive action between the spheres, each n or s inducing an s or n CH. SOLENOIDS AND ELECTRO-MAGNETS 331 respectively in the nearest portion of the neighbouring sphere. The total result .will be that each little sphere is magnetised, not along the lines of the field, but in a direction represented by n' s of fig. ii., lying between the direction of the field and the direc- tion of the cylinder. Thus each little sphere is acted upon by a couple tending to drag n' s' into the direction of the lines of the field (see Chapter II. 12). Hence there will be a couple acting upon the whole FIG. ii. cylinder, and there will be stable equilibrium' only when this lies along the lines of force. It is to be noticed that in this position the external and internal actions concur to give the maximum magnetisation. When the cylinder is perpendicular to the field there is also equilibrium, but unstable. In this position the internal induction acts against the external, and the magnetisation is at a minimum. We may therefore state that A magnetic cylinder tends to set along the lines of force of a uni- form field ; that is, to assume the position in which its magnetisation is at a maximum. (II.) A diamagnetic cylinder. In this case the external and internal induction will both be the reverse of what it was in case (I.). Thus we must interchange the letters n and s in the above given figures ; and must further remember that each n or s in one sphere induces an n or s respectively in the nearest portion of the neighbouring sphere. The total result will be that each little sphere will be magnetised FIG. iii. 332 ELECTRICITY CH. xx. somewhat as in fig. iii. Hence there will be a couple acting on each sphere : and this will be in such a direction that the whole cylinder will be urged to lie along the lines of force. In this position the internal and external inductions act against one an- other, and the magnetisation is at a minimum. When the cylinder is perpendicular to the field there is un- stable equilibrium, and the magnetisation is at a maximum. Hence a diamagnetic (see 6, end) cylinder tends to set along the lines of a uniform field ; that is, to assume the position in which its magnetisation is minimum. We may, however, add that with diamagnetic bodies the above action is very feeble ; and no observation has as yet detected any ' setting ' of such bodies in a uniform field. 9, A Long Body in a Non-Uniform Field, When a magnetic cylinder is suspended in such a non-uniform field as that between the two poles of a powerful magnet, the results of 7 and 8 concur to show that it will set along the lines running between the two poles. When a diamagnetic cylinder is so suspended, the action de- scribed in 7 would urge it to stand in the weakest part of the field ; i.e. at right angles to the lines of force between the poles. But the action given in 8 would tend to make it set along these lines. In all cases that occur in practice, the former action gives rise to the greater couple ; and so the diamagnetic cylinder stands at right angles to the lines running between the poles. To exhibit these phenomena, very powerful magnets are needed. We therefore proceed to consider how powerful tem- porary magnets may be obtained. 10. Solenoid With, and Without, an Iron Core. According to the view that appears to be most in accord with experiment, magnetic matter possesses innate magnetism. This magnetism is, however, 'molecular ; and we have evident magnetism, or an ex- ternal field, only when the molecules are suitably arranged. We see that, on this view, there is nothing very surprising in a moderate field * producing ' a strong magnet ; for the magnetism is in the soft iron already, and the field may be able to arrange the molecules suitably. We may contrast the magnetic action of an ordinary solenoid CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 333 with that of one in which there is a soft iron core. It is often said 'the core strengthens the solenoid.' But it is perhaps more in accordance with facts to say ' the solenoid renders evident the innate magnetism of the iron.' Experiments. (i. ) We can compare the action of the two on a balanced needle. (ii. ) We can examine the fields of the two respectively by placing them under glass and sprinkling steel filings above. It will be seen that in the case of the solenoid the field is weak, and that some of the lines of force ' leak out ' between the turns of the wire. In the case of the solenoid with iron core, the field is far stronger, and the strength is more concentrated at the poles. The iron is, from its symmetrical position, magnetised in the direction of the lines of force due to the solenoid ; i.e. in the direction of the solenoid's axis. The north pole of the core will be that at the north end of the solenoid, where the current, to one facing the end, appears to run counter-clockwise. 1 1. Electro-Magnets. Thus, when a soft iron core is wrapped round with many turns of wire, and a current is passed through the wire, the core becomes temporarily a magnet. Such magnets are called electro -magnets, and can be made far more powerful, mass for mass, than can any permanent steel magnets. We may regard the external field to be made up of two com- ponents ; the one due to 'evident magnetism ' now evoked in the iron, the other due to the spiral or solenoidal current. These two fields are superimposed upon one another. As long as the iron is far from saturation, the field due to it is approximately propor- tional to the field-strength (see 6) and therefore to the current- strength. But when saturation is reached, any further increase in current will only increase the comparatively insignificant com- ponent field due to the spiral alone. In winding the wire about an iron core, and in passing a current through the wire, it is necessary to have regard to the following considerations. (i.) The wire must not be too thick, or it will not be possible to give a number of turns sufficient for the production of a strong field with a current of reasonable magnitude. (ii.) The wire must not be so thin as to give great resistance and consequent loss of energy in heat. (iii.) It is of little use producing a field-strength greater than that necessary to magnetise the iron nearly to saturation. 334 ELECTRICITY CH. XX. (iv.) The distribution of the wire about the core must be adapted to the shape and dimensions of this latter. The accompanying figure shows one form of electro- magnet. Experiment. It is possible, in the case of a powerful electro- magnet, to trace the lines of force in a very striking manner. Instead of filings we may use small pieces of steel knitting-needles. If the field be powerful, the effect of gravity on these pieces of steel will be relatively insignificant ; and we may cause them to attach them- selves end to end, to one another, and so trace out the lines of force in any direction in space, while with permanent magnets we were able to trace the lines only over a horizontal plane. Note. Field due to an electro- magnet. In calculating the field due to an electro-magnet we can consider the solenoid, and the core which has become a cylindrical magnet, separately. But in general the field due to the former is relatively insignificant, and we need only regard the core. 12. Paramagnetic and Diamagnetic Phenomena. When powerful electro-magnets are employed it is found that all bodies are influenced by the magnetic field (see 6). If pellets of various materials are suspended, by means of a light and long thread, near one of the poles of such a magnet, it is found that certain substances are attracted by the pole, while others are repelled. Those bodies which are attracted are called paramagnetic, or magnetic; such are iron, nickel, cobalt, manganese, platinum, carbon, many salts of magnetic metals, solutions of such salts, and oxygen gas. Those bodies which are repelled are called diamagnetic ; such are bismuth, antimony, zinc, tin, mercury, lead, silver, copper, CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 335 gold, phosphorus, glass, quartz, alum, sulphur, sugar, hydrogen, nitrogen, water, alcohol, and most other liquids and gases not here named. Iron is the most strongly magnetic, and bismuth the most strongly diamagnetic, body known. If we make bars of various substances, those which are mag- netic will set axially, or in a line with the poles ; while those which are diamagnetic will set equatorially, or at right angles to the line joining the poles (see 9). The accompanying figure represents experiments with magnetic and diamagnetic liquids respectively. The liquid is placed in a watch-glass and rests on the poles. When the current is passed, the magnetic liquid B rises up in a heap over each of the two poles ; while the diamagnetic liquid A is repelled into a heap between the two poles. Such effects are very small, and must be magnified by means of re- flected light if they are to be made clear. The difference between the two classes of liquids is clearer if we employ thin glass tubes filled with the one or the other respectively, and observe whether these set axially or equatorially. It must, however, be remembered in this case that the glass itself is diamagnetic ; its action can be allowed for. 13. Pseudo-Diamag-netic Phenomena. In certain cases a bar may set equatorially when its material is magnetic, or axially when its material is diamagnetic, owing to peculiarity of structure. Thus, a bar may be made composed of short steel needles separated from each other, lying side by side, running transverse to the length of the bar. Such an arrangement will, as a whole, set equatorially ; each little needle lying axially. So again, if in a bar of bismuth the crystallisation have a certain direction with respect to the length of the bar, this may set axially. The repul- sion or attraction of pellets from or to a pole is the best way of dividing bodies into the two classes. 14. Relative Magnetism or Diamagnetism, By Archi- medes' principle we know that bodies immersed in any fluid medium appear to have a +, zero, or weight according as they 336 ELECTRICITY CH. XX. displace less than their own, their own, or more than; their own weight of that medium respectively. By this principle we can predict, e.g., whether a body immersed in water will sink, remain where it is, or be forced upwards. Similar reasoning applied to the case of a magnetic field leads us to predict what can be verified by experiment that when a body, whose coefficient of magnetisation with respect to vacuum is k, is immersed in a medium whose coefficient is h, the body will behave as though it were in vacuo and had a coefficient k' equal to k h. If this be true reasoning, then if k> h the body will appear to be magnetic ; if k = h it will be neutral, and if k < h it will appear to be diamagnetic. These predictions have been experimentally tested and verified. Thus, a weaker solution of ferric chloride appears diamagnetic when in the midst of a stronger solution, though in vacuo it is distinctly paramagnetic. 15. Is there Absolute Diamagnetism ? The question naturally arises: 'Is there then such a thing as true diamagnetism, or is it merely that some bodies are less magnetic than that which we call " vacuum " ? ' Some bodies which appear diamagnetic in the magnetic medium oxygen, may very well be found to be magnetic when tested in vacuo. But most diamagnetic bodies (e.g. bismuth) are still diamagnetic in vacuo. It can be shown that all phenomena of repulsion and of equa- torial-setting with which we are acquainted could be accounted for by supposing ' vacuum ' to be a medium slightly magnetic. But the phenomenon referred to in 16, viz. the contrary directions in which a ray of plane polarised light is rotated in different media, seems to imply an essential difference in the sign of k for the two classes of media respectively. Before, therefore, we can accept unreservedly the view that all phenomena come under the head of paramagnetism or relative paramagnetism, it will be necessary to show that such a view can be reconciled with the fact quoted. The whole question is at present unsettled. Quite recently (1886) experiments have been tried tending to show that bodies may change from diamagnetic to paramagnetic behaviour, or vice versa, according to the strength of the field in which they are- CH. XX. SOLENOIDS AND ELECTRO-MAGNETS 337 placed. It has even been suggested that perhaps these qualities are not permanent, but change with time under the action of a field. Curiously enough, Ampere's theory of ' molecular currents ' has been revived as a somewhat fruitful view. 1 6. Rotation of the Plane of Polarisation in a Magnetic Field. In giving some account of certain phenomena that show a remarkable connection between magnetic and electric stresses on the one hand, and radiant energy on the other, we must assume that the reader has some acquaintance with the elements of physical optics. If this is not the case he is advised to read enough of the subject to understand (i.) what is meant by a ray of light or of other radiant energy ; (ii.) what is meant by a plane polarised ray, and by the plane of polarisation \ (iii.) what a Nicols prism is, and how it is used to obtain a plane polarised ray; (iv.) how a second NicOl's prism can be used as an analyser to detect whether a ray is plane polarised, and whether the plane of polarisa- tion has been rotated or has changed its 'azimuth.' Now it is found that if a plane polarised ray be passed through .a transparent medium that ordinarily has no power to rotate the FIG. i. plane, and if this medium be placed in a powerful magnetic field, then the plane of polarisation of the ray is in general slightly rotated in its passage through the medium. It was Faraday who discovered this. The accompanying figure indicates how the z 338 ELECTRICITY CH. xx. ' electro-magnetic rotation of a plane polarised ray ' may be de- monstrated. If more exact results are desired the necessary experimental arrangements are less simple. M and N are powerful electro-magnets, provided with hollow iron cores, their unlike poles being opposed so as to give a powerful field at c. Here is placed the piece of glass or other transparent body, whose rotating powers, when subject to a powerful magnetic field, are to be observed. At b is \\\Q polarising Nicol, and at a the analyser. If (as we here suppose) the body at c has no innate rotating power as has (e.g.) quartz, then before the current passes the plane of polarisation of the ray is not rotated ; and no light can be seen through a when the principal planes of the polariser b and of the analyser a are at right angles to one another. But when the current passes so that a powerful field is produced at c, then some light is received through a, and we must turn the analyser a about the line a b as axis, in order to recover the initial darkness. When monochromatic light is used we can then readily measure the amount and direction of the rotation by the amount and direction of the rotation that has been given to a. If the light be not mono- chromatic, e.g. if it be white light, we can never recover the white light again ; this showing that the rays of different wave-lengths have been rotated to a different amount. By experiments that we have not space enough to give here, Verdet and others established several results. These results will be FIG. H. made easier to understand by the simple diagram here given. The dotted lines represent a magnetic field as usual ; A and B are two points in it at which the magnetic potentials are V A and V B respec- tively, the former being [in the case represented] at the lower potential ; A B or B A is the direction of the ray ; /3 is the angle that this direction makes with the lines of force of the field ; I is the field-strength ; / is the CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 339 distance in the medium traversed by the ray ; and 6 is the angle through which the plane of polarisation is rotated. It was found that (i.) 6 depended upon the field-strength, being directly proportional to it. (ii.) 6 is directly proportional to cos ft. (iii.) 6 is directly proportional to /. (iv.) 6 depends upon the wave-length of the radiation in question. (v.) The sign of 0, or the direction of rotation, is not the same in all media. If we call the rotation positive when it is as a right-handed screw would rotate as it advanced down the lines of force, then we may say that in vacuo there is zero rotation (as far as experiment indicates), in diamagnetic media the rotation is + , and in magnetic media (such as a solution of ferric chloride) it is . (vi.) This rotating power seems to be innate in the molecules pro- vided that they are in a magnetic field. A substance preserves its power when in solution or when mixed with other bodies, and so we can with certainty prepare mixtures of the two classes of solutions which shall give zero rotation. (vii.) The direction and magnitude of rotation are the same whether the ray pass from A to B, or in the reverse direction. Hence the rotation can be multiplied by causing the ray to be reflected to and fro many times up and down the field, this being equivalent to multiplying /. It would not be practicable to make / great in any direct manner, if we wished still to preserve a strong field. (viii.) With reference to (v.) we may add that all wave-lengths have the same direction of rotation in the same medium. Theory suggested by the above. The above phenomena suggest very forcibly that in a magnetic field the ether may be eddying about the lines of force. For, were this so, the wave would naturally be rotated in azimuth during its passage. Remembering how the lines of force pass down the axis of a solenoid or spiral, it would seem as though a spiral current caused the ether to eddy about the axis of the spiral. It has even been suggested that perhaps electricity is itself ether, so that the eddy is produced by a spiral current of ether through stagnant ether. At present, however, there is in this direction little but conjecture. Some day, no doubt, electro-optical phenomena such as the above will lead to the establishment of very important results as to the real nature of magnetic, electrical, and radiant phenomena. We may observe that the opposite directions of rotation in different media indicate a real difference between the two classes of bodies, or tend to show that diamagnetism is not merely a relative phenomenon. Z 2 340 ELECTRICITY CH. xx. 17. Other Electro-Optical Phenomena. There are other phenomena that show how the propagation of a wave of radiant energy is affected by the magnetic or electric condi- tion of the medium through which the ray passes. Kerr discovered that when a plane polarised ray is reflected from the polished surface of the pole of a magnet, the plane of polarisation is rotated. He believes that this is due, at any rate in part, to the action at the reflecting surface, and is not simply due to the air or other gas that fills the magnetic field. He also discovered that when a simply refracting transparent dielectric is subjected to electric stress, as e.g. is the glass of a charged Leyden jar, it becomes doubly refracting. The effect was on the whole similar to that which was produced if it was subjected to com- pression along the direction of the lines of force. It seemed, in fact, as if there was tension along the lines of electric force, tending to compress the glass. 18, The Electro-Magnetic Theory of light, The genius of Clerk Maxwell founded a theory which should con- nect all these phenomena that are usually classed under the heads of magnetism, electrostatics, and electrodynamics, on the one hand, with the phenomena of light, heat, and of radiant energy in general on the other. But before this theory could grow into a complete structure the author of it died, and the work has never been completed. In what follows we have tried to indicate in an elementary manner the nature of the theory, and also to show how much uncertainty still exists with respect to it. We have made use of the suggestions put forward by Dr. Lodge, Professor S. P. Thompson, and others, as such suggestions show the directions in which Clerk Maxwell's funda- mental views tend'. (a) The ether. By the use of the expression the ether is implied a belief, now general among men of science, in a universal medium per- vading all space and penetrating solids and liquids. The ether is not matter, but possibly all atoms of matter are merely indestructible vortex-rings of ether, so that this latter is the parent of matter. Again, the ether is not energy ; but yet it is only through the ether that energy can be transmitted from one group of matter to another. The ether penetrates all bodies, but it may be that in some cases the matter has, to a greater or lesser degree, a ' hold ' upon the ether that pervades its intermolecular spaces, and gives to it something of the properties of a rigid solid, while in other cases the ether may be lett quite free. CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 34! (3) Radiant energy ; heat and light. We may here remind the reader that the transmission of heat and light is believed to be effected by means of waves propagated in the ether ; the vibrations being at right angles to the direction of propagation, somewhat as in the case of a shaken rope. Further, it depends upon the wave-length of the radiation whether it give the impression of light when received upon the retina, or in what respect its action is most remarkable. Heat and light are partial expressions, radiant energy is the more general term for transmissions of all wave-lengths. (y) Nature of electricity. It has been suggested that if anything can rightly be called 'electricity,' this must be the ether itself; and that all electrical and magnetic phenomena are simply due to changes, strains, and motions in the ether. Perhaps negative electrification (as we believe has been suggested) means an excess of ether, and positive electrification a defect of ether, as compared with the normal density. (8) Electrostatic phenomena. It is possible that all electrostatic phenomena follow from strains in the ether, These strains (or defor- mations) can only occur where the ether possesses some degree of rigidity. On this view then it would appear that dielectrics are bodies which have what we may style so much 'hold' upon the ether that pervades their intermolecular interstices, that this is given, to a certain degree, the properties of an elastic solid. Thus the phenomena of the stored-up energy of a charged Leyden jar may imply that the ether pervading the glass, and probably the glass itself, is in a state of elastic strain analogous to that of a bent spring. Co?iductors, on the other hand, would be bodies in which the ether is free and, to speak from analogy, fluid, or in which it is incapable of retaining a strain. (e) Dielectrics are transparent, conductors are opaque. Now from mechanical principles we should say that through bodies in which the ether has some rigidity, waves of transverse vibration can be propa- gated ; while through those in which the ether is free and fluid, only waves of longitudinal vibrations can be propagated. From (#) and (8) we should therefore predict that dielectrics would be transparent to the transverse waves of radiant energy (i.e. to rays such as those of heat and light), while conductors would be opaque to the same. If we take into account such disturbing influences as those of internal and irregular reflexion, it can be fairly said that dielectrics are, as a rule, transparent ; while the conducting metals are certainly the most opaque of bodies. Here, then, is one fact that tends to confirm directly the theory. (f) Electric currents. There are reasons that would lead us to assume that when ' an electric current flows ' through a conductor, there is either a direct translation of the ether along the conductor, or 342 ELECTRICITY CH. xx, a propagation of longitudinal vibrations similar to those of sound in air. This view is consistent with the hypothesis that in conductors the ether is free and not rigid, and therefore not capable of trans- mitting waves of transverse vibrations. (77) Electro-motive force. We might then suppose that electro- motive force may be a kind of ' ether pressure ' due to un-uniformity of distribution, or to be, at least, of a nature analogous to what is sug- gested by such a term. (&) Electro-magnetic induction. Let us suppose a ' current ' to be sent along a wire, this wire being surrounded by a dielectric such as air. Now we shall see in Chapter XXI. that when any change in current-strength occurs, all neighbouring conductors are affected ; temporary currents being ' induced ' in them. Since the possibility of action at a distance is now denied, we must suppose that this action is due to some kind of transmission of energy through the dielectric ; and it is most consistent with the views expressed above, and with the observed facts of induction, to imagine that something of the nature of a wave in which the vibrations are transverse is propagated at right- angles to the wire carrying the current. If this is the case we should expect to find the velocity of propagation of electro-magnetic inductive action to be the same as the (average) velocity of radiant energy, since both are supposed to be waves of transverse vibrations, propagated in the ether. Now this is actually the case ; many measurements having shown that the rate at which electro-magnetic inductive action is pro- pagated is nearly the same as the mean velocity of light. In this agreement we have what is probably the strongest corroboration of the theory that we are discussing. (i) There is another piece of confirmatory evidence in the relation that certainly exists between the specific inductive capacity of a dielectric and its refractive index. (K) Lines of magnetic force. In 16 we pointed out how the phenomena there described suggested to us that lines of magnetic force are in some way axes of eddies formed in the ether. Here we may add that the eddy may be one in which the ether moves continu- ously round, or may be formed of circular vibrations in which there is no final displacement of the ether. Considering the wire bearing a current, the circular lines of mag- netic force about it, and the radiating lines in which electro-magnetic inductive action is propagated, we may say that probably the follow- ing relation holds. ' When there is a discharge of electricity in one direction the lines of electro-magnetic induction are at right-angles to this direction and radiate from it, while the lines of magnetic force are at right-angles to both these directions.' 343 CHAPTER XXI. ELECTRO-MAGNETIC INDUCTION. i. General Account of Induction Phenomena. Faraday, discovered that when a conductor moves in a magnetic field, or when the field in which a conductor is situated is caused to vary in strength, then there is in general a current induced in the con- ductor, this current lasting only as long as the movement or variation lasts, and ceasing when conditions are again stationary. Since a current implies an electro-motive force, we may say that such movements or changes give rise to induced E.M.F.s in the conductors. The ejection of the E.M.F., and of the conse- quent current, depends upon the nature of the movement or change. If there be an E.M.F. (and current) in the conductor already, the induced E.M.F. if in one direction will be added to the initial E.M.F., and in the other direction will decrease, reduce to zero, or even overpower and 'reverse' the initial E.M.F., according to the relative strengths of the two. It is soon noticed that the cases in which an E.M.F. is in- duced in a conductor are mainly as follows. (a) When a conductor cuts across the lines of force of a magnetic field. If a complete circuit so cut across the lines, it may happen that the E.M.F.s induced in the opposite sides of the circuit are equal and opposed, giving a zero resultant E.M.F. in the complete circuit. (I)) When a magnetic field is created in the midst of a cir- cuit, or is/ caused to cease. This case is shown in Experiment (i.). (c} When the field enclosed by a circuit is varied in strength. This case is shown in Experiment (iii.). The changes in question can be brought about in various ways : by the continuous movement of a simple rectilinear wire across a magnetic field ; by a movement of translation of a circuit 344 ELECTRICITY CH. XXI. across a non-uniform field ; by rotation of a circuit in any field ; by making and unmaking an electro-magnet (or solenoid) in the midst of a circuit ; By the approach of a current, solenoid, or magnet towards a circuit, or its withdrawal from the same. And in many other ways. In the following experiments, which are here given without further comment, we use coils instead of simple circuits in order to obtain more remarkable results. Experiment. (i.) A coil is made of a quantity of fairly stout insulated wire, and the ends of this wire are connected with the terminals c and d, so that a current can be sent through it when desired. This coil A is concealed in fig. i., but is seen in fig. ii. A second coil B is made ; it is hollow and encloses the coil A, but is entirely insulated from it. The ends of this latter coil are connected with the terminals a and b. A galvanometer is also connected FIG wit'h these terminals, so that any current in B will be detected. The effects are more marked if the wire of which B is made be very long. It, therefore, must be of fine wire, to admit of great length and many turns within a reason- able compass. The coil A is called the primary, and the coil B the secondary. When a current is sent through the primary, the galvanometer indicates a sudden current in the secondary ; this current lasting only until the primary current is established and steady. When the current is broken again we observe another current induced in B, this time in the other direction. Thus the creation or destruction of a field within B produces currents in B, the currents being in opposite directions respectively. There is no current in B, while the current through A is steady ; that is, while the field within B is constant. (ii. ) In the next figure the arrangements and connections are the same, but now A can be thrust into B or withdrawn from B. It is found that by thrusting A, while a current is flowing in it, into B, we induce in this latter a current in the same direction as would have been induced had we left A within B and had then sent the current through A. So also the withdrawal of A CH. XXI ELECTRO-MAGNETIC INDUCTION 345 answers to stopping the current in it. We find, also, that the currents induced are more or less violent according as we move A more or less rapidly respectively. FIG. ii. (iii.) Similar effects are produced if we thrust a magnet into B, or with- draw it again. We should expect this, having seen that a coil bearing a current acts as a weak magnet As regards the relation of the direction of the induced cur- rents to the direction of the primary currents, or to the sign of the 346 ELECTRICITY CH. XXI. magnetic pole which is presented to the coil B, enough will be said later on. 2. General Keason for ' Induced Currents.' The actual reason why such movements or changes in field-strength as those described should have the effect of inducing E.M.F.s in the con- ductors concerned, cannot be said to be truly known. For it must depend upon the real nature of electric currents and of magnetic fields, and upon their relation to 'the medium that pervades all space.* But we have certain great laws established by experiment, which have included and bound together in one theory all the phenbmena that have been noticed up to the present point. If we can show that these new phenomena of ' induction ' come also under these laws and could be predicted from them, we shall in a sense have shown the * reason for ' induction. The laws to which we refer are Conservation of energy, Faraday's laws (Chapter XII. 7), andfoule's law (Chapter XV. 4). Let us consider the case of a circuit A which includes a battery-cell, opposite to which is a magnetic pole N. (Instead of a single pole we might consider the one end of a very long magnetic bar whose other end is far away and need not be considered.) This circuit will attract or repel the pole N, according to their relative position ; since the circuit acts as a magnetic shell, and produces a magnetic field. If it attracts N, then work can be done by N as it moves up towards A ; and this work comes out of the system of the two bodies A and N, so that there must be an equiva- lent of energy lost from this system by ' Conservation of energy' How is this energy supplied ? If the two bodies A and N were a gravitation system, i.e. merely two masses attracting one another by the action of gravi- tation alone, we know that we should have lost an equivalent of potential energy ; that is, we could not restore N to its initial position save by doing upon it just as much work as we previously got out of it. In moving towards A it moves with the lines of force, in moving from A it moves against them. But in our electro -magnetic system of the circuit A and the pole N matters are very different. We can, without any work, make or break the current, i.e. create or destroy the magnetic field at CH. xxi. ELECTRO-MAGNETIC INDUCTION 347 will. Hence we could turn on the current when we are pulling N towards A, and thus get work done ; and could turn it off while restoring N to its initial position, and thus require to do no work on N. Thus, in a gravitation system the law of conservation of energy demands only that we consider the change in relative positions of the bodies, or ' the configuration of the system ' ; while in an electro-magnetic system we must, if the law is to hold, seek for some source of energy other than that depending on relative positions ; for we have shown how we could get un- limited work done and yet end up in the initial positions. We are driven to conclude then that the energy must come out of the circuit A, and shall be able (see 3) to see how the re- quisite energy can be supplied if we suppose E.M.F.s to be in- duced in the circuit A whenever work is being done on or by the pole N ; the E.M.F.s being in opposite directions in the two cases respectively. 3. More Exact Reasoning, in a Simple Case. Let us con- sider a vertical rectilinear current A, and a single pole N pivoted so as to be capable of movement about A. (We cannot in practice have a single pole, but we can contrive an arrangement in which a current acts on one pole only of a magnet, so as to produce continuous rotation.) When the current passes, the pole will be urged round and round the wire A continuously. Hence we can have unlimited work done, while the pole continually returns to its initial position. As argued in' the last section, there must be an equivalent of energy lost from the circuit of which A is part. We will employ a notation similar to that used in Chapter XV. Let E be the E.M.F. of the battery in the circuit ; R the total resistance of the circuit ; C the current that flows when the pole N is stationary, i.e. when no work is being done external to the circuit ; C the current when the pole N is moving, i.e., when external work is being done. [We here use absolute units.] (In order to understand better what follows, the student is recommended to read Chapter XV. 3, 4, 8, and 9 again.) Initially, i.e. before the pole N begins to move and to do work, Tf we have a current C = , energy lost in the cell at the rate of ix E C per second, and an exact equivalent of heat energy appear- ing in the circuit at the rate of C 2 R or E C per second 348 ELECTRICITY CH. xxi. Now let the pole move and external work be done. How can the circuit supply the necessary energy ? At first sight it seems simplest to suppose that C remains unaltered, while less heat is evolved ; the diminution of heat evolved per second being the exact equivalent of the mechanical work per second done by the pole. In this way the current C would be unaffected, only it would give us less heat with an equivalent of mechanical work. Such an hypothesis would save the law of ' Conservation of energy ' from being broken ; but it is neither in accordance with experiment, nor with Joule's law which is supported by experi- ment. For, by Joule's law, as long as the current is C so long E is there heat C 2 R given out per second. And since C = -n> this heat is equivalent to the whole energy E C given out by the battery per second ; so that there is none left for the external work. Hence, when the pole moves we conclude that the current cannot remain the same. Let us suppose, then, that it decreases and becomes C instead of C ; and let us see if such an alteration will give us the energy needed for the external work done in moving -the pole. The total activity of the cell is now E C, and the heat activity is C 2 R. Since E = C R, and since C is less than C , it follows that C 2 R is less than EC; or we have less heat activity evolved than is the equivalent of the cell's activity. Thus, if the current decrease from C to C while E remains the same, we have a balance of activity left to account for the mechanical work done on the pole. Now, since the battery and its E.M.F. E remain unaltered, and since R is itself unaltered also, it follows from Ohm's law that the current cannot fall from C to C unless an E.M.F. e arise contrary to E. If such an E.M.F. e be supposed to arise, we have (see Chapter XV. 9). E C = energy per second expended by battery. T7* T7* C = ^ , and is less than C which = -^. J R R | C 2 R, or (E e] C, = energy per second appearing as heat. E C C 2 R, or e C, = energy per second available to do the external work on the pole. CH. xxi. ELECTRO-MAGNETIC INDUCTION 349 The reader will notice at once the close resemblance of the above distribution of energy to that which occurs when a battery is employed to electrolyse any body, i.e. to do chemical work. In both cases there is a reverse E.M.F. e ; and we have total energy is E C, heat energy is (E e) C, and energy expended upon the chemical or mechanical work is e C. (All these reckoned per second.) If the pole N be caused to revolve in the other direction, we must do external work upon it, since we move it against the lines of force. Thus we keep doing work on the system without gain- ing energy of position. So we conclude that we must in some way be giving energy to the circuit. But we cannot, by Joule's law, give heat to the circuit so long as C remains constant. As before, then, we conclude that C must alter ; and, this time, must increase to some value C'. This implies an E.M.F. d in the same direction as E. And we have energy of battery E C', heat energy (E + e') C f , energy given to the circuit from outside e' C (all reckoned per second). In fact, we merely change the sign of the induced E.M.F. Thus we have shown how, by reasoning founded upon the laws quoted above, we should predict that E.M.F.s would be induced in conductors whenever movements or changes are made in which + or work is done by the system. The case above was chosen for its simplicity ; but the same reasoning applies to all cases. The main facts of 'induction' were dis- covered and investigated by experimental methods. But it is satisfactory to see how readily they fall into place under the same few great laws as are followed by all the electrical phenomena with which we were previously acquainted. 4. General Expression for Induced E.M.F. More generally, let there be any system of currents and poles, or of currents alone. Let + or work be done by movements occurring in this 'electro-magnetic system' ; the work being due to the existence of the currents. Then the same arguments as those given in 2 and 3 show us .:... .... . . .,-> . *. . . (i.) That the equivalent of this + or work is not to be sought iri, the change in potential energy due to the changed relative positions of the bodies forming the system, as would be 350 ELECTRICITY CH. xxi. the case with a gravitation system, but is found as a diminution of or addition to the electric energy of the system. (ii.) That, in consequence of Faraday's and Joule's law, this loss or gain of the electrical energy is possible only if an E.M.F. be induced in one or more of the circuits. (iii.) That if this induced E.M.F. be , and there be a current C in the circuit, then the loss or gain of electrical energy per second, answering to the work done per second in consequence of the movements referred to, will be measured by e C. In what has preceded we have for simplicity spoken only of cases where the work done per second is constant. When this is the case the induced E.M.F. e will be constant, as will also the current C which = - -_ (see 3) ; and we have a ix constant gain or loss of electrical energy C e per second, the equivalent of the work done per second. In the case of a circuit moved in a magnetic field, the work done between any two positions is measured in ergs by C (N, N 2 ), as explained in Chapter XIX. 3 ; the symbols having the meanings there given to them. Now let (n\ 2 ) be the change per second in the number of marked lines of force piercing the circuit. Then the work per second due to the movement is measured by C (n { n 2 ). But we have seen that the equivalent of this is e C. Whence it follows that .............. Or, the E.M.F. induced in a simple circuit is measured by the change per second in the number of marked lines of force piercing the circuit. Here e is in absolute units of E.M.F., not in volts. (In fact, whenever we do not state to the contrary the reader may take it for granted that in this Chapter absolute C.G.S. units are intended.) Note. We must remark that where a pole of strength ju. threads through a circuit and round again any number of times, returning to its initial position, the above formula is not sufficient by itself. This was shown in Chapter XIX. |. Thus for each complete circle made about the circuit by the pole /* we must consider 4 IT /j. marked lines of force to have been added to those piercing the circuit (see Chapter XIX. Ii). As a particular case, when the pole n CH. xxi. ELECTRO-MAGNETIC INDUCTION 351 revolves a times per second about a rectilinear wire, it is as though there had been added 4 a IT /i marked lines per second to those piercing the infinite plane circuit of which we may consider the wire to form the edge. In this case, therefore, we have e = 4 a IT ^, this being the form of the ' general expression ' that applies to this particular case. If the circuit be made of m turns of wire, all of them pierced by the lines of force, then the E.M.F.s induced in them all severally are added, and give a resultant E.M.F. in the circuit that is m times as great as that in one turn. Hence if a change (N\ N 2 ) in the number of marked lines piercing the coil occur in / seconds of time (where / may be a small fraction), we have . . . , , Resultant E.M.F. e= e ; zero work as long as E = e and C was zero ; and positive work e C when C had changed sign in consequence of e being now > E. [All reckoned per second.] Experiment fully confirms this conjecture, and proves the formula e = (n\ -~ n^ to be universally true for conducting cir- cuits ; e being added to or subtracted from the initial E.M.F. E if there be such, or appearing alone if there be no initial E. This occurrence of induced E.M.F.s in all cases, whatever be the initial E.M.F. E, must be regarded as a phenomenon due to the ' nature of things ' ; it could not have been predicted from the laws of ' Conservation of energy,' &c., alone. Note. In the simple case of 3, where a pole revolves about a wire that now we will suppose to bear no current, there will be an induced E.M.F. measured by 4 air ^ ; the symbols having the same meaning as in 4, note. 6. Direction of the Induced Currents ; Lenz's Law. We have stated that the direction of the induced E.M.F.s depends upon the nature of the movement. It is not hard to see how we can, from the law of 'Conservation of energy,' predict this direc- tion. . There are three main cases to be considered, (I.) Where the -electro-magnetic system does positive mechanical work. Under this head come all cases in which magnets, or wires bearing currents, are caused to move in consequence of electro- magnetic attractions or repulsions. We will take as a typical case that of a pole urged (by the electro-magnetic field) round a wire carrying a current ; the Case, in fact, of 3. Here there is mechanical work done upon the pole, since this latter is either CH. xxi. ELECTRO-MAGNETIC INDUCTION 353 gaining kinetic energy or perhaps is set to work some machine at a constant speed. According to the argument of 3, there is an induced E.M.F. e opposed to E. That is, there is an E.M.F. induced in such a direction as would alone give a current in the opposite direction to that which is actually flowing, and which is causing the pole to move. We may say then that the induced E.M.F. opposes the movement that induces it. In all similar cases where there is work done by the system and electrical energy expended, the induced E.M.F. e is opposed to the initial E.M.F. E ; and would, if acting alone, drive a current opposed to that actually running. But in electro-dynamics, when the current changes its direction, then attractions and repulsions are interchanged. Hence, in all cases coming under this head, the induced E.M.F. opposes the inducing movement. If, in the simple case considered, the pole be absolutely free to move, then it will move with increasing velocity until the in- duced E.M.F. , which (see 4, note) is measured by 4^77/1, becomes equal to E. As e approaches E in value the current C becomes smaller : since C = ^-^. Hence the force acting on the pole be- K. comes smaller and smaller. For this reason it will take an infinite time for e to become actually equal to E, though it may not take long for it to become nearly equal. Supposing it to have become exactly equal, then the current ceases ; there is no energy expended by the battery, and, since the pole is by hypothesis perfectly * free ' to move, it will continue revolving with constant velocity, and there will be no work done. Things, therefore, will have come to a standstill, from the conservation of energy point of view, but there is all the time an electro-magnetic inductive action, since there is an induced E.M.F. e equal to E. (II.) Where positive mechanical work is done by an external agent upon the electro -magnetic system. Where we alter the rela- tive positions of poles or circuits so as to do positive work against the electro-magnetic forces, there must be given to the electro- magnetic system electrical energy equivalent to the work done. This can only be, as was argued in 3, by the induced E.M.F. e being in the same direction as the original E.M.F. E. Hence, the induced E.M.F. tends to drive a current in the same direction as that originally flowing. But it is owing to this original current A A 354 ELECTRICITY CH. XXI. that we are (by hypothesis) doing work against forces that oppose the movement. Hence the induced E.M.F. is in such a direction as to oppose the inducing movement. Thus in the simple case of 3, if we whirl a pole about a wire bearing a current, in the direction opposed to that in which it is urged by the field due to the current, we do mechanical work per second on the system ; and this appears as electrical energy eC gained per second by the circuit. This induced E.M.F. e adds to the current which is opposing the movement of the pole. There is now given out in the circuit per second heat to the amount of EC due to the battery, and also of eC due to the work done on the system from outside. We feel resistance in moving the pole ; and we can regard e C as a kind of equivalent ' heat of friction ' given out per second in the circuit. (III.) Where there was originally no current. We have seen in 5 how there are E.M F.s induced by movements even when there are initially no currents. Considering the simple case of 3, it 'is easy to see in \vhat direction the induced E.M.F. must be. Let us whirl the pole about the wire, and let the induced E.M.F. , and current arising from it, be in such a direction as to oppose the movement. We then do work on the system, and gain electrical energy in the system ; or conservation of energy is maintained. But if we were to suppose the E.M.F. and current to be induced in the opposite direction, i.e. so as to urge the pole in the direction in which it is moving, we should then have elec- trical energy both giving out heat in the circuit and doing external work in urging the pole, and we should have created this energy out of nothing ; since now no work is being done on the system. And this is against conservation of energy. Hence we conclude that the induced E.M.F. is such as to oppose the inducing movement. From considerations such as those given above we should predict that -... The currents which are induced in consequence of movements or other changes in an electro-magnetic system are invariably in such a direction that they tend to oppose these movements or changes. This law was observed experimentally and enunciated by Lenz. It is CH. xxi. ELECTRO-MAGNETIC INDUCTION 355 called Lenz's law. The experiments that illustrate this law are endless in number. We give here only a few. Experiments to illustrate Lenz's /aw. In experiment (I.) of I we can, without much difficulty, show that when the current in the primary A is started there is induced a current in the secondary B in the opposite direction ; while, when the current in A is broken, there is induced in B a current in the same direction. In both cases, therefore, the induced current opposes the change, for it will repel that in the primary while t,he latter is rising from zero to a maximum, and will attract it while falling from a maximum to zero. Or, again, the creation of a current in A has the effect of thrusting lines of force into B in one or other direction, and the inverse induced current in B opposes this by thrusting out lines in the opposite direction, while, when the current in A ceases and the lines are withdrawn, the direct induced current in B creates lines in the same direction, tending (as it were) to pull back those due to A. (ii. ) So, again, it can be shown that in experiments (ii. ) and (iii. ) of I, the induced current in B is such that the coil A or the magnet is repelled by B when advanced, and attracted by it when withdrawn, respectively. Regarding the coils as solenoids, and remembering the rule of Chapter XX. 3, it is easy to verify this statement. Another way of stating the observed result is to say that when more lines are thrust into the circuit B, the current induced is such as to thrust opposed lines out ; while, when lines are withdrawn, the current induced opposes the withdrawal by creating lines in the same direction. (iii.) A circuit is arranged comprising a battery, resistances for regulating the current, a single rectilinear wire B, and a delicate galvanometer. When a wire A bearing a current is brought up to B, the decrease or increase in the deflexion of the galvanometer will indicate a current induced in B in the oppo- site direction to that flowing in A ; when A is withdrawn the galvanometer's increase or decrease in deflexion indicates an induced current in the same direction as in A. So, by Ampere's laws, the induced current is such as to oppose the movement. (iv. ) With the same apparatus we may rapidly pass a powerful magnet pole across the wire B, without touching it. The current induced can be shown to be such as tends, according to the law given in Chapter XVIII. 2 and 12, and Chapter XIX. 7, to oppose the inducing movement. 7. Constant Induced Currents. The question as to how we can obtain a constant induced current is of great importance. We shall, later on, discuss those machines by means of which we can induce currents for the purpose of electric lighting, &c. ; and we shall show how a practical constancy in the current induced can be obtained. In this place we propose only to show how an absolutely con- stant induced current could theoretically be obtained. A A 2 356 ELECTRICITY CH. xxi. (L) By revolution of a pole about a wire. When a pole n re- volves with constant velocity a times per second about a wire, the induced E.M.F. e is, as we have seen, constantly equal to 4 # TT p. An arrangement in which the wire revolves with uniform velocity about a pole, equivalent to the converse movement, is practicable, but is not a very serviceable machine for obtaining induced cur- rents. Since the resistance R of the wire is constant, it follows that the current is constant, being equal to ----, or ~ 7r ^. lx Jx (II.) By the rail-and- slider arrangement. In the figure ARE represents a conductor of resistance R, including a galvanometer. A C and B D are thick strips of copper of no resistance that is appreciable. C D is a cross piece, also of no resistance, sliding in perfect contact with the two strips. The dots represent a uni- form magnetic field in which we suppose the lines to run from above perpendicularly down into the plane of the paper, i.e. into the plane of the apparatus. If the slider move with uniform velocity in the direction of the arrow, we add to the circuit A C D B A a constant number of marked lines of force per second; say a numbers. Then the induced E.M.F. is measured by e = n ; and, since we suppose the resistance to lie in the part A R B only, the induced current is equal to ^ or -**- ix K. and is constant. In the case given it will be such as to oppose the introduction of more- lines down into the circuit. That is, it will be such as to thrust lines up out of the circuit ; i.e. the circuit will have a N polarity presented upwards, or the current will flow in the direction C A B D, If, other things remaining the same, we incline the field so that ihe lines make an angle 6 with the perpendicular to the CH. xxi. ELECTRO-MAGNETIC INDUCTION 357 plane of the circuit, it is clear that now we add only ncosO marked lines per second to the circuit. Resistance expressed as a velocity. We have seen in Chapter XVI 1 1. 5 that, in the electro-magnetic system, R is of the dimensions of a velocity. It happens that the above simple apparatus enables us to show how R can in fact be measured by a velocity. (Compare also Chapter XVIII. 4, note on method (ii.) of determining resistance.) Let us assume for simplicity that the other quantities concerned are each unity ; i.e. that the field is of unit strength and is perpendi- cular to the plane of the apparatus, so that we have one line of force piercing each i sq. cm. ; that the rail is of i cm. length, and that the current is of unit strength. The rail must move with such a velocity that, in the formula given above, n may be numerically equal to R. This means that we must add R sq. cms. to the area of the circuit each second ; or, since the rail is i cm. long, this must move with a velocity expressed by the number R, or with R cms. per second. Hence, under the above ' unit conditions ' we can measure R by the velocity of the rail required to give unit current. The velocity that would thus measure the absolute unit of resistance is that of i cm. per second, while io 9 cms. per second measures i ohm. 8. Changes that give Induced Currents. In considering the question as to whether in any particular case there will be an induced current, we have to remember two facts. (a) That when a conductor cuts across lines of force there is always an induced E.M.F. (If) That when there is a change in the number of lines piercing a circuit there is a resultant E.M.F. induced in the circuit. We will consider several cases. (I.) A circuit moving in a uniform field. If the circuit move parallel to itself (in such a way that the direction of movement causes the circuit to cut the lines of force), the number of marked lines embraced will not alter ; and there will be no resultant E.M.F., or current, induced. The two sides of the circuit, it is true, do cut the lines ; and so, by principle (a), there is an E.M.F. induced in each side. But these E.M.F.s are equal and opposed. There is no resultant E.M.F. in the circuit, but the top and bottom of the circuit, where the wire does not cut the lines of force, will be maintained at different potentials, as could be demonstrated by connecting them with a quadrant electrometer. 35^ ELECTRICITY CH. xxi. If, however, there be any movement of rotation, then the number of marked lines embraced is altered, and there is an E.M.F. induced. It is, in fact, a very usual way of obtaining induced currents to rotate a circuit or coil in a field that is more or less uniform. (II.) Movement of a circuit in a non-uniform field. Where the field is not uniform, movements of a circuit will in general produce a change in the number of marked lines embraced, and so there will be an E.M.F. induced. In certain cases the two move- ments of translation and rotation respectively may give as a result no change in the number of .lines embraced ; in which case there is no E.M.F. induced in the circuit as a whole. (III.) Movements of an incomplete circuit ; e.g. of a rectilinear wire, In considering the movements of an incomplete circuit, such, e.g., as a rectilinear piece of wire, we may adopt two courses. We may consider it to form part of a closed circuit, the rest of which is at an infinite distance, and is indefinitely remote from the magnetic field in question ; or we may consider the piece of wire by itself. It may be stated as a general law that there will or will not be an E.M.F. induced according as the wire cuts or does not cut the lines offeree, as strings are cut by a knife. Note. If this were absolutely true we should never have an E.M.F. induced when the wire either moved in its own direction, piercing the field end- foremost as a needle, or if it moved in any way in a plane in which lay the lines of force. Now, it is certainly true that in the latter case we never have an induced current. But in the former case we might have a current ; for, the wire might, so to speak, ' tunnel its way ' end-on along the axis into a cylindrical system of lines of force ; the self-induction in a rectilinear current, mentioned in the note to 10, being a case of this nature. While, therefore, the statement that When a conductor moves in a magnetic field there infill or will not be an indticed current according as it does or does not cut the lines of force is a good general rule, covering all cases of importance, still it is well to seek for a rule that shall cover all cases. Now, such a rule can be found from Lenz's law, if we remember the nature of the field given by a rectilinear cur- rent, and so by a current of any shape. We may say that When a conductor moves in a magnetic field there will or will not be an indticed E. M. F. in it according as the field due to a current produced by such an E.M.F. can or can not oppose the change in field due to the movement. (IV. ) Cases where the conductor (or circuit] is stationary, the field being altered. In such cases there is no obvious cutting of lines CH. xxi. ELECTRO-MAGNETIC INDUCTION 359 of force. But both theory and experiment tell us that all such cases of alteration in the field about a conductor, or circuit, can be regarded as equivalent to cases where the same change has been produced by movements on the part of the conductor from a weaker to a stronger part of the field, or conversely. We may, therefore, consider that all cases included under this head have been already discussed above. 9. Coefficient of Mutual Induction, or of Mutual Potential. Let us consider two simple circuits, A and A', carrying currents C and C' respectively. Each gives a magnetic field, and each is placed in the field due to the other. If n and n are the number of marked lines due to A' embraced by A, and due to A embraced by A', respectively, then the potentials on each circuit due to the other are Cn and C'ri , respectively. That is. it would require Cn ergs to bring up A from infinity to its present position, A' remain- ing stationary ; or Cn' ergs to bring up A', A remaining stationary. Now, if A be moved to infinity, we do Cn ergs work ; and if A' be moved after it into the same relative position as initially, we do + Cn' ergs work. But, on the whole, since things are exactly as they were, we must, by ' Conservation of energy,' have done no work. Hence it follows that Cn and Cn' must be equal ; and, therefore, two circuits exert on one another a mutual potential. Again, the number n of marked lines of force due to A' that pierce A are, cteteris paribus, directly proportional to the current C' of A'; this following from the fact that in our system of units we measure currents by the field-strength produced at constant distance. And the number n 1 of marked lines that pierce A' are in like manner directly proportional to C. Hence, since we have shown the potential to be { mutual,' this * mutual potential ' must be measured by some expression of the form CC ' M\ where M depends upon the shapes and positions of, and distance between, the two circuits, and not on the current- strengths. Thus we have Cn C'n' = CC'M. If we make C and C' both unity, we find that M = n or n'. Hence we see that when unit current flows in each circuit, the number of marked lines due to the other, piercing each circuit respectively, is the same. 360 ELECTRICITY CH. xxi. And we see, further, that the symbol M, whose meaning we had not given exactly, represents this number. The reader will notice that, for unit current, this number is, as we said, something depending upon the shapes and positions of, and distance between, the two circuits. It only remains to state that M is called the coefficient of mutual induction or of mutual potential. Hence, the coefficient M of mutual induction, or of mutual poten- tial, between two circuits is measured by the number of marked lines due to either that are embraced by the other when the currents are both unity. And when the currents are C and C' respectively, then the mutual potential will be expressed by CC' M. If we consider a current C sent through the one circuit, it will in con- sequence send C M marked lines of force through the other. If it take / seconds to establish the current (where t may be a small fraction of a second) then there are lines added per second to the second cir- cuit. Hence the induced E.M.F. e = 1VT . This shows us (i) the reason for M being called the coefficient of mutual induction as well as that of mutual potential ; and (2) how, in such an arrangement, the magnitude of the induced E.M.F. e is inversely proportional to the time taken to establish or destroy the primary or inducing current. 10. Self-Induction. The 'Extra Current,' Let us consider a circuit comprising a coil of many turns and a battery; this circuit being so arranged that it can be made or broken at will. When the circuit is made the current does not rise to a maxi- mum at once. If we consider only the positive current, or the flow of + electricity from the 4- pole of the battery to the pole, we find that it takes time to rise to a maximum ; and that in so rising the turns of wire that are more remote from the + pole of the battery always lag behind those that are nearer to it. Each turn of wire, as the current in it increases, thrusts an increasing number of marked lines through the adjacent turns of wire. It, therefore, acts inductively on these, and, by Lenz's law, the induced E.M.F. is in such a direction as to oppose the rise in current- strength. There is, in fact, an inverse induced current. Thus the coil, as a whole, offers an inductive obstruction to the rise in current that is quite distinct from resistance ; and that will, as far as it is CH. xxi. ELECTRO-MAGNETIC INDUCTION 361 due to the above given cause, disappear if the coil be unwound and laid out as a straight wire. Note. In a straight wire there is also induction, but to a less degree. In this case each bit of wire gives a field of circular lines of force, not only about itself, but also, to a much smaller degree, about the wire in front and behind ; and any increase in this field is also opposed by induction (see 8 (III.), note}. When the circuit is broken, induction in the inverse direction, i.e. so as to oppose the cessation of the current, ensues ; and there is an induced current added to the original current just as the latter is ceasing. Now, the primary current can be broken much more abruptly than it can be made; and hence, since e is inversely proportional to the time taken to effect the change in field, the E.M.F. of this direct induced current is much greater than is that of the inverse induced current ; it may, indeed, be made very great indeed. This direct induced current that occurs in the coil when the circuit is broken is called the extra current. Experiments. (i.) In the figure B i; a coil in circuit with a battery whose poles are E and E' ; this circuit can be broken at E. The two points A and C are connected by a short circuit that includes a galvanometer G. When this circuit is complete the current flows as shown, and the galvano- meter is deflected as indicated. When the circuit is broken at E, it is easy to see that a direct extra current in the coil would now pass round the galvano- meter in the contrary direction to that in which it before passed, although in the coil B it flows in the same direction as before. Hence the extra current 362 ELECTRICITY CH. xxi. would be indicated by an opposite direction of deflexion in G. In order to distinguish this reverse swing from the fall back to zero, which would ensue merely from the fact that the battery circuit is broken, it is necessary to keep the needle at zero by means of stops placed so as to prevent the former deflexion, but so as to permit of the second deflexion if such there be. Such a deflexion is, in fact, observed ; and it demonstrates the existence of the extra current. (The present writer has found that very marked results may be obtained with two Leclanche cells as battery, an ordinary rough astatic galvanometer at G, and a powerful electro-magnet instead of the simple coil B. ) (ii. ) If there be in circuit a powerful electro-magnet and a sufficient battery, and if the circuit be broken by a person holding in his moistened hands the two ends of the circuit, a shock will be perceived. The high E.M.F. of the induced extra current is sufficient to drive some current through the human body that has suddenly been interposed in the circuit. We see, then, that when a circuit is made, the current rises slowly, and research has shown that its rise is oscillatory. It then continues uniform as long as the battery is constant. When the circuit is broken, there is a sudden leap in the magnitude owing to the extra current ; and finally it ceases again in an oscillatory manner, much more abruptly than it began. All this is very readily exhibited by means of a curve, the abscissae measuring, time and the ordinates magnitude of current. n. Induced Currents of Higher Orders. It is found, as might indeed have been predicted, that induced currents will themselves act as inducing currents. We may arrange a series of coils somewhat as follows. First, a primary A, and round it a secondary B. B may then be in circuit with a coil B' at a distance from A, so that currents induced in B will circulate also in B' ; and round B'is placed another coil C upon which B' can act inductively, while A is too remote to have any direct influ- ence. We will use the words ' direct ' and ' inverse ' when the currents are in the same direction as, or in the opposite direction to, the original current in A respectively. We find that when the current is made in A, we have an inverse current induced in B and therefore passing in B'. This current, as it rises in strength from zero to a maximum, induces in C an opposed current, which will therefore be direct ; and, as it falls again to zero, it induces in C a current in the same direction, which will therefore be inverse. So, when the current in A is broken, we have a direct current induced in B ; and in C, an inverse followed by a direct current. CH. XXII. CHAPTER XXII. ARAGO'S DISC, RUHMKORFF's COIL, AND OTHER CASES OF INDUCTION. i, Induced Currents ('Eddy Currents') in Solid Metallic Masses moving in a Magnetic Field, When a conducting mass moves in a magnetic field currents are induced. Here, as always, the currents are such as to oppose the movement ; work must be done on the masses in order to move them, and we have developed in the mass equivalent electric energy, which finally runs down into the form of an equivalent of heat. Such currents will, as a rule, run in eddies ; but if we connect two points in the mass by a conducting wire, these two points being so chosen as to be at different potentials, we thereby modify a portion, at least, of the induced * eddy currents ' into a current running round a definite circuit. Experiments. (i. ) If a copper cube be caused to rotate between the poles of a powerful electro-magnet while this latter is as yet ' un-made,' and if we then ' make ' the electro-magnet by sending a current round it, the cube will be visibly retarded or stopped in its movement. (ii. ) A disc caused to rotate in the field will get perceptibly heated. (iii. ) In 3 we shall see how a current may be ' collected ' from the disc. (iv. ) When a person cuts through the field between the poles of a very powerful electro-magnet with a copper knife, it will appear as though he were cutting through soft cheese, so strong is the opposition due to the induced currents. 2. Arago's Disc and Magnetic Needle There is one case of the above that is of especial historic interest, and though in no way peculiar, will be described at some length. In 1824 Arago discovered the ' damping ' effect produced by the presence of copper and of other conducting masses on magnetic needles oscillating near them. If a needle oscillate very close over a copper disc, and still more if it oscillate between two copper discs, 364 ELECTRICITY CH. xxn. it will very soon come to rest. This is due to the induction of currents in the copper, these currents being such as to oppose that motion of the needle which is the origin of the induction. In the figure we have a copper disc caused to rotate with great velocity under a magnetic needle ; a sheet of glass between the two obviates any disturbance due to air-eddies caused by the rotating disc. The needle is deflected in the direction of rotation of the disc, and will, if the velocity be great enough, finally rotate also. This motion of the needle is not difficult to explain. In consequence of the rotation of the disc in the magnetic field due to the needle, currents are induced in the former. These currents are in such a direction that they oppose the relative motion of disc and needle. They would be induced equally if the disc were stationary and the needle rotated. We have thus a reaction between needle and disc that tends to stop the relative motion ; and so, if the needle be free to move, it will be urged round in the same direction as the disc. If slits be cut radially these interfere with the induced currents, and therefore with the actions described; but if they be cut in circles whose centres lie on the common axis of rotation of disc and of needle, their presence makes much less difference. 3. Continuous Current Collected from Barlow's Wheel, In the figure the dots represent a field of force, supposed to be running down into the plane of the diagram. The circle is a copper disc, revolving in the direction of the arrow. O B C A O CH. XXIT. ARAGO'S DISC AND RUHMKORFF'S COIL 365 is a circuit, of which OCA is a wire connected at O with the axis of the disc, and having sliding contact with the edge of the disc at A ; this circuit is completed by whatever radius of the disc happens to lie between O and A. If a current from an external source be sent in the direction A O B C A, the disc will rotate in the direction of the arrow. We can consider the field as acting always on the moveable radius O A, urging it (by the law given in Chapter XIX. 7, and elsewhere) to the left. When so used, the disc is called 'Barlow's wheel' If there be no external source of current, but the disc be forcibly turned with the arrow, there will be a current induced in the circuit in such a direction as to oppose B the motion, i.e. in the direction O A C B O ; the seat of the in- duced E.M.F. being in the shifting radius O A, which is always cutting the lines of force. The current thus induced gives lines of force rising perpendicularly upward from the plane of the diagram, opposed to those of the inducing field. With a powerful electro-magnet this experiment is very easily performed. If the rotation be performed in the opposite direction, a very remarkable result follows. The current is of course, in any case, in opposite directions in the portions O A and B C, which we may for simplicity suppose to be parallel. Hence the motion of the disc (when this turns in the contrary direction to that of the arrow), tending as it does always to move O A towards B C, will be against the electro-dynamic repulsion of the parallel and opposed currents in O A and B C respectively. This will give rise to in- duction opposing the motion; i.e. the opposed currents in O A 366 ELECTRICITY CH. XXII. and B C, or the current of the circuit, will increase in strength. Thus, if a current be started in the circuit, and if then the external field be caused to vanish, induction will continue, owing to the action between the portions O A and B C. We thus have the phenomenon of a current maintained solely by work done on a system of copper conductors ; there being no external magnetic field, no battery, and no electricity due to friction. 1 If the disc turn as in the figure, the inductive action between B C and O A tends again to oppose the motion ; and in this case the effect will be to lessen the current in the portions O A and B C, i.e. the current of the circuit. 4. Induction in the Earth's Field. We can obtain induced currents by rotation of a coil in the earth's field. In general, when a coil so rotates the number of lines of force piercing it is varied, and there will be E.M.F.s induced. In the figure, S R is a coil of many turns of insulated wire, and XY represents the direction of the earth's lines of force. The coil is turned by means of a handle M. The whole is mounted upon a stand in such a way that the axis of rotation n:ay lie in any direction whatever. If the coil be initially in the position shown, viz. perpendicular to the lines of force, then as it is turned 1 The apparatus as so used is called ' Sir W. Thomsons electric current accumulator. ' CH. xxn. ARAGO'S DISC AND RUHMKORFF'S COIL 567 the number of lines piercing it will be diminished, becoming zero when the coil presents its edge to the lines of force. As it is further turned the lines will begin to pierce the other face, so that, between the initial position shown and that which is 180 from it, the number of lines piercing the one face of the coil will pass from a + maximum, through zero, to a maximum. All this will induce an E.M.F. and current in one direction. Thus, if the lines offeree run from X to Y, this rotation through 180 causes, first, a withdrawal of the -j- lines piercing the face that is initially uppermost, and then an introduction of lines into this face. Both these changes, which may, indeed, be regarded algebraically as a continuous diminution in the -f lines piercing this face, are opposed by the face in question becoming S in polarity ; for this polarity opposes the weakening of the + field piercing the face, and also opposes the introduction of lines. Hence, in this case, the current will run clockwise. But as the coil continues to rotate through the remaining 180 back to its initial position, similar reasoning shows us that an opposite current will be induced. Hence, as the coil rotates steadily, there are equal and opposite currents induced each half-turn. To collect these into one current that may be caused to deflect a galvanometer, &c., a com- mutator is arranged at a. This is a simple contrivance, to be described later in Chapter XXIII. 4, by means of which the two ends of the coil interchange their connection with the two ends of the external circuit exactly 'at the time when the current changes its direction ; i.e. just when the coil is passing the position shown in the figure. Experiments. ^.*} It can be shown that the current is greatest when the change per second in number of marked lines embraced, for the same speed of rotation, is greatest. This will be when the axis of rotation is perpendicular to the lines of force ; since then the lines pass from the maximum possible to the minimum possible. (ii. ) When the axis lies along the lines of force, there is no current. For now no lines pierce the face, and there is no change produced by rotation, the lines lying always in the plane of the coils. (iii. ) The current is greater as the velocity of rotation is greater. This apparatus would, theoretically at least, enable us to de- termine the earth's elements at a place. For that direction of the axis which gives us zero induced current on rotation, is the direc- 368 ELECTRICITY CH XXII. tion of the earth's lines. And if the axis be now placed at right angles to these, the magnitude of the induced current for a given velocity of rotation would, if the * constants ' of the instrument are known, give us the earth's field-strength. 5. Induction Coils ; General Plan. We have seen in Chapter XXI. i how currents may be induced in a ' secondary ' coil by the making and breaking of a current in another coil, called the ' primary,' placed inside the former. We have further seen how the magnitude of the induced E.M.F. depends mainly upon three conditions (see Chapter XXI. 4), viz (i.) The field-strength given by the primary circuit when the current flows through it. (ii.) The number of turns of wire in the secondary that are pierced by the lines of force due to the primary. (iii.) The suddenness with which the primary current is made or broken. In order to fulfil condition (i.), the primary is provided with a soft iron core, and the current sent through the primary is strong. Thus we make or destroy a powerful field each time we make or break the primary current. For condition (ii.), it is necessary to have very many turns of wire in the secondary. . The wire, therefore, has to be very thin, and we get a high E.M.F. but a great resistance. For condition (iii.), there are special arrangements devised. These will be given in 8 and elsewhere. By paying proper attention to these three conditions, we may obtain induced currents of very high E.M.F. but of very short duration. Such currents give sparks, produce shocks, and pro- duce other phenomena not obtainable from the primary current. For certain purposes this modification of the energy of the primary battery is desirable. An arrangement consisting of a primary coil with its own iron core, a secondary coil of many turns of wire, and a contrivance for making and breaking the primary circuit, forms what is called an induction coil. If N be the total number of marked lines of force due to the primary that pierce the m turns of the secondary, and if it take /! and / 2 seconds respectively to break and to make the primary, then we have CH. xxii. ARAGO'S DISC AND RUHMKORFF'S COIL 369 On breaking, the direct induced E.M.F. e L = N x m . *i On making, the inverse induced E.M.F. e 2 = >< -^. By ' making ' or ' breaking ' we mean the increase from zero to the maximum, and decrease from the maximum to zero, respec- tively. The times / t and / 2 will be some small fraction of a second of time. 6. Practical Difficulties to be Overcome. The main diffi- culty encountered in the contriving of an induction coil which shall give induced currents of high E.M.F., lies in fulfilling the last of the above three conditions. When the current is made there is an inverse current induced in the primary itself, and this causes the rise to a maximum to be relatively gradual. Hence the E.M.F. e<> of the inverse current induced in the secondary will be relatively small, since the time *> ( see 5) is relatively great. (We may here remind the reader that when a coil has an iron core the inverse and direct currents of self-induction are far greater than when there is no iron core, since the field due to the core is added to that due to the current.) This difficulty cannot be overcome j and hence the devisers of induction coils have turned their attention entirely to rendering the break of current very abrupt, or /, very small, and so obtaining in the secondary a direct induced current of high E.M.F. When the primary current is broken, there is in the primary a direct extra current due to self-induction, the presence of the iron core making this effect greater. This direct extra current has a high E.M.F., so high, indeed, that the current breaks disrup- tively across the break just made in the circuit. Thus, when we break the primary the extra current gives a spark at the break ; and this both prolongs the current, or increases /,, and therefore diminishes e lt and also may fuse and injure the metal at the key by which the ciicuit is made and broken. This difficulty is met and overcome by means of the condenser described in 8. Another obstacle in the way of abrupt break is the induction of eddy current in the iron core ; these currents, of course, oppo- ing the cessation of the primary current. This is obviated by B B 370 ELECTRICITY CH. xxn. slitting the core longitudinally or by making it of a bundle of soft iron wires insulated from one another. Again, the iron core takes a certain time to become magnetised and demagnetised. This is not such an important obstacle ; but even this is nearly entirely obviated by means of the condenser, as will be explained in 8. Again, so great is the E.M.F. induced in the secondary that fliere is danger of a spark penetrating the insulating coating of this wire. This danger is obviated by a special method of winding. 7. Ruhmkorff's Coil. Fig. i. represents a form of coil called Ruhmkorff's coil, after the inventor. Here N and P are FIG. i. the terminals of the primary connected with the battery ; the primary coil is not shown, but its core A, composed of a bundle of soft iron wires insulated from each other so as to obviate eddy currents, is seen slightly projecting from the coil ; C is an ar- rangement for turning the current off, or on, permanently ; b is a piece of metal called the hammer, which alternately is attracted up to the iron core A, thereby breaking the current, and falls back by its own weight, thus making the current again ; B is the secondary coil ; p and/' are its terminals ; inside the stand K is the con- denser, whose function will be described later. As a special instance we may quote the dimensions of Spottiswoode's large coil. This was constructed on the same principles as the above ; but the arrangement for making the circuit was worked separately, and was not the automatic hammer contrivance shown in the figure. CH. xxn. ARAGO'S DISC AND RUHMKORFF'S COIL 371 In this large coil the primary was a wire of about i inch diameter and 660 yards in length, wound in about 1,350 turns. The secondary was a wire of about ^~ inch diameter, and 280 miles in length. The whole coil was 4 feet in length and 20 inches in diameter. The maximum spark obtained was about 42 inches long. This coil is the largest yet made. In fig. ii. we have a section of a coil very similar to that of fig. i., though the two do not correspond in all their details. Initially the hammer b lies on M, and the primary current from the battery X passes by P into the primary coil D, out of this again to the metal piece L, and so by the hammer b through M and N to the battery X again. By the passage of the current the core A is mag- netised, increasing many-fold the field due to the primary coil alone. This creation of a field inside the secondary coil B induces an inverse E.M.F. in this coil ; and, if the terminals //' be in contact or very near to one another, an inverse current will pass. This inverse current is, as we have explained, of no very great E.M.F. The moment that A is magnetised, the hammer b is attracted upward, and thus the primary circuit is broken. This causes the core A to be unmagnetised ; and, in consequence of the with- drawal of the field, there is a direct E.M.F. induced in the secondary coil B. This E.M.F. is high, since the break of the primary is very abrupt, and /, (see 5) is very small. We have not yet described the part played by the condenser F F, as this matter deserves a section to itself. We may here re- mark that L and N are connected by the route N M b L only, there being no connection through q and s. Caution. It may be well to warn the student that exposure to shocks from the induced currents of an induction coil may be very B B 2 FIG. ii. 3/2 ELECTRICITY CH. xxn. pamful or even fatal. He should not take shocks from any but small coils, say those giving -inch sparks as their maximum. As regards the winding of the secondary coil, we may say that this wire is wound in sections, separated from one another by ebonite partitions ; these sections being such that any two portions of the wire which are at a great difference of potential at the make or break, are separated from each other by one of these partitions. In fact, great care is taken to prevent the coil being ruined by internal discharge. The primary may be made and broken in many ways ; the ' ham- mer' arrangement is only one of many forms. Thus the 'make- and-break ' is sometimes effected between a platinum point and an alloy of mercury and platinum, under the surface of alcohol ; altera- tion of the surfaces due to the extra-current spark being thus obviated. The speed of the make-and-break may be regulated by hand, by clock- work, or in other ways. Thus the primary currents are more under control, and the inverje and direct induced currents may be observed separately. 8. The Part Played by the Condenser. Connected with the metal piece L q are a series of sheets of tin-foil ; and connected with NJ are another series. These two sets lie as shown in the fig. ii. of the last section, being separated by sheets of waxed paper or of other insulating material. The whole thus forms a condenser of very large surface, the set connected with L forming one plate, and that connected with N forming the other. Now when the primary current is broken by the rise of b, an extra current is self-induced in the primary. Were theie no condenser, this would leap across the space between b and M in disruptive discharge ; thus both prolonging the time /, of break, and injuring the surfaces of contact. But with the condenser the case is different. Before the extra current can give a spark across the gap between b and M, it must raise the two large condenser plates connected with b and M, to the necessary difference of potential. These plates have a very large capacity, however. Hence the extra current is employed in charging the condenser, and does not give a spark across M b at all. This is the main use of the condenser ; it prevents ' spark- ing,' and thus permits of .an abrupt break to the primary. Another use is as follows. While b is still in mid air between M and A, the extra current not only charges the condenser, but again rebounds (as it were) ; and, traversing the primary in the CH. xxii. ARAGO'S DISC AND RUHMKORFF'S COIL 373 contrary direction, helps to reduce A more abruptly to the neutral state. A further effect of the condenser is to lower the E.M.F. of the inverse current that is induced in B on ' making ' the primary. For the current has to charge the condenser ; and hence its rise to a maximum is delayed. We have said that it is on the direct induced current that we must depend for a high E.M.F. ; and we shall show in n how it is an advantage to reduce as far as possible the E.M.F. of the neglected inverse current. The con- denser, therefore, will have* been of service in a third way. 9. Condition of the Secondary Circuit when Closed. Referring to the formula of 5, we see that in the coil just described the number N is the same for both induced currents, but the time / 2 is greater than /*,. From this it follows that the inverse induced E.M.F. e 2 is less than the direct E.M.F. with the axis O r such that C is pro- portional to tan <, or to ^- i , as explained in Chapter XIII. It is clear that in this case the points A and B have a difference CH. XXIV. DYNAMOS AND MOTORS 413 of ordinates that is .less than the ordinate xy. This means that the AV between the terminals will not measure the total E.M.F. E of the dynamo, but will be less than this latter ; there being some fall of potential down the internal resistance R b according Direction of current a' FIG. ii. to Ohm's law. We may conveniently designate the AV between A and B by the symbol E B . It will not be hard to find what relation E B bears to the total E.M.F. E. Let Rj be the internal resistance ab, and R 2 the external re- sistance ba' . Then, since there is in the whole circuit a total fall of potential measured by E, there will in the portion R t be a fall measured by ^ . E, by Ohm's law. This expression can be written TT> . also as R! C, since C = - hR 2 * Hence, E falls short of E by the amount R^ C ; or . " . . . We can therefore find out the value of the E.M.F. of a dynamo (supposing it not to be shunt-wound) when it is running and when the circuit is closed, by finding the AV between its terminals and by adding to this the product of its resistance into the current. If measures be made in amperes, ohms, and volts, there will be no 'constants ' involved in these expressions. (iii.) Dynamo and motor ; a reverse E.M.F. In fig. iii. we have the case of a circuit that comprises a dynamo and motor. A and B are the terminals of the dynamo, C and D those of the motor. The resistance R t of the former is measured by the 414 ELECTRICITY CH. XXIV. abscissa ab, and the resistance R 2 of the latter by ca. The re- maining abscissae bc&!\&da' together represent the resistance r of the connecting wires. A x, y z, and w A', are of course parallel. Direction of current c d FIG. iii. The ordinate xy represents the E.M.F. F of the dynamo, while zw represents the reverse E.M.F. e of the motor. The current is given by the expression J ~~ e - ; or, in the dia- - ! + -K-2 + r "" w gram, it is proportional to Using the same notation as in (ii.) above, we may say that the AV E B between the terminals A B is related to E by the equation ................... E = E* + R! C. In a similar way we have for the AV E D between the terminals of the motor the relation .............. E^ = e + R 2 C, since here the fall of potential due to Ohm's law is added to the fall measured by e. The reader should notice the fact, at first sight somewhat para- doxical, that in the dynamo the E.M.F. (represented for convenience by the abrupt rise xy} appears to be opposed to the current ; while in the motor the E.M.F. (similarly represented by zw] appears to be acting with the current. These apparent difficulties disappear upon a little consideration. It is the work done on the dynamo that keeps, as it were, pumping CH. xxiv. DYNAMOS AND MOTORS 415 up the electricity from x to y\ and it is the advantage thus gained that enables the current to flow in the direction indicated by the arrow. Hence the driving E.M.F. E is represented suitably in the diagram by the rise x y. In the case of the motor we may observe that if the current is to do work, it must be by being, as it were, let down an f electrical hill.' Hence zw must occur in the opposite direction to that in which xy occurs ; the current must fall down z w. Moreover, such a fall as 2 w rightly represents a reverse E.M.F., or one which opposes the current, for the following reason. The current is proportional to tan $ (see Chapter XIII.) ; and hence the current is diminished, if the inclina- tion of the lines Ax, yz, and wA', to the axis Or be diminished. Now the fall zw does diminish the angle <, as is easily seen if we compare fig. iii. with a figure in which z w is removed, xy and a a' remaining unaltered. Hence e is rightly represented by the fall ziv. ii. Work done per Second upon a Dynamo as Related to the Velocity v of Rotation. The mechanical work per second or activity expended in driving a dynamo is, if we neglect friction, measured in watts by the product EC. The manner in which the magnitude of this product depends upon the velocity v of rotation of the armature varies according to the nature of the dynamo. (i.) Case of magneto, or other constant-field, machine. Here we have, by the formula of Chapter XXIII. 2, that E is proportional to 77 ; and hence that, when there is no other E.M.F. in the circuit, C is also proportional to v. Therefore the work is proportional to ir. (ii.) Case of series, or other vary ing-field, machines. In these the E.M.F. is first affected directly by v, and then is affected also in- directly, inasmuch as increase of current increases the field-strength. Hence the work is proportional to some higher power of v than the second power. Since however the increase in field-strength, cor- responding to a given increase in the current, depends upon the degree to which the cores of the field-magnets are already saturated, it is not possible to express in any simple and yet exact manner the relation between the activity E C and the velocity of rotation v. 416 ELECTRICITY CH. xxv. CHAPTER XXV. VARIOUS APPLICATIONS OF ELECTRICITY ; TELEGRAPHS, TELEPHONES, MICROPHONES, ELECTRIC LIGHTING. i. Introductory, In the present Chapter we shall describe, but necessarily in a very brief manner, various applications of ' electricity ' ; these applications being for convenience divided into groups, as indicated in the heading of the Chapter. From the nature of the subject there will not be, as there was in most of the previous Chapters, any order or progress from group to group. The whole forms a somewhat miscellaneous collection, illustrative of various principles explained earlier in the Course. TELEGRAPHS AND ELECTRIC SIGNALLING. 2. General Principle of Telegraphy, Let us consider two stations A and B, more or less remote from one another, con- nected by an insulated wire, the circuit being completed either through a return wire, or by means of large metallic plates buried in the soil, through the earth itself. It is clear that from either station can be sent currents that will pass through the other station. And if each station be provided, not only with a battery and with a suitable instrument for making or breaking current, but also with a receiving instrument on which the current acts to deflect a needle, sound a bell, or in other ways attract attention, it is clear that we have the means of exchanging signals and messages between the two stations. Note on the l return wire.' We may remark that as a rule a return wire is not employed, the circuit being completed by means of large metallic plates buried in the earth, as is seen in the diagram to 4. It is not necessary to suppose that the current does actually return through the earth. For the plates will be kept at what will be approximately zero potential (or the same CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 417 potential), since each has a large surface in contact with a practically un- limited conductor. And, as long as each end of the line is kept at zerv potential (or the same potential), so long will the battery send a current through it. (i) The electric bell. The accompanying figure represents a very simple form of signalling instrument : it is similar in principle to the automatic make- and -break de- scribed in Chapter XXII. 7. When in the position shown here (see fig. i.) the circuit is complete. But as soon as a current passes, the electro- magnet attracts the iron keeper a, causing the hammer to strike the bell , and the circuit is broken at the place where the spring C touched the keeper a. The current ceases, and with it the magnetism. Hence the keeper a is no longer attracted ; and so, acted upon by the spring to which it is attached, it flies back again and makes contact once more with C. The process is then repeated. As long, therefore, as the current is kept on the line, so long will the hammer vibrate and the bell sound. This instrument is used only for attracting attention, not for messages. FIG. i. FIG. ii. ^2) The electric sounder. This instrument (see fig. ii.) is very simple in construction. The current from the sending station passes E E 418 ELECTRICITY through the coils of the electro-magnet, and the iron keeper a is attracted downward. This causes the other end of the lever to hit with a sharp sound the screw d. When the current ceases, the spring S pulls the lever down again, and this hits the lower screw e. When the current is made and broken very rapidly, these two sounds occur almost simultaneously, as a sharp double ' click.' If the current be not broken again at once, FIG. the two sounds are separated by a corresponding interval, and the sound is more deliberate. These two sounds can be readily distinguished ; and by suitable combinations of these we can, as will be explained in 3, represent all the letters of the alphabet, and so form a code for messages. It is to be noticed that if there be passing between the point d and the lever a strip of paper, moved by clock-work at a uniform CH. xxv, , VARIOUS APPLICATIONS OF ELECTRICITY 419 rate, then each sharp make-and-break of current will cause a dot to be made on the strip ; while each slower make-and-break will keep the point pressed against the paper sufficiently long to make a dash on it. When the signals are thus to be printed, the in- strument is modified, the lever being provided with a pointed style, or with a small sharp-edged wheel, and d being replaced by a smooth roller. (3) The needle tele- graph. -By sending cur- rents round any form of galvanometer in opposite directions, we can give to the needle deflexions in opposite directions respec- tively. On this principle is constructed the com- mon ' needie telegraph/ of which one form is shown in fig. iii. Here the coils and the needle are both vertical, and the motion of the needle is limited by two ivory stops. In fig. iv. we see at the base of the instru- ment the commutating key by means of which the current can be made or broken, and sent in either direction. The current passes first through the sending instrument, deflecting its needle in one or the other direction. It passes thence through the line wire, and affects in a corresponding manner the needle of the receiving instrument. By means of a suitable code both this and the last instrument are used to transmit verbal messages. 3. Telegraphic Alphabets, In all those instruments that are in extensive use there are only two elementary signals : the dot, to which we agree to consider a deflexion to the left to be the FIG. ELECTRICITY CH xxv. corresponding signal in the needle instrument ; and the dash, to which corresponds the deflexion to the right of the needle. The signals answering to the letters of the alphabet are various combinations of the above elements, the only principle observed being that simple signals shall represent constant recurring letters. SDTGIE sixcra PRINTING. :NEEDLE. PHTNTCTG. TSTEDIE. A y N A B /XV I/I c AA P ^_ Jk D . Ax Q , IIJ E - i R vA 1 F \\A S \NN G /A T / H --,,-- NX\\ TJ VN/ I -- \\ V XNS/ J V/// W S// K u X AN/ L x/xr Y A// M II Z /As This system is called, from its inventor, the Morse code. The reader will notice how readily this code, intended originally for dots and dashes in the Morse instrument described in 5, adapts itself to other methods of signalling ; as by the needle instrument, by the voice, by flashes of light, by flags, &c. 4. The Needle System of Telegraphy. We will now de- scribe briefly the simplest manner in which two stations may be united on the needle system. In the figure, A and B represent the two needle instruments (see 2, fig. iii.) at -the two stations respectively ; jthe batteries are represented as usual. E and E' are large plates buried in the earth, serving to keep the two ends of the line at the zero potential ; K and K' are the two .commutating keys by means of which each station can send a current in either xxv. VARIOUS APPLICATIONS OF ELECTRICITY 421 direction through the other station ; the line wire (shortened out of all proportion, of course) is seen at the top of the diagram. Action of the commutator key. The terminals of the battery are connected with two strips of metal, marked white and black respectively in our diagram. These are insulated from each other. The pieces / and r are two keys of metal, that are kept pressed against the upper (or white] strip by means of a spring. In this, A B which is the normal condition of things, there is a complete circuit through A, the upper strip, the line, B, the other upper strip, the earth plate E', the other earth plate E, and so to A again. The batteries are cut out of the circuit ; since the lower (or black) strips, and with them the negative poles of the batteries, are insulated. Now let the key / be depressed ; thus leaving the upper, and touching the lower, strip. The connections make it evident that this will send a current along the .line to B, through E' and E, and back to the pole of the battery by A and /. If the other key be depressed, the current will pass in the other direction. If similar 'keys be depressed at both stations simultaneously the circuit will be broken, and no message can be sent. 5. The Morse System. The Morse instrument is in all essentials the sounder of 2, fig. ii. ; but it is modified so as to print dots or dashes on a strip of paper. In the figure we show the simplest manner of connecting two stations by means of Morse instruments. The two Morses are at M and M' ; K and K' are simple keys, not commutators ; G and G' are galvanometers to indicate to the sender whether or no the current actually passes, or whether the line is stopped ; this being 422 ELECTRICITY CH. xxv. needed, inasmuch as the current does not pass through the sender's Morse. In the position shown, the keys, kept in place by a spring, make contact at 3 and 4, so that each station is ready to receive a message. On either key being depressed a current is sent along the line and through the other instrument ; so that dots or dashes will there be printed, according to the duration of the contact made by the sending key. If both keys be depressed at once no message can pass. A preliminary signal is usually sent, so that the receiving clerk may set going the clock-work that moves the paper. The Morse is often worked by sound, the ear soon acquiring the power of reading off the message in this way. 6. Relays. On long lines the current will usually be too feeble to work the printing Morse. In such cases a relay is em- ployed. This is an arrangement consisting of a local battery in circuit with the Morse, and a key that can make or break this circuit ; this key being worked by the feeble line current through the intermediency of an electro-magnet. In the figure on the next page the feeble line current works the electro-magnet R ; this acts upon the soft iron piece a, and by means of the lever makes contact at i, 2 ; the local circuit is thus completed, and the Morse S is worked by the strong local current. 7. Transmission through Cables, under Water. When the line connecting two places has to be laid under water, it then consists of one or more copper wires thickly coated with some insulating material ; round this again is generally twisted stout iron wire in order to give the strength necessary, and this is coated CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 423 with some waterproof material. The whole is called a cable ; the messages conveyed by it are sometimes called cablegrams. Difficulties occurring in transmission by cable. It was soon found that it was not possible to send, through a cable of any considerable length, the clearly separated dots and dashes required by the Morse receiver. The sharp tap on the key at the sending end is represented at the receiving end by a feeble current that slowly rises to a maximum and again slowly sinks to zero. With such currents we cannot use any of the instruments described above. Had not the Thomson's reflecting galvanometer, or some similar instrument, been invented, cablegraphy would hardly have been practicable. With this instrument, however, the reading of a message is easy. The principle of the reflected beam of light gives the means of magnifying the motion to any desired extent. The needle is of very small mass and possesses but little moment of inertia about its axis of suspension ; and hence its deflexion will follow, with but an inappreciable lag, the varying undulatory current. And the extreme delicacy of the instrument enables very 424 ELECTRICITY CH. xxv. small currents to be perceived. Finally, the small mass of the needle, its method of suspension, and the command of it that is given by the controlling magnet, renders it possible to use a modified form of this instrument on board ship during the laying of the line. Reading the messages. On a land line we are able to give to the needle very rapid deflexion to right and to left of the zero mark. But on a cable the currents last so long that it is not practicable to wait for the dying away of one current before sending another. Hence the currents overlap one another, and allowance has to be made for this in reading the message. The clerk at the receiving end watches the spot of light, and observes, not its move- ments to right or left of the zero mark in the middle of the scale, but its jerks to right or left wherever it happens to be. In fact, a shifting zero mark is used. Cause of retardation of signals. The origin of the alteration of the sharp signals sent, into the slow and gradual undulations received, lies mainly in the fact that the cable forms a long con- denser of very great capacity. In the case of an overland wire, the sharp making of contact effected by striking the key is followed by the passage along the wire of what we may call a tide of electricity ; this tide having a relatively abrupt front, as kas the well-known ' bore ' that is some- times seen on a tidal river. The receiving instrument is affected, almost instantaneously, with the full force of the current. But in the case of the cable matters are very different. The tide of electricity rushes into the cable, but as it proceeds it is to a great extent detained to charge electrostatically the cable ; this being a condenser in which the wire and the sea-water form two coatings separated by the insulating di-electric. Thus the rise to a maxi- mum occurs very gradually, as would the rise of tide in a stream out of which ran many side channels that must be filled by the tide before it can advance up the river. (We need not say that the analogies here given are very incomplete.) Hence at the receiving instrument the current rises very slowly to a maximum. 8. Earth Currents. Condenser System of Working, It is found that currents are continually flowing from one point of the earth to another, these currents being due to thermo-electric CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 425 or to other unknown causes. When the two earth plates of a telegraphic line happen to be buried where the earth is not at one potential, a current will flow along the line wire. The greater the A V of the two places at which the plates are sunk, the stronger will be the E.M.F. driving the current along the line. During those disturbances of the earth's magnetic field which are known as magnetic storms, these earth currents become very strong, and may entirely over-ride and nullify the ordinary currents by which the messages are transmitted ; such ' storms ' occur, for example, during brilliant displays of the aurora borealis, and during any noticeable changes in the surface of the sun. These earth currents give great trouble on long lines, and especially in long cables. There are two methods of meeting the difficulty. (i.) The return wire method. Where it is practicable, a return wire may be used instead of an ' earth.' If the clerks find the earth current to be so strong on any particular day as to give trouble, they may temporarily complete the circuit by means of any wire that does not happen to be in use on that day. (ii.) The condenser system. The diagram here given represents the principle of the condenser method. In this system the line is not continuous, but the line from the sending station is connected with one condenser plate, while from the other plate passes away the line to the receiving station. In the simple arrangement shown in the figure, a depression of the key would send a charge into the condenser plate a. This would call up an equal and opposite charge in the plate b ; and the effect upon the instrument G would 426 ELECTRICITY CIT. xxv. be the same as though the current sent had passed on. This action of course ceases when the condenser is charged, but, as this latter is always of very great capacity, the current due to the electrostatic induction lasts as long as is necessary. When the key is again released, the plate a is discharged to earth ; and hence there will be a discharge of b also, and therefore a reverse current through the instrument G. In practice there is a condenser at each end, the cable being totally insulated. There is double induction, but, by the electro- static principles discussed in Chapter X. and earlier, the instrument at the receiving end will be influenced by currents passing in the same direction as if they had come direct from the sending station. The condensers employed consist usually of many sheets of tin-foil separated by insulating sheets of paraffined paper ; the sheets of tin-foil being connected up into two alternate sets, answering to the two plates of a condenser. 9. Insulation of Wires. If we cut connection with the earth at one end of a line, and send a current into the wire at the other end, a galvanometer in the line will settle down to some steady deflexion. In fact, the c insulated ' line wire leaks to earth along the whole distance ; the insulators acting as conductors of very great resistance. The longer the line, the more is the leak- age and the lower is the resistance offered by the insulators collec- tively. It is usual to fix upon some definite resistance (in ohms) as the resistance per mile that ought to be offered by any given line of wire. If, without any special cause, such as fogs, rain, or snow, the insulation fall below this standard, then it is suspected that some accidental connection with earth has been made. .io. Duplex Telegraphy. In 4 and 5 we noticed that it was not possible to send a message from each station to the other respectively at one and the same time. Could this be done, it is evident that we could get twice the use out of a line. We can readily devise a method in which each clerk could transmit his own message through his own instrument ; the problem to be solved is much harder, viz. how the message sent from A shall be indicated by the instrument at B only, while that sent from B shall be indicated at A only. This problem has been solved in two ways, one of which we will briefly describe. en. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 427 (i.) The differential method. In the figure, E and E' represent the two Morse instruments (or relays), the electro-magnet in each being wound with two equal coils. The line from the key K branches at a ; one path is through the one coil of E, through the line /, through the one coil of the receiving instrument E', and so to earth ; the other path is through the second coil of E, through an adjustable resistance box r, and so direct to the other terminal of the battery. The same may be said with respect to the paths from the other key K'. In order to work this system it is necessary that the two paths from either key offer equal resistances. In this case, since there are in each instrument two equal and opposite coils, the depression of the key K alone will not affect E, but will affect E'; while the depression of K' alone will not affect E', but will affect E. In the diagram the keys are shown in their normal position. This fact enables us to test for the desired equality of resistances ; r and r 1 being adjusted until the desired end is attained. It is to be noticed that in the action on the Morse instruments or relays we need not consider the direction of the current. Let us now suppose that, while A is sending a message to B* by depressing the key K, the key K' is also depressed. This will destroy the balance previously existing. Through the one coil of E there will now flow less current than before, or zero current, or even a reverse current; this depending upon the relative E.M.F.s of the two opposed batteries. So long, then, as K and K' are both depressed, the instrument E is worked by this difference of 428 ELECTRICITY CH. xxv. currents in its two coils ; and, if K be released, E will be worked directly by the current from? B alone. Hence, under both condi- tions E will attract its armature, or release it, in obedience to the movements of the key K' at the station B. The same holds good with respect to the instrument E'. Thus, by means of a system in which checks to the currents sent are recorded as currents received, the problem of duplex telegraphy was solved. (ii.) The WheatstonJ s bridge method. In another method a somewhat different principle is employed. Here each instrument is placed in a ' bridge,' and is unaffected as long as the extremi- ties of the bridge are at the same potential. When matters are properly adjusted, this state of equilibrium at the one station A is destroyed by the depression of the key at the other station B; and this is the case whether the key at A be worked or no. Hence duplex working is rendered possible. TELEPHONES. ii. Telephones. Introductory. Telephones are instruments by means of which it is possible to transmit between two stations, more or less remote from each other, musical notes or other sounds, and even articulate speech ; an instrument at the one end is spoken to or sung to, and the instrument at the other end gives out the words spoken or the tune sung. [The same result can be obtained for comparatively short distances by means of speaking tubes ; but to such a system, which is merely a case of reflexion of sound, the word telephone is not applied.] The only telephones much used are those in which the trans- mission is effected electrically. There are mechanical telephones, sometimes used, in which mechanical impulses are transmitted along the connecting wire ; but in electric telephones it is an undulatory electric current that passes. In all electric telephones we have a transmitter into which the message is spoken, and a receiver which again utters the message : these two instruments may be identical or different in form. In some telephones the speaking of the messages causes an undulatory current to spring into existence : and this again causes the receiver to utter the message. In such, there is no external CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 429 source of current needed. In other telephones there is an external source of current, and the speaking of the messages causes this previously -existing current to become undulatory. 12. The Bell Telephone. The most complete form of tele- phone, for general use, is that invented by Graham Bell, called, after his name, the Bell telephone. In this instrument no external source of current is needed, and it can act either as receiver or as transmitter. In the figure we have a section of the instrument ; from this the construction can be explained. E is the mouthpiece into which we speak; M is a bar-magnet of about four inches long; B is a long coil of very fine insulated wire surrounding the pole of the magnet; C C are the terminal screws; and finally, D is a vibrating membrane made of soft iron and very thin, When we speak into the mouthpiece, the soft iron diaphragm vibrates in exact accord with the air-waves impinging upon it. This acts inductively upon the steel magnet, and alters the number of marked lines of force piercing the coil of fine long wire. This gives rise to induced currents in the coil; these will be feeble, but of high E.M.F. Hence, when we speak into the transmitting instrument, we cause undulatory currents to pass along the line wire, these undulations answering to the air vibrations, and therefore answering to the words spoken. These currents arrive at the receiving instrument, and there pass round the coil. This alters the magnetism of the magnet's pole, and this again causes the iron diaphragm to vibrate. Finally, this iron diaphragm, vibrating in exact accord with that of the transmitter, will cause the air to vibrate, and, when the ear is held close to the instrument, the words of the message will be heard. 43 ELECTRICITY CH, xxv. It is to be noticed that the vibrations of the diaphragm of the transmitter to and fro will give rise to induced currents in opposite directions respectively ; and that in the receiver these currents will strengthen or weaken the magnet pole, and therefore attract further or release somewhat the diaphragm of the transmitter respectively. Hence, the to and fro vibrations of the one diaphragm will be re- produced in the other. Sensitiveness of the telephone. This telephone is of remarkable sensitiveness. By it we can detect currents too faint to affect any ordinary galvanometer, provided that the currents are undulatory, or at least subject to abrupt variations in strength. Hence, in many cases the telephone can be used as a very delicate galvano- scope. If a telephone (protected by a bridge when necessary) be in a secondary circuit such as that of a RuhmkorrTs coil, the note that it emits will indicate the number of induced currents passing per second; or the number of makes-and-breaks per second in the primary circuit. It has been used in investigations regarding the stratifications that appear in vacuum tubes; and it has thrown light upon the relation that these bear to the number per second of induced currents, and also to the steadiness of the contact maker in the primary. In the circuit of a ' Gramme,' the telephone shows plainly, by .emitting a note, that the current is really undulatory, though for many purposes it acts as if continuous. Induction disturbances on telephonic lines. The great sensitive- ness of telephones is a source of difficulty in their practical use. If the return be made through the earth, conversation cannot as a rule be carried on at all. For, what with earth currents and leak- age from other circuits, the instrument emits a continuous bubbling or frying sound that drowns the faint 'speech.' Return wires are therefore always used. But even then there is much induction if the wires run anywhere near ordinary telegraph wires ; and it re- quires a very special method of laying the telephone wires to reduce the Babel of sounds, due to these induced currents, to comparative silence. Details as to the precautions taken will be found in more technical works. CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 431 13. Telephones with External Source of Current, The Bell telephone is a wonderful instrument, certainly ; it acts as transmitter or receiver, and it requires no battery to work it. But it must be remembered that the currents transmitted by it are very small indeed, and that consequently the message is delivered by the receiving instrument in a very faint way. To obviate this difficulty, telephones have been invented which make use of strong currents driven by an external source, such as a Leclanche cell. Edison's carbon transmitter is an example of this class of telephone. This instrument has a mouthpiece and a vibrating metallic diaphragm, but no magnet or coil. At the back of the diaphragm is a button of carbon, this resting against a piece of metal. As the diaphragm vibrates the carbon will make better or worse contact with the pieces of metal touching it on the two sides ; better contact when it is pressed harder, worse when it is somewhat released. Now the two pieces of metal and the carbon between them form part of the circuit of a Leclanche or other cell. Hence, as the diaphragm vibrates, the current flowing in this circuit will vary in strength, the variations corresponding exactly to the original sound vibrations. Now this current forms the primary to a small induction coil, the secondary wire of which is in circuit with the line to the distant station. No current will flow in the secondary, and therefore none through the line, so long as the primary is steady. But when some one speaks to the transmitter and thus causes variations to occur in the primary, then will there be currents induced in the secondary, in exact accord with the air vibrations of the speech uttered to the trans- mitter. These currents traverse the line to the distant station, and the receiver (which we may suppose, e.g., to be a Bell telephone) will reproduce the original speech. 14. Microphones. Professor Hughes found that when in a circuit there is a loose contact, or still better a group of loose contacts, the current is exceedingly sensitive to even very slight mechanical disturbances, such, e.g., as those produced by sound- waves impinging upon the loose contact or upon the stand sup- porting the same. Undulations are produced in the current that is passing, and these undulations are so exactly in accord with the mechanical vibrations to which they are due, that a telephone 432 ELECTRICITY CH. xxv. included in the circuit will reproduce with greater or less distinct- ness the sounds spoken near the loose contact. An instrument constructed on the principle and for the object indicated above is called a microphone. It is found that loose contacts of carbon give the most striking results, and so in the usual form of instrument a carbon rod rests loosely between two other carbon pieces. If a fly walk over the stand of this instrument, the faint jarring due to its tread produces undulations in the current passing, and a telephone in the circuit will give out what we may, not however very exactly, call the ' sound ' of the fly's tread. 15. Properties of Selenium. The Photophone, It is well known that in general the electrical resistance of bodies depends upon their temperature, and varies with it either inversely or directly. There seems no reason to doubt but that whenever radiant energy (whether this consist of rays whose chief action is heating, or of rays remarkable chiefly for the impressions of light that they produce when they impinge upon the retina) impinges upon a body, the molecules of that body are to a greater or less degree affected ; and that consequently the electrical resistance of the body is affected. Now it is found that the body selenium, at least when in one of its several physical conditions, is affected in this way to a very remarkable degree. Even when the longer waves, those whose action is par excellence ' heating,' are sifted out of the ray, so that the radiation left is remarkable mainly for its action upon the retina and for its chemical action, still the effect produced upon selenium is very great. Hence one often hears of ' the action of light on selenium.' In order to exhibit or use this variability in conductivity, we must construct what is called a selenium cell. In one form of this there are two separate spirals of wire wound round a cylinder of hard wood or of other insulating material. These spirals are separated from each other, but run parallel to, and close to, one another through all their length. The one spiral is attached to the one end of the battery circuit, the other spiral to the other. So that the circuit is, as far as we have described it, broken, but the two halves run close to each other for a considerable length and in a compact manner. The space between the two CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 433 spirals is now filled up with selenium, which is melted on. We thus have the circuit completed by what is equivalent to a strip of selenium of very small length for the current to traverse, but of very great width. Thus the resistance is small, while the action of the radiant energy upon the selenium ought to produce a relatively great effect upon the current, there being such an extent of selenium to be acted upon. With the help of such a 'selenium cell' a piece of apparatus, called a photophone, has been constructed, by means of which sounds, and even articulate speech, can be transmitted from a distance. It is so contrived that, on speaking to the transmitting instrument, a beam of light reflected from it to the receiving station varies its intensity in exact accordance with the vibrations due to the voice. The beam falls on a selenium cell, which is in a circuit with a battery and a telephone. The resistance of the cell varies with the variations in the beam, and so also does the current. Hence the current at the receiving station varies in accordance with the vibrations due to the voice at the transmitting station, and the telephone thus reproduces the sounds uttered. ELECTRIC LIGHTING. 1 6. General Account of Electric Lighting. Illumination by means of electricity is effected in two main ways. We will give a brief preliminary description of each of these, and will then proceed to some detail. (i.) Incandescent lighting. In the one system a thin conducting filament is heated to whiteness by the passage of a current. The filament is usually or always enclosed in a vacuous glass vessel, and hence is not consumed. Here we have simply a case of Joule's law : if the resistance be R and the current be C, we have given out per second radiant energy measured in watts by C 2 R. It is necessary to make the filament small and highly resisting, so that the radiant energy may be intense in quality, i.e. so that the temperature may be very high. (ii.) Arc lighting. If we have a battery of high E.M.F. and a circuit of not too great resistance (forty Bunsen's cells of fair size will fulfil these conditions), and the current be passed through two rods of hard carbon which first touch one another, the point of contact will soon glow and attain a dazzling brilliancy ; and if now F F 434 ELECTRICITY the carbons be slightly separated, the current will continue to flow across the interval in the intensely dazzling continuous discharge known as the electric arc. The illumination seems to be due to the intense ignition of a stream of carbon particles that are con- tinually transported from the + to the terminal. The arc can be deflected by a draught of air, or by being acted upon by a magnetic field. Its length depends upon the current-strength ; i.e. upon the E.M.F. of the battery and the total resistance in the circuit. It is found that the arc is the seat of a reverse E.M.F. e, against which the current is driven by the battery, and that also it offers resistance R as does a conductor. Hence the radiant energy evolved per second in the arc is expressed in waits by the sum of the two activities C e and C 2 R. Here then we have not a simple case of Joule's law. 17. The Incandescent 'Lamp.' The substance chosen for the 'filament' spoken of in 16 (i.) is now invariably carbon. This substance possesses the great advan- tages of being practically infusible and of being a very good radiator. The main difficulty encountered in its use was that of obtaining a filament that should be at once thin enough, and at the same time tough and not liable to break. Edison found the suitable material in the prepared fibres of a certain kind of bamboo. Swan and others acted upon cotton thread or fine strips of card with strong sulphuric acid, and then carbonised the tough threads thus obtained. In the figure is shown a Swan lamp. It will be observed that, in order to give a greater amount of light within the same space, the filament is given a twist into one or more spirals. Near the ends the filament is thicker ; this is to obviate any great heating near the points of support, as this would lead to the breakage of the filament. If the globe be filled before exhaustion with a hydrocarbon gas instead of with air, the slight residue that always exists even in CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 435 a good * vacuum ' will act rather to strengthen the filament by deposition of carbon than to weaken it by combustion. It is found that the carbon filament is slowly dispersed and otherwise weakened by continued use. Each lamp has, in fact, a definite ' life ' that is usually expressed in ' hours of use with so many amperes j or in ' ampere-hours.' ' Candles per horse-power^ in incandescent lamps. It is of great interest and importance to know the average relations be- tween the candle-power of an incandescent lamp, and the horse- power expended upon the lamp. We know that the activity in watts expended upon the lamp is C 2 R, where C is the current in amperes and R is the resist- C 2 R ance of the lamp in ohms. Or it is measured by -'^ (approx.) in English horse-power. The unit by which we usually measure candle-power is very arbitrary and somewhat uncertain ; it is the illumination given by a wax-candle of a certain form in which 120 grains of wax are consumed per hour. It is fairly evident that the desired relation must be established by experiment, and cannot be calculated. In 'Nature,' vol. xxiv. p. 270, may be found a series of results obtained by Sir W. Thomson and Mr. J. T. Bottomley. It is found that if we wish to obtain such a degree of ignition only as shall give the lamp a chance of a fairly long life, we must be content with from 200 to 250 candles per horse-power. It must be noticed that we obtain, for the same horse-power ex- pended, better luminous effects if we employ high temperatures. The total radiant energy emitted each second from several lamps moderately heated may equal that emitted from one lamp intensely heated ; but the radiation from the former may be useless from the luminous point of view. We should therefore raise the lamp to as high a temperature as is compatible with its safety. From what we have said, then, it is evident that the expression C 2 R does not give us a measure of the light, but only of the total radiant energy, emitted per second by the lamp. The resistance of a lamp varies. As the temperature of carbon rises v its resistance decreases. Hence the resistance R of the F F 2 436 ELECTRICITY CH. XXV. lamp is not constant. In order to calculate the watts or horse- fiouer expended on the lamp when any definite current C is running, it is necessary to measure the difference of potential EB existing between the terminals of the lamp at the time. The expression C x Eg gives us the required number of watts without error. We can measure C by means of a galvanometer of low resistance included in the circuit, and E B by means of a potential galvanometer of high resistance (see Chapter XIV. n) con- nected up with the terminals of the lamp. 1 8. Arrangement of Incandescent Lamps in Parallel. The simplest manner of arranging a number of incandescent lamps of equal resistance is to connect them in parallel with two stout conductors that are connected with the two terminals of the dynamo respectively. The lamps are of equal resistance, and possess no reverse E.M.F. Hence the current between the two stout conductors (which we may add are called leads) is divided equally between the lamps. If one lamp be turned off, the current that now flows is divided between the others. Referring to the case of * m equal branches,' discussed in Chapter XIII. 9, II., let us suppose that here the resistance of each lamp is R; that of the rest of the circuit, including the dynamo, is negligible ; and that the E.M.F. is E. Then, when the m lamps are arranged in parallel, the equivalent resistance p will equal 5 and the current will be C= = ; . Now the !R R /;/ current flowing in each will be C, or will be . Therefore, m R under the not quite attainable conditions assumed, viz. of zero resistance in circuit and in dynamo, the turning off or on of any lamps would not alter the current flowing through each of the remaining lamps ; for, if m lamps were left, we should have for each lamp a current that is -J-th of , or should have as before. m R R We may put the matter somewhat differently. In order to supply each lamp of resistance R with a constant current C, it is necessary that the terminals of each lamp be kept at a constant AV measured by E, where E = C R. Now the stout conductors CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 437 between which the incandescent lamps are slung have, we assume, zero resistance, and are therefore each at one potential throughout its length. We shall then attain our end if we keep these copper leads at the required SV measured by E. But, by Ohm's law, if we have zero resistance in the dynamo and leads, it follows that the whole fall of potential in the circuit will occur across the lamps, and none will occur through the rest of the circuit. That is, the leads will be maintained at the full AV measured by E, and the current through each lamp will be constant. In practice, however, we are not able to construct an armature, though we can make leads, of zero resistance. If the armature have a resistance B, while the resistance of m lamps in parallel is R, then the total resistance of the circuit is B + R. Hence m m by Ohm's law there will be through the armature a fall of potential T> measured by . E, while the AV between the leads will B + z - R m -'-R be measured by the remaining fall m _ . E. Now as m in- B + lR m creases this expression is not constant, but decreases. Hence, in practice it will not do to supply a set of lamps in parallel from a machine of constant E.M.F.^L for the current in each lamp will decrease as the number of lamps increases. We must have a dynamo in which E increases somewhat when m increases, and conversely. This condition is fulfilled by some dynamo of the series nature. For, as m (i.e. the number of lamps) increases, the total resistance which is measured by B + R will decrease ; m hence C increases, and therefore [from the nature of a series- dynamo] E will also increase. 19. The Economy cf Incandescent Lighting. Supposing that we have 100 lamps, each of 50 ohms resistance when hot, ar- ranged in parallel. This will give an equivalent resistance of but 5 ohm. (We will neglect the resistance of the leads.) If we wish to use 90 per cent, of our energy in the lamps, and waste only 10 438 ELECTRICITY CH. XXV. per cent, indicating the dynamo, then by Joule's law the resistance of the dynamo must bear to that of the lamps the ratio of i to 9. That is, the resistance of the dynamo must be but '05 ohms. Hence, for incandescent lighting we must have dynamos in which the current passes only through very thick wire, dynamos of very low resistance. If the current demanded by each lamp be i "2 amperes, then we must maintain the leads at a AV measured by 60 volts ; since by Ohm's law we have . . . .& E = C . R = 1*2 x 50 = 60, for each lamp. Again, this AV will be T 9 o of the whole fall of potential given by the dynamo ; or the E.M.F. demanded of the dynamo must be y x 60 = 66 volts. We use activity measured by (i'2) 2 x -5 watts ; and we waste activity measured by (i'2) 2 x '05 watts. If, as we have done, we neglect all other waste, then we shall require ^ 2 ' x ' 5 + 5j horse -power to do work at the rate here 746 required. The relation between activity and candle-power is not an exact one, since the latter depends upon the quality of the radiant energy emitted per second as well as upon its quantity. But it will give some idea of the relations between the two if we mention that according to Mr. Swan (' Nature,' vol. xxvi. p. 358) we can obtain at least 200 candle-power for each horse-power of activity evolved in the lamp (see also 17). Another authority states that, allowing for 10 per cent, waste in heating the armature, we can run nine twenty-candle Swan lamps per horse-power. This estimation agrees practically with the last, since in that no deduction was made for waste in the circuit. We must further, if we aim at economy, run the lamps at as high a temperature as is consistent with their safety ; as was explained in 17. 20. The Arc Lamp. In the figure is shown a simple form of arc lamp, such as is now used only for lecture-room purposes. Initially the two carbons, made of hard gas coke or of some other very dense form of carbon, are in contact. When, however, the current is well established, by which time the carbons at the point of contact will have been raised to a very intense white heat, a small CH. XXT. VARIOUS APPLICATIONS OF ELECTRICITY 439 electro-magnet that is in the circuit acts upon a lever ; and this, by means of a clutch that acts by friction, raises the rod bearing the upper carbon. Thus the carbons are slightly separated and the arc is established. If by the using up of the carbons the space between becomes too great and the current becomes enfeebled, the electro-magnet's hold on the lever and clutch becomes relaxed ; and the latter allows the upper carbon rod to slide down nearer to the lower. When the current again rises to its proper value, this slipping movement is checked by means of the electro-magnet. If the arc be entirely broken, it can be restored only by the car- bons coming again into actual contact. Appearance of the arc. With the naked eye the arc cannot be studied ; the extre- mities of the carbons and the arc itself presenting a dazzling liquid brightness that, with the consequent irradiation effect, baffles analysis. Through dark glass we can, however, study the arc; and we then see that the source of the light lies in the extremities of the carbons, whence the arc springs across, and in the arc itself. This latter presents a flame-like appearance. Mechanical transport of carbon. When thus examined, or when a magnified image is thrown upon the screen by means of a lantern, it is seen that one carbon becomes hollow after some time. By actual measurements of weight it seems that it is the + carbon that wastes most, there being a continual transport of particles from it across to the carbon ; and that the hollow appearance of this latter is due to the fact that this stream of transported par- ticles is deposited upon it in a crater-like manner. Of course both carbons are also consumed slowly when the arc is exposed to 443 ELECTRICITY CH. xxv. the action of the air. The transference from + to carbon is therefore best shown when the arc is used in vacuo. This mechanical transport is obviated by the use of alternate current machines. Temperature oj the arc. According to some experiments ot M. Rosetti, of Padua, the + carbon attains the higher tempera- ture. These temperatures vary when the current varies. The experimentalist in question believes that we can regard the carbon to attain 2,500 C, and the + carbon 3,200 C, at least. Definition of equivalent resistance. If under any given condi- tions of dri\mg-.M.jF. E, &c., we remove the arc and substitute such a resistance p that the current remains unaltered, then p is called the equivalent resistance of the arc. The reader will re- member (see 1 6) that the arc offers a reverse E.M.F. as well as a resistance. Hence the expression ' equivalent resistance ' has not the simple meaning of Chapter XIII. ; its exact meaning will be dis- cussed in 22. It must carefully be noticed that we do not here in the least imply that under all conditions we could substitute p for the lamp without altering the current ; but only that under the given conditions the one is equivalent to the other. 21. The Reverse E.M.F. of the Arc, If by means of some form of quadrant electrometer we examine the two carbons after the arc has been established, we find a large AV existing between them, varying in general from 25 to 55 volts. Now if we consider any two points in a circuit, there will be a certain AV between them. If r a be the resistance between these two points, V a the AV between them, and C the current, and if there be no source of JZ.M.F. between these points, then by Ohm's law we have V a = r a . C. In such a case we get double of V a if we choose points such that the resistance between them is double of r at e.g. if on a uniform wire we take two points sepa- rated by double the former distance, C being constant. But in the case of the arc we find that the AV is nothing like doubled if, when C is maintained constant, we double the length of the arc. In fact it is found that the AV between the carbons, or V M consists of two parts ; the one part is a reverse E.M.F. e M while the other part follows Ohm's law and is measured by r a C, where r is the true resistance of the arc. M. Edlund, in his ex- CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 441 periments, found that the total A V between the carbons was thus composed. We may then write .......... understanding that of the two constituents of V a by far the larger is^ a . M. Edlund even detected the reverse E.M.F. e a as existing for a fraction of a second after the current had ceased to flow. It is worthy of remark that we have here a somewhat new case of transformation of energy. We expend activity to the amount of C . e aj against the reverse E.M.F. e a ; and we get an equivalent, not in chemical activity as in an electrolytic cell, nor in me- chanical activity as in a motor, but in heat activity. 22. 'Equivalent Resistance' in an Arc lamp. Consider the terminals A and B of an arc lamp, the AV between these terminals being V fl . Now if E be the E.M.F. of the dynamo, the fall V a occurs in the lamp, and the remaining fall (E V fl ) occurs in the rest of the circuit. If we remove the lamp from between its terminals and insert the equivalent resistance (.^"definition in 20), the AV V a must remain unaltered, otherwise the fall of potential through the rest of the circuit, and therefore the current, would be altered, which would be contrary to the definition of equivalent resist- ance. In the first case, therefore, we have ........ ,. V a = e a + r a C; and in the second case we have ............ Whence p = = Now it is found that V a does not vary in proportion to the current C, but that, within certain limits of current, it remains faiily constant. Hence, as we hinted in 20, p is not a constant at all, but it varies inversely as. C approximately, within certain limits of C. Example. One form of arc lamp used in the Brush system 442 ELECTRICITY CH. xxv u requires ten amperes current. The AV between its poles, or V M is about fifty volts. Here, then, we have ': 4 p = ^=-5 ohms. 10 Hence, on the assumption that a current of ten amperes is to be maintained, we may in our calculations substitute five ohms resistance for each lamp. 23. Series Arrangement of Arc Lamps. Since the reverse E.M.F. of a lamp varies from zero, when the carbons are in con- tact, to fifty volts or so when the lamp is in full power, it would be impossible to arrange arc lamps in parallel. We could not insure any equality of distribution even for a moment, and the system would be eminently unstable. We must, therefore, arrange the lamps in series, so as to insure the same current passing through each. Even then there are difficulties that have been overcome by means of many very ingenious regulating contrivances. We must consequently employ machines of very high E.M.F. ; and, if we desire to avoid the waste that ensues if the armatures have a high resistance, we must obtain this high E.M.F. in part at least by exceedingly rapid rotation of the armature (see formula, Chapter XXIII. 2). It is usual to employ alternate current machines, as these can be readily constructed to give high E.M.F.s, and as we thereby waste the carbons equally. As an example let us suppose that we wish to light twenty Brush lamps in series. With the necessary current of ten amperes we may count each lamp as equivalent to five ohms. Hence, if E be the required E.M.F., we must by Ohm's law have . . . . >,..> E == C R = 10 x (5 x 20 + R') Tolts, where R' is the resistance offered by the dynamo and the rest of the circuit. If R' be about 10 per cent, of the total resistance, we must have ( I00 + } = \ 9 / E = 10 9 / 9 Dangers from currents of high E.M.F. This explains the dangerous character of the currents employed in arc lighting as compared with those employed in incandescent lighting. If a person touch with one hand a wire conveying a current at such a CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 443 high E.M.F. as the above, he may form a branch circuit in conse- quence of leakage to earth ; and at this high E.M.F. a fatal current may pass through him. The effects of alternate currents on the human frame are far more serious than are those of con- tinuous currents. Of course contact with two hands is even more dangerous. Instant death has occurred in both these ways. There are of course fire risks, due mainly or entirely to the igni- tion of wires by the passage of successive currents. For an account of the various means by which arc lamps are regulated, we must refer the reader to more technical works. To such works we must also refer him for descriptions and figures of other forms of electric lamps, as \he Jab loch koff candle, Jamins lamp, Werdermanris lamp, the Sun lamp, &c. 24, Further Remarks on the Use of Arc Lamps. For the reason given before, viz. that it is only the radiant energy of high temperature that is useful for illumination, arc lamps are more economical than incandescent lamps ; but of course this holds only when the areas to be illuminated are sufficiently great to demand such powerful centres of illumination. Small arc lamps are not economical, nor indeed can they be readily kept steady ; hence for houses the incandescent is the only practicable light. On the same principle it is more economical, though often not at all convenient, to expend our horse-power on one large lamp than on several small ones. One obvious consideration is as follows : that as we must have arc lamps in series, and as two smaller lamps in series will give a combined reverse E.M.F. that is very much greater than that of one large lamp, the same current that will maintain the latter in full action may be totally unfit to maintain the former. In order to insure economy in working it is, as a rule, better to construct dynamos specially for the particular work for which they are intended, and not to use the same dynamo for a large and for a small number of lamps in series. MORE RECENT MEASURING INSTRUMENTS. 25. Current-meters or Ammeters. As was observed at the end of Chapter XVII., practical measuring instruments must be independent of the earth's magnetic field, and must not be affected by such magnetic disturbances as are of usual occur- rence. 444 ELECTRICITY CH. xxv. We require two classes of such instruments. Firstly, the necessarily more delicate ' standard instruments,' by means of which we can from time to time test and calibrate the ordinary * working instruments ' ; and secondly, a rougher and more portable type of instrument, whose construction can be varied to suit various requirements. Standard instruments. Our standard instrument should be delicate, uninfluenced by external magnetic disturbances, and capable of measuring large or small currents. Moreover its de- flexions should be proportional to the currents causing them ; and its ' constant ' should not change perceptibly in (say) a few months. All these requirements are met very fairly indeed by a modi- fied form of a permanent-magnet instrument, an ' improved Deprez-d' Arsonvale galvanometer. ' In this instrument a coil of fine wire, wound on a copper frame for damping purposes (see 27), is suspended between the poles of a permanent horse-shoe magnet whose lines of force lie more or less in the plane of the coil. The current is conveyed to and from the coil by means of very fine wires [Professors Ayrton and Perry use very fine flat wires of phosphor-bronze] which give practically no torsion-couple for usual deflexions of the coil. Embedded in the coil, and lying along the lines of force when the coil is in its zero position, are a number of small needles of soft iron. When a current passes through the coil, the field acts on the coil with a deflecting couple tending to set it at right angles to its 'zero-position' ; and it also acts on the soft iron needles with a couple that tends to restore the needles and coil to their zero- position ; and the final deflexion, indicated, as shown on p. 284, by lamp, minor, and scale, is such as gives equilibrium between these two couples. The particular value of getting rid of a tor- sional restoring couple, and substituting for it a magnetic restoring couple, is that even considerable changes in the magnetism of the permanent magnet produce no change in the value of the deflexions ; since both deflecting and restoring couples are affected alike. The galvanometer is usually ' shunted ' off the main circuit ; and, by varying suitably the resistance of the shunt and that in CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 445 the galvanometer circuit, we may arrange the instrument so as to measure currents of any strength in the main circuit [see Chapter XIII. 10]. The instrument is calibrated by electro-chemical means ; details of such calibration will be found in practical text- books. It then serves to caliorate other instruments ; these instru- ments being placed in the main circuit, currents of various strengths measured by the [shunted] standard instrument being passed through them, and their readings being noted. Ammeters in which soft iron is used. In a very large number of ammeters the field due to the current causes movement in a piece of soft iron, and this causes an index to move. The de- flecting moment so caused is balanced usually by a torsional couple, or by the moment of a 'weight' about an axis. In most of such instruments there may be very large errors due to resi- dual magnetism, if currents of very different magnitudes be measured soon after one another. Such errors can best be obviated by having the iron used both very soft and very thin. Solenoidal Ammeters. More accurate, though usually less port- able and more expensive, are instruments in which current passes through two coils, one of which is movable. Such an instrument was alluded to in Chapter XVII. 15. Here the action is between the two coils only ; there is no soft iron, and therefore no errors due to residual magnetism. Sir W. Thomson has devised accurate instruments based upon this principle. ' Deflexional ' and ' zero' instruments. We may for some purposes divide measuring instruments into two classes : (i. ) those in which we observe the deflexion of an index, (ii.) and those in which a measured force or couple brings the index back to zero. Concerning (i.) it may be said that they enable us to see at once the magnitude of the current [or other quantity to be measured], and to observe changes in it. But the range of movement of the index, and therefore the range of measurement, is usually not great ; and the divisions on the scale are usually not very proportional to the current [&c.j, since the arm of the couple acting on the coil or piece of soft iron usually changes owing to its movement in the field. 446 ELECTRICITY CH. xxv. Concerning (ii.) it may be said that we get a far larger range of measurement by a suitable choice (e.g.) of torsion wire ; and, as there is no movement in the field, the deflecting couple, and therefore also the restoring couple, is proportional to the current or to the square of the current. But it is inconvenient not to be able to observe at once changes in the current. By means of a ' magnifying spring ' Professors Ayrton and Perry have contrived to make deflexional instruments in which the index has a long range of movement, and yet the piece of soft iron moves but very slightly. Thus these instruments possess in a large measure the two main characteristic advantages of the ' zero type ' of instrument. 26. Voltmeters. Most voltmeters are practically simply ammeters in which the resistance is very great as compared with the resistance lying between the points whose A V we wish to measure [see Chapter XIV. n]. It is clear that if the resist- ance of the voltmeter changes, the A V between the terminals that answers to a certain deflexion will change also. This is a source of error in all such voltmeters. The Cardew Voltmeter. If we have a wire so shielded that it is not subject to variable currents of air, then its rise in tempera- ture and consequent elongation will depend only upon the magni- tude of the current traversing it ; that is, upon the A V existing between its terminals On this principle Cardew constructed his ' heated wire volt- meter.' One of its great advantages is that it can be used for alternating currents also. [It is evident that if the wire is to change in temperature readily, and to heat and cool rapidly, it must be very fine, and therefore be of high resistance. Hence the Cardew instrument, based upon the above principle, is adapted for a voltmeter, but not for an ammeter]. By another ingenious application of their ' magnifying spring ' Professors Ayrton and Perry have contrived to produce an instru- ment of this type that is of much smaller size and greater sensi- tiveness, and in which the waste of electric energy is much less. 27. < Damping ' in Ammeters and Voltmeters. It is usually of considerable importance that the index of the instrument should rapidly come to rest CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 447 The method of 'damping' by means of induced eddy- cur- rents [see Chapter XXII. 2] is often employed. Thus ordinary needle-galvanometers usually have copper plates under the needles; in some instruments the needle is of such a shape that it can be entirely enclosed in a copper cylinder ; and, in the Deprez- d'Arsonvale instrument, described in 25 of this chapter, the movable coil is wound on a copper frame that moves with it in the magnetic field. In some of their more recent instruments Professors Ayrton and Perry have made use of the ' viscosity of air ' ; and, by attaching to the index a vane fringed with fine feathers, have succeeded in getting a very dead-beat action. 28. Wattmeters. It is often desirable to measure the watts expended in some particular portion of a conductor, as e.g. in an incandescent lamp. We might measure separately the AV in volts between the extremities of the conductor, and the current in amperes flowing through it, by means of voltmeters and ammeters respectively. But instruments have been devised which will give us the watts more directly. As it is necessary to see such instruments in order to under- stand them thoroughly, we here give merely a representative diagram, illustrating the principle. Let L represent an incandescent lamp, and A B its terminals ; let Eg be the AV between its terminals, and C the current flowing through it. F is a coil of fine wire of relatively very high resistance ; so that the current C' that flows through it is insigni- ficant as compared with C. Of course C is proportional to E, assuming that the resistance of F does not alter perceptibly. E is a coil cf stout wire, through which passes the whole cur- rent (C+C). Kl 1 1 1 1 _ _ OF 448 ELECTRICITY CH. xxv. In reality, the coils F and E are part of one and the same instrument ; and one of the two, say F, is movable. If the instrument is of the 'zero type' we can bring back F to its zero- position by a measured force or torsion-couple. The electro- magnetic force that we thus balance and measure is of course proportional to the product C x (C + C') ; that is, very nearly to C x C', since C' is relatively small so that (C ) 2 can be neglected. But C' is proportional to E^. Hence our measured force or torsion-couple is proportional to the watts, E^ x C, expended in the lamp. Mr. Swinbourne has devised a means of eliminating the small error mentioned above. As regards the fine coil F, the wattmeter is subject to error due to change of resistance with changing temperature. 29. Integrating Instruments. Coulomb-meters. Though it is generally recognised that it is energy, or (quantity) x (difference of potential), that should be measured and paid for, and not quantity alone, yet instruments designed to measure the total quantity supplied with varying current-strength are worth noticing. We can, however, give but a brief description of one type only. Of course if the A V between the * leads,' where they enter the house, remains constant, then the total quantity supplied will be proportional to the total energy supplied ; and this may very well be the case where the house is supplied for incandescent lighting only, since for this the A V should be constant. Dr. Arorfs 'Pendulum' Coulomb-meter. This instrument is in detail somewhat complicated ; but the principle of its action, which is all that we can give here, is easily explained. A clock, that under ordinary conditions keeps accurate time, has its bob composed mainly or partly of a. permanent magnet of very hard steel. The current passes round a solenoid of thick wire, so placed as to give a vertical magnetic field symmetrical with respect to the pendulum-bob when this latter is in its lowest position. The direction of the current and the polarity of the bob are such that when the current passes there is a force directed vertically upwards, acting on the bob. When the current passes, then, the practical effect is to change the gravitational constant ^ into some slightly smaller value (gk\ The upward magnetic force acting on the bob must be very small CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 449 as compared with its weight ; or k must be very small as compared with g. And the field must not be so strong as to alter the strength of the magnet. When these two conditions hold it can be shown that the increase of the time T of a complete oscillation [due to the weakening of the total downward pull on the bob] is (approximately) in direct proportion to the current-strength. Hence the amount of 'time lost' by the clock during each minute is also directly proportional to the average current-strength during that minute. And so, finally, the total 'time lost ' by the clock during the day will be a measure ot the total quantity of electricity that has passed during that day. [In infinitesimal nota- tion it will measure C.dt.] 30. Integrating Instruments. Energy-meters It is, how- ever, a far more important matter to measure total energy supplied. Professors Ayrton and Perry have devised an instrument by means of which this may be done. In general respects it is like the instrument last described ; but the bob of the pendulum is a coil of fine wire of relatively very high resistance [see Chapter XIV. n]. The terminals of this coil are connected with the two ' leads ' by which the electrical energy to be measured is being supplied ; and thus at each moment the current C' passing through this fine coil is directly proportional to the A V between these ' leads.' The main current C, in comparison with which C is negligible owing to the ' relatively very high resistance ' of the fine-wire bob, passes through two fixed coils of very thick wire situated one on each side of the bob when this latter is in its lowest position. Again, as in 29, we have the practical effect that the gravita- tional constant g is altered into (gk). But in the present case the value of k is proportional to the product CC', and not to C alone ; that is, is proportional to the product * * .* (current) x ( A V between the leads) ; or to the watts supplied to the house at each moment. Hence the ' loss of time ' during each minute is a measure of the watts supplied during that minute ; and the total ' loss of time ' in the day is a measure of the total electric energy supplied during the day. [In infinitesimal notation it will measure I W. dt, where W represents the watts at the time /.] G G 450 ELECTRICITY CH. xxv. 31. Testing of Coulomb-meters and Energy-meters. By means of ammeters and voltmeters it is not difficult to see how the instruments just described can be tested and their constants found. But a description of the manner in which this is done in practice would be out of place here indeed, the student should go through a laboratory course if he wishes to understand the details of the use and calibration of measuring instruments. DISTRIBUTION BY ALTERNATING CURRENTS AND 1 TRANSFORMERS.' 32. General Principle of the System. One of the most important problems in electrical engineering is how to supply a district with electrical energy from a central station, without either an undue percentage of waste in the ' leads ' from having too thin wires, or undue expense from having these too massive. Now, in Chapter XXIV. 8 we saw that power could be transmitted from a distance along thin leads with economy and on a sufficiently large scale, provided that we used a high ' driving E.M.F,' in the dynamo, and also a high ' reverse E.M.F. ' in the motor. And, again, we could attain the same desired result in the case of incandescent lamps if only their total resistance were very great as compared with the resistance of the leads. As, however, incandescent lamps are worked 'in parallel,' their combined resist- ance will not be relatively high if the leads be long and thin. The system named above aims at solving the problem in what we may call a third way ; and we will briefly describe the principle of this system, it being now of considerable importance. Let us suppose that there are upon the same soft iron core two coils of wire ; one composed of many turns of very fine wire, and the other of fewer turns of very thick wire ; and let an alternating current be sent through the fine-wire coil. Then we have an instrument that resembles a Ruhmkorft's coil, in which the long fine coil is chosen *s> primary, and the short thick coil as secondary. Instead, therefore, of obtaining small cur- rents of high E.M.F. by means of large currents of low E.M.F., we conversely obtain large currents of low E.M.F. by means of small currents of high E.M.F. When the secondary circuit is closed there will be [by mutual CH, xxv. VARIOUS APPLICATIONS OF ELECTRICITY 45 1 induction] a reverse E.M.F. induced in the primary. Thus if E be the E.M.F. of the dynamo, and the reverse E.M.F. in the primary be an< ^ ~io per cent, of the energy of the dynamo's current is lost in the form of heat. Find the work per second of the motor in kilogram-metres per second. 4. A dynamo has a constant E.M.F. of 200 volts, and is in circuit with a motor, the total resistance of the whole circuit being 6 ohms. Find .... (i.) the current when the motor is at rest ; (ii. ) the current, activity used in the motor, total activity, and activity wasted in heat, when the motor is run at such a rate that there is 'maximum work ' per second done ; (iii. ) the current, total activity, and activity used in the motor, when this is run at such a rate that the efficiency is 90 per cent. 5. It is desired to measure the resistance of a certain incandescent lamp when the carbon is rendered white-hot by a current of I -3 amperes. It is found that then there is a difference of potential of 65 volts between the terminals of the lamp. What is the required resistance ? 6. A cable a mile long, whose ends are A and B, has a total resistance of 3-59 ohms. It is put into water, and a small hole is pierced to the core at a certain point. When the end B is insulated and the resistance is tested from the end A, this is found to be 2'8i ohms. When A is insulated, the resistance tested from the end B is found to be 276 ohms. Find the distance in yards of the puncture from the end A, and its resistance in ohms* xl ELECTRICITY ANSWERS TO QUESTIONS. CHAPTER I. (4) Float the two on a piece of cork, on water. CHAPTER II. (I) 2222?. (2) |. (3) (4)

'nes ; 2i (5) 4000 dynes ; 6 = tan ~ T CHAPTER III. (I) 960. (2) 150. (6) Deflect each to same angle with lines of earth's 'horizontal field.' (7) The moments of inertia of the needles enter into the formulae of calculation. (8) -v/63 = 8, nearly. (9) n, nearly. (10) 26-6. (n) '2 dynes ; there is attraction. (12) By means of a large magnet suitably placed. (14) Ratio is 256 : 225 ; magnetic moment of needle assumed constant. (15) _L; _L,. (16)24. CHAPTER IV. (7) Attraction, 2 dynes ; repulsion, '25 dyne. CHAPTER V. (3) (i.) A and C are at a + potential, B at a zero potential, and the leaves diverge with +. (ii. ) BC all at zero potential, and the leaves hang together. CHAPTER VI. (3) Both outside and inside are raised in potential, their difference of potential remaining (practically) constant. ANSWERS TO QUESTIONS xli CHAPTER X. (i) B is at a + potential, but lower than A's. That, as we charge A, the potential of B rises more slowly than does that of A. (a) Q* = te <3 (4) Q = 10 ; total K = 6 ; common potential = I '6. (5) Divergence increases ; divergence decreases ; leaves collapse. (6) K = 255. (7) 10 ; 10 a ; no ; 880. (8) Diameter = Z5 centimetres. (9) That of A rises, that of B falls, somewhat. (10) The force on a + unit at a given distance will be less; hence th& * marked lines ' will be fewer, and the equipotential surfaces further apart. Q2 11 I ) It will decrease. (12) -t- ergs. (15) Merely substitute magnetic unit pole for electric unit quantity, and take infinity, not ' earth,' as region of sero potential. (16) ii CHAPTER XL (1) Its potential will have fallen. (2) Copper is deposited by chemical substitution, and then the whole acts as a closed cell. (3) We have a closed cell, and the nascent hydrogen sets free antimony on the platinum plate. (4) More copper is set free by the nascent hydrogen. (5) Hydrogen will be set free from the platinum plate as in an ordinary closed cell. (7) If the current be too * dense,' i.e. too great per unit area of the plates. CHAPTER XII. (1) (i.) Copper deposited on copper dish, lead dioxide on the lead plate, (ii.) A back current, and a return (at least partial] to the initial condition. (2) A deflexion, and then a return to zero. A deflexion the other way, and a return to zero. (3) A continued deflexion. (5) By suitable changes in method of coupling. (8) Same method as in Question 5. CHAPTER XIII. (i) Easily proved by formula. (5) Use formula for current in each case. (6) In six main ways. 4 in series and 3 in parallel. xlii ELECTRICITY (7) Depends on the respective internal resistances, as well as on E.M.F.s. (8) Ratio is 3 : 5. (9) 15. (10) Either 6 in series, or 3 in series and 2 in parallel. (12) , and , amperes respectively. (14) Limit is C = -. J 3 : 3 B (15) There is no limit. Prove by formula. p (16) No. Prove by general formula. (17) Limit is . r , ON EA tan a, + tan a n (18) _* = --. (20) i volt. E B tan a, - tan a 2 (21) (i.) -5 ohm ; (ii. ) from *5 ampere to f ampere ; (iii. ) f volt. CHAPTER XIV. ^ Mass of copper . _ 3-3 Mass ot aluminium 2 '3 (3) The two answers are B = '5496, and B -= -5494, ohms respectively. (4) i -608 + io fi , nearly. (5) Ratio is 4 : i. (6) 5 cells in series ; -027 amperes, nearly. (7) E.M.F.s have ratio of 479 : 654. (8) -187 centimetre, nearly. (11) 7 37'- CHAPTER XV. (3) 11-32 h. p., nearly. (4) 115 /Af. nearly. (5) (i.) Heat in first case : Heat in second case = 1& E- : - l6 E 2 = 2 i 3 3 (ii.) In the first case 3 of the heat is internal and external, while in the second case is internal and external. (6) 4 in series, 2 in parallel ; 1-8432 calories per second-, 77,414,400 ergs per second. (8) In parallel. ( 9 ) Heat in silver wire = 2_ Heat in iron wire 3' (lo) Reverse E.M.F. = 40 volts. ( 1 1 ) 400 watts ; 24 calories per second. (12) (i.) 90 -watts\ (ii.) 972, 4-86, and 4-86 calories per second respec- tively ; (iii. ) 9 watts. CHAPTER XVIII. (I) The needle is unaffected. (4) Consider the equivalent magnetic shell. (5) From E to W. (6) It will vibrate vertically, making and breaking contact alternately. (7) 9-425, nearly. (8) tan = 47-124, nearly. (9) Strength = pt = C. ANSWERS TO QUESTIONS xliii ( 10) p x i = n C, or p = n C. ( 1 1 ) Let t be the common thickness, expressed in centimetres, of coil and shell. Then p t = t n C, or p = n C. CHAPTER XX. (2) + n C and n C at the two ends respectively. (3) p = ^irn C x k. (4) (i.) n C A ; (ii. ) nC A + (5) (i.)47rwCA; (ii. ) 4 TT * pole- strength = (4*)- x CHAPTER XXI. (2) Consider the earth's field and the magnetisation of the soft iron due to this. (6) This is an example of Lenz's law. CHAPTER XXII. (2) Eddy currents arise. (3) Induction ; Lenz's law. (4) By magnetic induction the field due to the magnet is altered. (6) Eddy currents arise, and Lenz's law holds. (7) (i. ) Hardly any extra current; (ii. ) decided extra current ; (iii.) still greater extra current. (8) Slower vibration. Compare with Question 6. (9) Still slower vibration. (10) Observe that with suitable pole downwards the vibrations may even be stopped altogether. CHAPTERS XXIII. , XXIV., and XXV. (1) Small internal resistance, low E.M.F. (2) Much depends on whether' or no the dynamo can ' leak ' to earth. (3) 37'96 kilogram-metres per second. (4) (i-) 33i? amperes ; (ii.) i6f amperes, l666| watts, 3333^ watts, l666| watts ; (iii.) 3^- amperes, 666| watts, 600 watts. (5) 50 ohms. (6) 892 yards, nearly; '99 ohm. INDEX. ACC A CCELERATION, dimensions of, 13 t x Accumulator, electric current, 366 wa*er-dropping, 102 Activity, unit of, 239, 298 in the voltaic circuit, 240 et seq. in the circuit of a dynamo and motor, 405 et seq. in cases of induction, 347 et seq. jEpinus's condenser, 76 experiments with, 78 Air, its specific inductive capacity taken as unity," 121 Alphabet, Morse, 420 Alternate discharge, 83 Amalgam, electrical, 46 Amalgamation of zinc plate in galvanic cell, 173 Amber, earliest observations of its electric properties, 42 Ammeter, or current galvanometer, 291 calibration of, 293 different kinds of, 443 - ' damping ' in, 446 Ammonium chloride, electrolysis of, 185 Ampere, the practical unit of current, 191, r:22, 298 Ampere's discovery of the equivalence of a magnetic shell and an electric circuit, 306 laws of the actions between currents, 311 ; between elements of currents, 313 rule for ascertaining directions of lines offeree, 273 theory of magnetism, 326 Angles, solid, 304, 322 Anions, 184 Anode, 183 Arago's disc and magnetic needle. 363 Arc, electric, 434 appearance of, 439 reverse E.M. F. of, 440 lamp, 438 Arc-lighting, series arrangement of lamps in, 442 dangers of, 442 Armatures of electrostatic induction machines, 104 et seq. of dynamos. 385 ct seq. Clark's, 386 Siemens's, 387 Gramme, 389 Siemens-Alteneck, 395 Brush, 395 BUN Armatures wound 'for E.M.F.' and 'for current," 394 Aron, coulomb-meter of, 4^8 Astatic system of needles for galvane- meters, 281 Atmosphere, electrostatic potential at a point in the, 117 ' Attracted-disc ' electrometer, 153 Attraction and repulsion in magnetism, 2 ; in electrostatics, 43, 49 occurring in magnetic and diamag- netic phenomena, 330, 334 Aurora borealis [set: references given ia Preface, p. viii.] Axis, magnetic, definition of, 29 correction for error in the, 33, 36 BA. ohm, 222 Barlow's wheel, 364 Battery cell, general view of, 171, 196, 238 various forms of, 177 et seq. secondary or ' storage,' 196 et seq. the economy of, 247 resistance of, 230 _ methods of coupling, 180, 207, 219 arrangements of, with respect to heat evolved, 245 constant, 177 molecular interchanges in, when cir- cuit is closed, 187 standard, 177 Bell, electric, 417 Bell telephone, 429 Bichromate cell, 177 Bismuth is diamagnetic, 329 Bismuth-antimony thermo-electric pile, 254 cross, experiment with, 204 Boats, electro-motor, 411 Bohnenberger's electroscope, indication of differences of potential by, 153 'Bound,' the term, as applied to electric charges, 81 Boxes, resistance, 223 Bridge, Wheatstone's, 226 Brush discharge, 99 Brushes, collecting, of Gramme machine, 39 2 Brush machine, 395 Bunsen's cell, 178 xlvi INDEX CABLE, telegraphic, 422 Cablegram, 423 Calcium chloride used to keep electro- scopes dry, 43 Calibration of ammeters and volt-meters, 2 93. Calorie, unit of heat so called, 239 Capacity, electrical, 68, 74 formulae for, 142, 146, 148 Carbon plates of constant battery-cells, 181 rods of arc lamp, action of current on, 439 respective temperatures of, 440 transmitter, Edison's telephone, 431 Card piercing by electric discharge, 92 Cardew, volt-meter of, 446 Cascade arrangement of Leyden jars, 88 energy of discharge in the, 152 Casella, dip circle of, 37 Cavendish's method of finding specific in- ductive capacities, 122 Cells [see under Battery-cells, Thermo- cells, and Electrolysis] Centimetre, the unit of length. 12 Centimetre-gramme-second (C. G. S.) sys- tem, 12 Charge, electric, distribution of, 55, 56 density of, 55 nature of, 90, 161, 341 - density of, distinguished from poten- tial, 144 energy of, 150 sign of, distinguished from sign of potential, 74 residual, 91 bound and free, 81 exemplified by alternate discharge, 83 * Chemical ' theory of voltaic cell, 170 Circuit, closed, action of a pole on, when circuit is complete, 308 ; incomplete, 310 closed, laws regarding, 201 magnetic field due to, 271 et seq. equivalent to a magnetic shell, 306 closure of, 174 fall of potential through the, 215 primary, making and breaking, 368 secondary, of induction coil, its condi- tion when closed, 373 ; with air break, Circuits, movement of, general law, 318 ; when the field is uniform, 318 ; when the field is not uniform, 318 ; when the cir- cuit is incomplete, 319 divided, 211 magnetic potentials due to, 316, 322 potential on, due to magnetic field, 317, 3 2 3 mutual potential of two, 359 Clarke's dynamo-, or magneto-, electric machine, 385 Clark's (Latimer) standard cell, 177 potentiometer, 234 Clouds, electrical potential of, 117 Coefficient of mutual induction, or mutual potential, of two circuits, 359 w of magnetisation, 329 Coils, induction, 344, 368 et seq. caution respecting their use, 371 resistance, 222 Combination, heats of, as connected with E.M.F.s, 249 CUR Commutator, simple, 386 key in needle system of telegraphy, 421 Compass, ordinary, 29 mariner's, 29 Condenser, yEpinus's, experiments with, 78 Condensers, 76 comparison with an isolated body, 83 capacities of, formulae, 149 ' investigated by ballistic galvano- meter, 287 investigated by other methods, 123, 124 charge of, dependent on the nature of the dielectric, 83, 121 et seq. of induction coil, 372 used in secondary circuit of induction coil, 375 use for obviating effects of earth cur- rents in cables, 425 spherical, the capacity of, 145 plate, the capacity of, 148 Conductivity, 206 Conductor, perfect, definition of, 70 electrical capacity of, 68 hollow, acting as an electrostatic screen, *39 Conductors and non-conductors, 44 Conductors of an electric current, 183 alterations in capacity of, 74 distribution of electric charge on, 55 ; according to their shape, 55 specific resistances ot, 235 are opaque, 341 lightning. 113 for transmitting power from dynamos, 410 Cone, distribution of charge on, 56 Contact potential series for metals, 165 E.M.F.s, 267, 269 ' Contact ' theory voltaic cell, 170 Copper-plating, 193 Copper sulphate, electrolysis of, 185 Cores of electromagnets, 333 Coulomb, the practical unit of electric quantity, 191, 298 Coulomb-meters, 448 Coulomb's torsion balance, 20 et seq. uses in magnetism, 21 et seq. uses in electrostatics, 50 determinations of distribution on a disc, 56 Couple, on a magnetic needle in a uniform field, 16 Couples, 14 Couronne des tasses^ 169 Crookes s high vacua ; experiments in, 377 et ?eq. f experiment in a low vacuum, 379 Cube, distribution of charge on, 55 Current, 202 follows closure of circuit, 174 energy of the, 240 heating effects of, 182, 241 effects of, on unequally heated metallic conductors, 264 electro-magnetic unit of, 297 element of a, 295 law of its action on a magnetic pole, 296 reaction of a pole on, 308 elements of, actions between 313 INDEX xlvii CUR Current, making and breaking, in induc- tion coil, 368 magnetic actions of, 221, 271, 294 instruments for measurement of, 276 et seq. meters for household distribution, 443 accumulator, 366 practical unit of, 222, 298 circular, field due to a, 274, 302 continuous, collected from Barlow's wheel, 364 - dynamos furnishing a, 387 obtained by induction, 355 the ' extra,' 360 maximum with a given number of cells, 219 Currents, action of, on magnetic poles, 271, 294, etc. attraction and repulsion between, 311, 321 continuous rotations of, 313, 322 alternate, dynamo-electric machines giving, 399 - induced, 343 et seq. under \\hat conditions obtained, 357 disturbances caused by, in telephone lines, 430 acting as inducing currents, 362 constant, 355 direction ot, 352 in conducting masses; 'eddy currents,' 363 local, in impure zinc plate, 175 mobile, action of the earth's field on, 311 sinuous, action of on a magnetic pole, 307 Curves, magnetic, 17 of work, dynamo-electric, 408 Cylinder, distribution of electric charge on, 56, 143 capacity of, 149 divided, induction experiment with, 54 electric machine, 60 Cylinders, magnetic and diamagnetic, setting of in magnetic fields, 330 ' T^AMPING/observationof, by Arago, J .363 applied to ammeters and volt-meters, 446 Daniell's cell, 179 molecular interchanges in, on closing circuit, 187 Declination, magnetic, 30 method of measuring, 31 Declinometer, 32 Decomposition of electrolytes, 184 primary and secondary, 188 simultaneous, 189 ; exemplifying Fara- day's laws of electrolysis, 190 ot water [see Water] Deflexions, method of, as used in mag- netism, 38 comparison of, to measure resistance, 225 Degradation of energy, 238 De la Rive's floating battery, 306 Diamagnetic bodies, list of, 334 Diamagnetism, relative, 335 whether there is absolute, 336 Dielectric, nature of -a, 83, 162, 341 strain in a, 90 5 forms of, 98, 377 et seq jndary circuit of induction coil, ELE Dielectrics are the seat of electrostatic potential energy, 161 specific inductive capacities of, 121 et seq. Dimensions of units, 12, 301 Dip of magnetic needle, 30 how to find, 36 Dip-circle, 37 Disc, distribution of charge on, 55 Discharge, alternate, 63 by Bunsen's flame, 48 mechanical eftecis of, 92 magnetic effects of, 92 heating effects of, 93 induction effects of, 94 duration of, 96 energy of, 150 ; in the cascade arrange- ment of Leyden jars, 152 examples in, 159 various forms of, 98, 377 et seq in secor 376 et seq. Distribution of electric charge, 55 laws of, illustrated by the ice-pail, 57 from the potential point of view, 143 investigated by aid of electrometer, 158 by alternating currents and ' trans- formers,' 450 Dots and dashes, Morse signals, 420 apparatus for printing, 421 Dynamo, work done upon, in relation to velocity of rotation, 415 Dynamos, various lorms of, 385 et seq. activuy expended upon, 405 as motors, 402 distribution of potential in circuit of, 412 general remarks on, 404 Dyne, definition of, 13 EARTH, magnetic field of, modified by the presence of iron, 30 magnetic elements of the, 30 ; how to measure, 31, 37 action of, on a magnetic needle, 28 Earth's field, observation of changes irr the, 40 action of, on mobile currents, 311 total field resolved into components, 34 magnetism, 28 Ebonite, a j?o >d 'electric. '43 how to prevent it deteriorating, 63 Eddies in the ether about lines of mag- netic force ('!) 339, 342 _ Eddy currents due to induction, 363 Edison's telephone, 431 Eel, electric, 169 Efficiency, dynamo-electric, 408 Electricity, atmospheric, 113 distribution of, on conductors, 55 electrostatic induction machines for, 100 et seq ' fluid ' theories of, 48 frictional, machines for, 60 measurable as a quantity, 65 specific heat of, 265 practical applications of, 416 et seq. statical, phenomena of, 42 et seq. ' current ' ; in what sense to be under- stood, 163 xlviii INDEX ELE Electricity, suggested identity of, with the ether, 341 positive and negative, 45 ; produced simultaneously and in equal quantities, 46, 47 Electrics and non-electrics, 45 Electrification of persons, 64 Electrodes, 184 polarisation of the, 173, 194 Electro-dynamometer, 291 Electro-etching, 19* Electrolysis, experiments in, 184 Faraday's laws of, 189 industrial application of, 191 nature of, 186 Electrolyte, molecular interchanges in, during passage of current, 187 Electrolytes, 183 Electro-magnetic theory of light, 340 Electro-magnets, 33 3 paramagnetic and diamagnetic, ex- periments with, 334 Electrometer, measuring E.M.F. of open cell by, 232 Electrometers, 153 attracted-disc lorm, 1=13 quadrant, Sir W. Thomson's, 155 et seg. uses of, 158 Electromotive force [see E.M.F.]. 201 Electro-motor, distribution of potential in circuit of, 414 Electro-motors, 402 general remarks on, 404 Electronegative, the term, 184 Electrophori, continuous-acting, looet seg. Electrophorus, 59 Electro-plating, 191 Electropositive, the term, 184 Electroscope, condensing, 98 gold-leaf, 43 charged by induction, 54, 73 Electroscopes, differences of potential in- dicated by, 153 Electrostatic fields offeree, 69, 131 * induction machines, 100 et seg. Electrostatics, duality of phenomena of, 42 the three laws of, 49 ; expressed in one formula, 52 'Elliott '-pattern quadrant electrometer, *5y, 43 ule's law and Joule effect, 241, 263 ~17 ATHODE, 183 J\_ Keepers, magnet, 9 Kerr's electro-optical discoveries, 340 Key, Morse, 421 irchhoff's laws, 216 ite, Franklin's experiment with the, 113 ADD'S 'self-exciting' dynamo, 387 Lamellar distribution of magnetism, T La c c, 438 es arrangement of, 442 iSKdescent, 434 arrangement of, in parallel, 436 'Lead,' in Gramme machine, 393 -the metal) used in secondary batteries, 197 Leclanch^'s cell, 179 Lenz's law, 352 Level, electrical [see Potential], 66 Leyden jars, 85 energy of discharge of, 151 -- cascade arrangement of, 88 energy of discharge in the, 152 spark, 99 -- nature of the charge in, 90 used with induction coil, 375 Light, electro-magnetic theory of, 340 rotation of plane of polarisation of, Lighting, electric, two systems of, 433 Lightning-conductors, 113 flashes, length of, 116 sheet, forked, and globe, 120 Lines, isoclinic and isogonic, 31 _ offeree [see Force] Liquids, specific resistances of, 236 Laclestone, magnetisation of iron or steel by, i ( Lupton's tables of constants, 236 MACHINES, electric, 60 Magnet, controlling, used with galvanometers, 282 cylindrical, constitution of, 324 long thin, constitution of, 2 completion of molecular circuit in, 9 Magnetic moments of needlts, i-, 26, 40 shell, 303 Magnetisation of steel or iron bars, i coefficient of, 329 methods of, 10 by electrostatic discharge, 92 Magnetism, Ampere's theory of, 326 duality of phenomena of, 42 earth's, 28 evident, distribution of, along a bar- magnet, 23 induced, 7 of iron ships, 30 lamellar distribution of, 303 laws of, 26 ; proved by torsion balance and by method of oscillation, 27 molecular theory of, 3 relative, 335 affected by heat, 7 temporary and residual, 6 Magnetometer, 40 Magneto machines, 395 as motors, 402 Magnets, how to make, i, 10 result of breaking, 3 use of keepers for, 9 field-, 387 hollow, contrasted with solenoid, 327 Mance's method of measuring resistance of a battery-cell, 231 Mariner's compass, 29 Mass, as distinguished from weight, 13 Maximum current with a given battery, 219 work with a motor. 405 Maxwell's (Clerk) electro-magnetic theory of light, 340 Mechanics, preliminary knowledge of, needed, 12, 128 Megohm, multiple of ohm, 2^5 Mercury, use in amalgamating zinc plate of galvanic cell, 173 cups for connecting wires, 175 Meridian, magnetic, plane of the, 30 Metals, contact potential series for, 165 thermo-electric cells composed of, 252 et seg_ specific resistances of, 236 Meters for ' total current ' supplied, 443 Microhm, fraction of ohm, 235 Microphones, 431 Molecular theory of magnetism, 3 Molecules, interchanges between, in elec- trolysis, 187 Moment, magnetic, of a needle, 15 how measured, 17, 26, 40 Moments of single forces and couples, 14 Morse alphabet, 420 system of telegraphy, 421 Motor [see Electro-motor] Mutual potential of two circuits, 359 N EEDLE, magnetic, action of the earth on a, 28 INDEX 1. NEE RES Needle, inagnetic, fracture experiment with, 2 axis of a, 29 moment of, 15, 26, 40 telegraph, 419 system of telegraphy, 420 Needles, astatic system of, 281 Neutrality, electrostatic, 47 magnetic, 4 Nobili, thermo-pile of, 254 ' North pole,' caution as t<9 the use of the term, 2 OERSTED'S experiment to show mag- netic field about a current, 274 Ohm, practical unit of resistance so called, 221, 298 Ohm's law, 201, 203 - example of its application, 207, ^19 graphic representation -of, 209 Kirchhoff's extension of, 216 Opposition method of measuring E.M.F. of open cell, 234 resistance of battery cell, 230 Oscillations, method of, 24 used to compare two magnetic fields, 25 used to compare two magnetic moments, 26 used to prove magnetic ' law of inverse squares,' 27 used in measuring -earth's mag- netic field-strength absolutely, 38 PARAFFIN, specific inductive capacity Of, 121 anhydrous, use with ebonite, 63 ' Paramagnetic ' means same as ' magnetic,' 334 Paramagnetic bodies, list of, 334 Peltier effect in thermo-cells, 263 theory of, 268 E.M.F.S, 267 electrometer, 77 Photophone, 433 Pile, Volta's, 167 Zamboni's, 168 dryness of, 168 Pith-ball, experiments with, 43 Plane of the magnetic meridian, 30 Planters secondary cell, 197 Plate electric machines, 63 condenser, 147 Plumbic acetate, electrolysis of, 186 Polarisation in voltaic cells, 175 methods of obviating, 176 of electrodes, 194 -*- plane of, of light, rotated in a magnetic field, 337 Polarity, magnetic, 2 of molecules, 4 Pole, magnetic, action of, upon a closed circuit, 308 reaction of, on an element of current, 308. unit magnetic, 14 Poles, magnetic, law of variation of force between, 26 Poles, magnetic, mutual attraction and repulsion of, 2 action of currents on, 294 action of element of current on, 296 positive and negative, 15 potentials on, 316, 323 of the voltaic cell, 172 Pole-strength, ' dimensions ' of, 300, 301 Potassium hydrate, electrolysis of, 185 Potential, electrostatic, 66, 131 at a point in the atmosphere, 117 methods of measuring, 118 results of observations on, 119 contact series, 165 curve of, round the axis of the Gramme armature, 392 difference of, 66, 131 distribution of, in the circuit of a dynamo and motor, 412 how to measure by work, 67, 131 as distinguished from electrostatic density, 144 measurement o, 131, 153 rate of change of, 135 of an isolated sphere, 141 of thunder-clouds, 115 fall of, through the circuit, 21$ sign of, distinguished from sign of charge, 74 magnetic, due to circuits, 316, 322 due to magnetic shells, 304 mutual, of two circuits, 359 Potentiometer, 234 Poundal, the term, 13 Power, transmission of, from a distance^L 409 Proof-plane", 52 QUADRANT electrometer, formula for, 157 potential of atmosphere me 119 uses of, 158 RAILWAYS, electric, 411 Rainfall after a lightning flash, 117 Bay, plane polarised, rotated in a mag- netic field, 337 Relay, the, 422 Replenisher, electrostatic inductive, 104 Resinous electricity, the term, 46 Resistance, 202 on what conditions dependent, 204 unit of, 221, 298 measurement of, by method of substitu- tion, 224 by comparison of deflexions, 225 by Wheatstone's bridge, 226 et seq. _ equivalent, of electric arc, 440, 441 of a 'multiple circuit,' 206 expressed as a velocity, 357 of a battery-cell, 230 of a galvanometer, 229 specific, 235 coils, 222 boxes, 222 Resistances, table of specific, 236 Hi INDEX RES Resistances compared by differential gal- vanometer, 286 Rheostat, 224 Riess, his observation of electrostatic dis- tribution on a cube, 55 Rigidity, molecular; same as 'coercive force,' 6 Ring-armature, 388 Ruhmkorflf's coil, 370 SATURATION, magnetic, 6 of cores of electro-magnets, 333 Screens between two electrified systems, 58, 139 Second, or time unit, 12 Secondary battery-cells, 196, 247 Selenium cell, 432 Series-dynamos, 397 as motors, 403 Series, electrostatic, 165 thermo-electric, 254 I Shellac, specific inductive capacity of, 121 Shells, magnetic, equivalence of electric circuit to, 306 fields and potentials due to, 302, 304, 322 potentials on, 317, 323 it Ships, iron, how to render their magnetism symmetrical, 30 Shocks, ' return,' 115 from induction coils, caution, 371 Shunt-dynamos, 398 .mts, 214 mens's armature, 387 mens-Alteneck armature, 395 Silver-plating, 194 Sine galvanometer, 279 Smee's cell, 177 a, cannot decompose water, 248, 251 Sol^Mhd, ideal and practical, 324 dV n of, with and without iron core, resemblance of, to a solid cylindrical unagnet, 326 contrasted with hollow cylindrical mag- net, 327 Solenoidal ammeters, 445 Sounder, telegraphic, 417 Spark, electric, duration of, 96 various forms of, 99, 376 et seq. passage of, the term, 92 of induction coil, use with spectroscope, 37 6 37 8 Spark-board, Wheatstone's, 96 Sparks, electric, discharged from persons, 64 Specific inductive capacities, 121 etseq., 147 resistance, 235, 236 Sphere, isolated, distribution of electric charge on, 55 electrostatic capacity of, 142 potential of, 142 Spheres of different radii, relation between potential and charge of, 143 Stands, insulating, 53 Steel, hard, duration of magnetism in. 6 ; why used for magnets. 10 at a white-heat not attracted by a mag- net, 7 TOR Steel, magnetisation of, i by electrostatic discharge, 92 Stool, insulating, 64 Storage cells, 196, 247 Storms, magnetic, 31 atmospheric signs of, 425 Substitution, method of, applied to measurement of resistance, 225 Sulphate of sodium, electrolysis of, 185 Sulphur, specific inductive capacity of, 121 Sulphuric acid, action of, in the voltaic cell, 173 use of, in decomposition of water, 184 Surfaces, equipotential, electrostatic, 69, *33 Swan lamp, 434 T AIT'S thermo-electric diagram and formula, 259 262 Tangent galvanometer, 276 et seq. Telegraphy, general principle of, 416 alphabet used in, 419 interferenceof earth currents with, 424 duplex, 426 Morse system of, 421 needle system of, 420 submarine, 422 Telephone, of Graham Bell, 429 Edison's carbon transmitter, 431 Telephones, 428 worked by an external battery, 431 Telpherage, 411 Theory, electro-magnetic, of light, 341 of thermo-cells, 265 et seq. of Peltier and Thomson effects, 268 of electro-magnetic induction, 346 et seg, Thermo-cell, theory of, 265 Thermo-cells, 252 Peltier effect in, 263 Thermo-diagrams, 259, 262 Thermo-electric neutral point, 257 power, 256 series, 254 Thermo-pile, 254 Thompson's diagrams, showing activities of dynamo and motor, 409 Thomson effect, 264 - theory of, 268 (Sir W.), his electric current accumu- lator, 366 note graded current galvanometer, 290 graded potential galvanometer 289 guard-ring for the attracted, disc electrometer, 154 method of measuring galvano- meter resistance, 229 mirror galvanometer, 283 quadrant electrometer, 155 reflecting galvanometer, impor- tant use of, in submarine telegraphy, 423 inductive replenisher, 104 water-dropping accumulator, 102 Thunder-clouds, high potential of, 115 Toision balance, Coulomb's, 20 strength of two magnetic poles com- pared by, ?.? INDEX TOR Torsion balance, distribution of evident magnetism observed by, 23 laws of magnetism proved by, 27 laws of electrostatics proved by, 50 Tram-cars, electric, 411 'Transformers,' or instruments of distribu- tion, 451 Tubes of force, 138, 139 UNIT magnetic field, 15 - jar, 86 magnetic pole, 14 Units, electro-magnetic, absolute and practical, 221, 297, 298 determination of, 299 and electrostatic ; ' dimensions ' of, 301 electrostatic, 66, 69, 131, 142, 301 mechanical, 12, 127 et seq. VACUO, diamagnetic bodies in, 336 Vacuum tubes, phenomena of dis- charge in, 377 et seq. Vane, electric, 63 Variations of earth's magnetic elements, 31 Varley's induction machine, 103 Velocity, dimensions of, formula, 13 Velocity ' ratio ' ; connecting electrostatic and electromagnetic units, 300, 301 Vitreous electricity, the term, 46 Volt, the practical unit of E.M.F., so called, 221, 297 Volta, views and experiments of, 165 Volta's cell, 169 ; theory of, 171 nature of chemical action in, 173 pile, 167 Volt-meter galvanometers, 233, 289, 446 calibration of, 293 - 'damping' in, 446 Voss induction machine, 105 ZIN WATER, decomposition of, r8 4 not possible by a single Smee's cell, 248, 251 dropping accumulator, 102 molecular interchanges in, during de- composition, 187 Watt, practical unit of activity so called, 240 Watt-meters, 447 Weber's electro-dynamometer, 291 Weight as distinguished from mass, 13 Wheatstone's bridge, 226 slide form of, 228 resistance-box form, 229 use of, in duplex telegraphy, 428 rheostat, 224 spark-board, 96 Winter's electric machine, 63 Wire, size of, used in making electro-mag- nets, 333 network, as a lightning protector, 115 return, in telegraphy, 416 use of, against earth currents, 425 Wires, conducting ; heat evolved in, 243 et seq. resistance of, 204 for resistance coils, 222 telegraphic, insulation of, 426 Work, unit of, 127 examples of, 128 in relation to force overcome, 129 formulas for, in dynamo-electric ma- chines, 405 maximum, with a motor, 405 per second, or activity, 239, etc. ZINC, solution of, in the voltaic cell, 173 heat evolved during, 182, 249 impure, local currents in, 175 PRINTED BY SPOTTI3\VOODE AND CO., NEW-STREET SQUARE LONDON RETURN CIRCULATION DEPARTMENT TO** 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 1 -month loans may be renewed by calling 642-3405 6-month loans may be recharged by bringing books to Circulation Desk Renewals and recharges may be made 4 days prior to due date DUE AS STAMPED BELOW APR 1 3 1981 AUG C UNIVERSITY OF CALIFORNIA, BERKELEY "FORM NO. 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