The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 35 of 35

The Magnetic Force at any Point in the Plane of a Circular 'Current. — The magnetic force at any point in the plane of a •circular conductor conveying a current may be found by an elegant geometrical method due to Mr. A. Kussell (see Tlw Electrician, Vol. XXXI., p. 284).

FIG.

Let P be any point in the plane of a circular current, and let F be the magnetic force at P due to a current of strength i in the wire. Let d s be an element of length of the circle, and let OR be the radius of the circle. Take any point R on the circumference of the circle (see Fig. 1), draw the diameter through 0 P, and at R draw a tangent to the circle. Then draw the radius 0 R, and through P draw P N perpen- dicular to the tangent. Join PR. Let OP = a, OR = R, P R = r, P N =p, and the angle R P N = <£.

698 APPENDIX.

Then, by Ampere's law, the magnetic force F at P due to the element d s of the circuit at R is equal to ^-^ ^ . Hence

But 2}ds =

-r = V R2 - a2 sin2 <f> - a, cos £,

j , V B' - a' ritf^+a COB »

2 - a2 sin2 4 d d> + •

The second integral, taken between the assigned limits, is

_

i/Ba-aasin2<£d& is called an .

elliptic integral of the second order, and represents the length of the circumference of an ellipse which has 0 for its centre,, P for one of its foci and 2 R for its major axis.

Hence, if we describe an ellipse on the diameter of the circle, with 0 as its centre and P as its focus, the magnetic force F at the point P due to a current i in the circle is equal to the value of ™^— 2 x *ne length °f foe circumference of thia ellipse.

In practice we can easily describe this ellipse by means of two pins and a thread, and then measure the length of its- circumference with a measuring-wheel, such as is used for .measuring distances upon a map, or by laying a thread round the ellipse and measuring its length.

In this way a practical measurement can be made of the magnetic force due to a circular current at any point in its plane.

Up to the limit of «=0'8R, the length I of the circum- ference of the ellipse can be calculated approximately from the formula —

APPENDIX. 599

NOTE B. (See page 35.)

The Total Induction in a Circular Solenoid.— If we consider a circular-sectioned ring to be closely wound over with turns of wire, we obtain a circular solenoid.

Let a be the mean radius of the circular cross-section of the solenoid, and let R be the radius of the circular axis of the solenoid. The whole solenoid may be considered to be resolved into elementary solenoids. Let d S be the cross- section of one of these, and let x be the radius of the circular axis of the circular elementary solenoid.

Let N be the total number of turns of wire on the ring.

Then, for the elementary solenoid, there are ^ — turns per unit of length.

The magnetic force in the interior of the elementary solenoid is H, and 4B-NI_2NI

Zwx~~ x ' where I is the current in the solenoid.

Hence, if the medium is non-magnetic, the surface integral of magnetic force is the measure of the induction. Hence the magnetic induction in the interior of the elementary solenoid is

2 N I

— d S. Hence the whole induction Q through the whole

solenoid is obtained by integrating this last expression over the area of the solenoid, viz., n-a2. Therefore between proper limits

Now the integral / — taken over the circular cross-section can J x

be shown to be equal to 2:r {E -

and hence Q = ^ N I {E -

This, then, is the expression which should be employed for

the total induction over the circular cross-section of the ring,

if the mean radius of cross-section a is not very small

compared with the radius of the circular axis E.

If — is a small quantity, then, since

€00 APPENDIX.

we have E- v'R2— a2 = ^-p nearly when ^ is small, and hence

NI ,

or 4- times the current turns per unit of length of solenoid multiplied by the area of cross-section of the solenoid ; and for a very large thin circular solenoid the magnetic force in the centre is 4?r times the current turns per unit of length.

NOTE C. (See page 52. )

The Calibration of the Ballistic Galvanometer. — A galvano- meter consists generally of two parts — a magnet and a coil of wire. The magnet may be fixed and the coil suspended and movable ; or the magnet suspended and movable and the coil fixed. If the arrangements are such that the movable system when disturbed is brought to rest without vibration, or with very few vibrations, the galvanometer is called a damped or aperiodic galvanometer. If, however, the resistance to motion is so small that the movable part, when disturbed or made to oscillate, continues for a long time to oscillate with very slowly decreasing amplitude of vibration, the galvano- meter is called a ballistic galvanometer.

If a body is suspended so as to vibrate about a vertical axis, and if when given an angular displacement about that axis it returns when released to its original position, and oscillates about it, the body will in general execute vibrations of decreasing amplitude. If I is the moment of inertia of the body about that axis, and if /* is the torque or couple which tends to restore the body to its original position if displaced round the axis by a unit angle, then, if at any instant the angular velocity of the oscillating body is u, the equation of motion, neglecting for the moment resistance to motion, is

where 6 is the angle of displacement at the instant when the angular velocity is u>.

i n

Since o> = — , we have as the equation of motion, dt

. APPENDIX. 6

and the time t of one complete vibration of the body will be

/*

This gives us the periodic time of oscillation of the body, assuming that the resistance to motion due to friction is nothing.

If there is any small resistance to the motion, such as air resistance or other causes of energy dissipation, the amplitude of the swings of tbe vibrating body will gradually decrease. If the experiment is tried of setting the needle or coil of a ballistic galvanometer in motion, and observing the amplitudes #i» a'ai xs, &c., of successive swings to the right and left, it will be found that xlt .c.2, u?s, &c., form a descending geometrical progression, and that the values of log a, log x.2, log #,, &c., form a descending arithmetic progression.

If the difference of the logarithms of the amplitudes of two successive swings, one to the right and one to the left, is taken, this difference, denoted by A, is called the loyarithwic • decrement of the galvanometer, and it will be found that this difference is approximately constant for any two successive swings right and left during the progress of the decay of the excursions. That is, log x1 - log x = A, also log a?a - log xa = A, &c. Hence, if the logarithm of the amplitude falls off or decreases by an amount A in tbe course of the passage of the needle or coil of the galvanometer from the extremity of one swing to the right to the extremity of one swing to the left, it may fairly be assumed that, if the coil or needle is started from rest by a sudden blow, the excursion x0 it would make if there were no resistance is related to the excursion xl it does actually make with the actual resistance existing by the relation

If the base of the logarithms is 10, or the logarithms are ordinary ones, then by the exponential theorem we have

C02 APPENDIX,

where M is the modulus 2-308, or the multiplier for converting ordinary logarithms into Napierian logarithms. M is the logarithm of 10 to the base e, the base of Napierian logarithms.

If - is small, we may neglect — and higher powers in com-

parison with -, and write 2

• A

10 2 =1+M^, nearly.

Hence |° = 1 + M * nearly,

or XQ =Xil+ M ~ , nearly.

In other words, we can find what the excursion xd would be if there were no air resistance by multiplying the actual first

excursion ^ by a factor ( 1 + M -\ called the correcting factor.

Hence, if the movable system of a ballistic galvanometer receives a sudden blow or impulse when at rest, and if it then- makes an excursion the angular magnitude of which is xlt and if such excursion is slightly resisted by air friction, we can eliminate the results of this and correct for the frictional resist-

ance by multiplying a^ by the correcting factor f 1 + M- j.

If the logarithmic decrement is measured directly in. Napierian logarithms, then the correcting factor is simply

K)

where A. equals the Napierian logarithmic decrement, or the logarithm of the ratio of one swing to the next.

Eeturning to the equation of motion of the needle or coil, since we have at any moment

let us consider a small magnetic needle of magnetic length I hanging in the centre of a coil so wound that when a current flows through the coil there is a uniform magnetic field due to the current in all the region within which the needle lies. Let the normal position of the needle when at rest be such

APPENDIX. 603

that the magnetic axis of the needle is at right angles to the direction of the field due to the current in the coil.

Then let M be the magnetic moment of the needle, H the strength of the controlling field, the direction of H being at right angles to the field due to the current in the coil.

Let the needle be set oscillating by a sudden impulsive couple acting upon it when at rest. Let 6 be the angular displacement of the magnetic axis of the needle at any instant t.

The restoring couple acting on this needle at that instant is M H sin 6, and, by the equation of motion,

dt

Hence, Irfo> — = MHsin Od6\

d t

dO but -=o,

Hence

Integrate the above equation from the instant when the needle leaves its position of rest with a finite angular velocity Ji until it reaches a displacement 6 and has a zero angular velocity. We have

if

or £

therefore fl = 2^ sin . ... (1)

If the impulse which starts the needle from rest with a finite velocity is that due to the flow of a quantity of electricity through the coil surrounding the needle, which flow is all over before the needle has had time to move sensibly from rest, we can obtain a relation between the quantity of electricity so sent through and the excursion of the needle.

For let i be the current in the coil at any instant, and let G be a constant depending upon the form of the galvanometer

€04 A PPEXDIX.

coils, such that G i is the magnetic field due to the current. Then, since M is the moment of the needle, M G i is the whole couple acting on needle, and the impulse of this couple is M G i d t in a time d t. If the whole impulse is over before the needle has had time to move, then the whole impulse of the couple must be equal to the total gain of angular momen- tum I d w taking place in the time d t.

Hence M G i d t = I d « ;

but if dq is the quantity of electricity which has flowed through the galvanometer coils in the time d t, we have

im£&

dt and

But since the whole impulse is over before the needle has time to leave its position of rest, we have, by integrating from w = 0 to w = fi, the equation

where S2 is the angular velocity with which the needle leaves its position of rest. But by equation (1),

therefore Q - jl sin |,

f\

or Q = /,- sin - .

Hence we see that, when a quantity Q of electricity is dis- charged suddenly through the coils of a ballistic galvanometer, the whole discharge being over in a very short time compared with the period of free vibration of the needle, the quantity of electricity is proportional to a constant, k, called the ballistic constant, multiplied by the sine of half the angle of excursion of the needle.

If the galvanometer has a logarithmic decrement A, which is moderately small, say not more than 10 per cent., then to a clo.se approximation

APPENDIX. 605

To determine k for any galvanometer, the easiest way is to charge a condenser having a capacity of c microfarads to a potential of v volts by placing it in contact with a battery of known voltage, and then discharging this quantity c v micro- coulombs through the ballistic galvanometer. If this quantity c v gives a •' throw " 8, we have

2 Hence k is determined.

Such a calibrated ballistic galvanometer can immediately be employed to measure a magnetic field. For if a loop of wire having N turns is placed in a field of induction so that the induction is linked with the circuit, and if the loop is connected with the ballistic galvanometer, then on suddenly withdrawing the loop from the field we shall get a " throw " of the galvanometer which can be interpreted to mean so much quantity in microcoulombs passing through the galvano- meter. Then, by the principles explained on page 31, the total change in induction linked with the circuit is equal to the product of the resistance of the circuit and the quantity set flowing through it, or

Induction x linkages = resistance x quantity.

If induction is measured in microwebers — one weber being 10s C.G.S. lines or units of induction, and one microweber therefore 100 C.G.S. units — we have the rule —

Microwebers x linkages - microcoulombs x ohms ;

and if one microweber of induction passing through the loop linked once with this circuit, is suppressed, it will cause one microcoulomb of electric quantity to flow through the circuit if its resistance is one ohm.

In employing the ballistic galvanometer to measure mag- netic induction, it must be observed that the condition of application of the above principles is that the whole change of induction must be completed in a very short time compared with the periodic time of the needle of the galvanometer.

To suggest, as is sometimes done, that the induction in the field magnets of a dynamo can be measured by surrounding

€06 APPENDIX.

them with a loop of wire connected to a ballistic galvanometer, and then short-circuiting or breaking the field circuit, is to presuppose a most unlikely event, viz., that the time during which such change of induction takes place is small compared with the periodic time of an ordinary ballistic galvanometer.

If the galvanometer is one with a movable coil of the d'Arsonval type, then it is better to standardize the galvano- meter by means of a coil of known length, turns and resistance, placed in the axis of a long cylindrical coil, the turns per unit of length of which are krfown. The reason for this is that if the galvanometer circuit is always closed the movement of the coil in the strong magnetic field induces currents in the galvanometer which resist its motion. Hence a powerful damping action comes into play from this cause alone. To ascertain what induction change caused a given galvanometer throw we proceed thus : —The standardizing secondary coil should be joined up in series with the galvano- meter coil and with the secondary or exploring coil, and it should be placed in the axis of a long coil, of which the field can be calculated. If, then, under the influence of an unknown induction linked or unlinked with the exploring coil we obtain a galvanometer throw 0, we can find out what was the induction causing this throw by interrupting or reversing a known current in the long standard field coil, and thus linking or unlinking a known induction with the standard secondary coil. If the current through the standard iield coil is altered until it gives a similar throw 6 on being interrupted or reversed, we know at once that the value of the unknown induction linked or unlinked with the secondary or exploring coil which gives an equal throw, must be the same as that calculable induction linked with the standardizing secondary coil.

INDEX.

Action at a Distance, 10

lotions taking place in Transformers, 518

Admittance of Condenser, 186

Alternating Current Curve Tracers, 522

Alternating Current Curve Tracing, 530

Alternating Current Flow in Conducting

| Circuits, 492

Alternating Current Flow, Theory of, 293

Alternating Currents, Distribution of, in

\ Conductors, 306

Alternating Currents, Propagation of,

through Conductors, 292 Alternative Path, Experiment of the, 401 Ampere's Discoveries on Electromagnetism,

Ampere's Fundamental Law, 15 Amplitude Factor of a Periodic Curve, 583 Analysis of Compound Periodic Curve, 90 Analytical Theory of Transformer, 586 Apparent Power, 157 Apparent Power given to Circuit, 205 •Ballistic Galvanometer, 51, Appendix Note

C., 600

•Ballistic Galvanometer, Use of, 51 Bernstein's Researches on Induction, 251 Blaserna's Researches on Induction, 246 .Branch Circuits, Impedance of, 163 Cardew Voltmeter, 155 Circuit, Inductive, 110 Circuit, Magnetic, 33 Circuit Non-inductive, 110 Circular Current, Magnetic Force at the

Centre of, 18

•Circular Solenoid, Magnetic Force in In- terior of, 23

Classification of Transformers, 517 Clock Diagram for Inducing and Induced

Circuits, 177 Clock Diagrams, 141 Closed Circuit Transformer, 515 Closed Magnetic Circuit, 38 Coefficient of Mutual Inductance, 121

Coefficient of Self-induction, 119

Coil Induction, Theory of, 232

Complex Periodic Functions, 202

Compound Periodic Curve, 84

Compound Periodic Curve, Harmonic Analysis of, 91

Condenser Equation, 183

Condenser, Flow of Current into, 182

Condenser, Time-Constant of, 184

Conducting Power for Lines of Force, 40

Conduction Current, 335

Confirmation of Maxwell's Theory by Hertz, 477

Correcting Factor of Wattmeter, 168

Current and ElectroinotiveForce Curves, 81

Current Diagram of Transformer, 561

Current Equation, 126

Current Flow in Circuits having Capacity, Inductance and Resistance, Initial Con- ditions of, 199

Current Flow in Inductive Circuits, Initial Conditions of, 194

Current Growth in Inductive Circuits, 123

Current Sheet, 339

Curve of Hysteresis, 60

Curve of Induction, Determination of, 528

Curves, Logarithmic, 130

Curves of Current and E.M.F. of Various Transformers on Open Secondary Circuit, 538, 539, 540

Curves of Magnetisation, 51, 55, 58

Curves of Power and Hysteresis of Trans- former, 551

Curves of Primary and Secondary Alternat- ing Electromotive Force of Transformer, 542

Cycle, Magnetic, 60

Delineation of Periodic Curves of Current and Electromotive Force, 519

Derived Curves, 103

Description of Simple Periodic Curve, 96

Description of Transformer Diagrams, 535

COS

INDEX.

Determination of Values, "v," 359 Dielectric Constants of Various Substances,

351-353

Discovery of Electromagnetic Induction, 2 Discovery of Induced Currents, 2 Displacement Current, 334 Displacement Currents and Displacement

Waves, 338

Direction of Magnetic Force. 14 Dove's Experiments on Induction, 262 Effect of Closing Secondary Circuit of

Induction Coil, 271 Effect of Form of Curve of Electromotive

Force on the Efficiency of Transformers,

579 Efficiency Curves of a Transformer taken

on Various Alternators, 582 Efficiency Curves of Transformers, 557 Efficiency of Transformers, 555 Electrical Oscillations, 372 Electrical Researches of Faraday, 1 Electric Currents produced by Magnetism, 5 Electric Displacement, 334 Electric Elasticity, 334, 333 Electric Surgings, 412 Electric Waves in Wires, 461 Electrodynamic Induction, 3 Electromagnetic Energy, 120 Electromagnetic Gyroscope, 324 Electromagnetic Induction, 14 Electromagnetic Induction. Discovery of, 2 Electromagnetic Medium, 333 Electromagnetic Momentum, 118 Electromagnetic Radiation, 478 Electromagnetic Repulsion, 307 Electromagnetic Repulsion, Theory of, 315 Electromagnetic Rotations. 320 Electromagnetic Theory, 332 Electromagnetic Waves in Air, 469 Electromotive Force Curves, 81 Electromotive Force Diagram for Inductive

Circuit, 144

Electromotive Force of Induction, 72 Electromotive Force of Rotating Coil, 76 Electromotive Intensity, 337 Electro-Optic Phenomena, 478 Electrostatic Induction, 1 Electrotouic State, 7, 118 Elihu Thomson's Experiments of Electro- magnetic Repulsion, 313 Energy Dissipation by Hysteresis, 64 Equation for Charge of a Condenser, 183 Equation for Periodic Current in Inductive

Circuit, 135 Equation for Rise of Current in Inductive

Circuit, 127

Equipotential Surface, 46 Ether, 333

Experimental Determination of Electro- magnetic Wave Velocity, 499

Experimental Determination of Inst; taneous Value of a Periodic Current. 62C

Experimental Determination of the Form of Periodic Curves of Transformer. 527

Experimental Measurement of Periodic Currents and Electromotive Forces, 154

Experimental Proof of Existence of Oscilla- tory Discharge, 381

Experimental Researches on Alternating Current Transformers, 558

Faraday's Copper Disc Experiment, 5

Faraday's Discoveries, 1

Faradav, Discovery of Self -Induction byr 111 "

Faraday's Electrical Researches, 1

Faraday 'sExperiment with the Iron Ring, 3

Faraday's Law of Induction, 31

Faraday's Ring Coil, see Frontispiece

Faraday's Theories, 6

Flow of Simple Periodic Currents into Condenser, 182

Flux of Magnetic Force, 26

Force, Magnetic, 43

Form Factor of a Periodic Curve, 583

Form Factor of Various Curves, 584

Fourier's Theorem, 85

Function of the Condenser in an Induction Coil, 390

Galvanometer, Ballistic, 51, Appendix Note C., 600

General Analytical Theory of the Trans- former. 586

General D.scription of the Action of Transformer, 514

Geometrical Illustrations, 138

Graphic Representation of Periodic Cur- rents, 139

Harmonic Analysis of Transformer Dia- grams. 550

Harmonic Curve, Description of, 87

Hedgehog Transformer, Current Diagram of, 562

Helrnholz's Researches on Induction, 252

Henry, Discovery of Electro-magnetism by, 11

Henry, Joseph, 11

Henry Joseph, Researches of, 207

Henry's Discovery of Induced Currents of Higher Orders, 219

Henry's Electromagnets, 11

Henry's Experiments on Self and Mutual Induction, 207

Henry's Experiments with Electromagnets. 12

Henry's Investigations, 11

Henry, The, 122

INDEX

Hertz's Confirmation of Maxwell's Theory,

449

Hertz's Electrical Papers, 461 Hertz's Experiments of Propagation of

Alternating Currents in Conductors, 492 Hertz's Researches on the Propagation of

Electromagnetic Induction, 418 Hertz's Resonator, 426 High Frequency, Effect of, in changing

Distribution of Current in Conductor,

294

Historical Introduction, 1 Hopkinson's Researches on Hysteresis, 65 Hughes' Experiments on Transmission of

Alternating Currents, 281 Hughes' Induction Balance, 274 Hughes' Induction Bridge, Theory of, 285 Hughes' Sonometer, 276 Hysteresis Curves, 60 Hysteresis Curve of Ganz Transformer, 553 Hysteresis, Dissipation of Energy by, 64 Hysteresis, Magnetic, 59 Hysteresis, Magnetic Variation of, with

Temperature, 68

Hysteresis, Value of, for complete Cycle, 62 Hysteresis, Values of, for various kinds of

Iron, 63

Impedance, 136 Impedance Diagram, 146 Impedance of Branch Circuits, 163 Impedance to Sudden Rushes of Current,

416 Impressed and Effective Electromotive

Forces, 143

Impulsive Discharges, 399 Impulsive Impedance, 417 Index of Refraction of Various Substances,

350

Induced Currents, 239 Induced Currents, Discovery of, 2 Induced Currents, Duration of, 246, 247 Induced Currents, Duration of, Helmholz's

Researches on, 254 Induced Currents, Felici's Researches on,

243

Induced Currents of Higher Orders, 219 Induced Currents, Investigations on, 226 Induced Currents, Leuz's Researches on,

243

nduced Currents, Magnetic Effect of, 231 Currents, Various Effects of, 230 ....jce, 108, 119

stance and Inductive Circuits. 107

iductance and Inertia, Analogies of, 109

iductance, Annulment of, by Capacity,

189

Inductance Bridge, 116 Inductance, Discovery of, 111

Inductance, Mode of exhibiting, 113 Inductance of Various Circuits, 123 Induction at a Distance, 2l4 Induction Balance, 274, 279 Induction Coil, Theory of, 232 Induction, Electromotive Force of, 72 Induction, Faraday's Law of, 31 Induction in closed Magnetic Circuits, 36 Induction in open Magnetic Circuits, 37 Induction, Magnetic, 24 Induction Mutual, 211, Induction of Electric Currents, 3 Induction Phenomena in open Circuits.

424

Induction Phenomenon in Dieletrics, 449 Induction, Prevention of, 259 Induction, Willoughby Smith's Researches

on, 261

Inductive Circuit, 110 Inductive Circuits, Division of. Current

between, 159

Inductive Effects by Leyden Jar Dis- charges, 224 Inductive Effects by Transient Electric

Currents, 222

Initial Conditions at Starting Current, 194 Instantaneous Value of Periodic Current,

135 Integral Calculus, Important Theorem in,

90

Intensity of Magnetisation, 42 Interference of Electric Waves, 471 Interference Phenomenon of Electric

Waves, 464

Iron Ring, Permeability of, 53 Joubert's Instantaneous Contact Maker,520 Kelvin, Lord, on Flow of Alternating

Currents in Conductors, 303 Kunz's Researches on Magnetic Hysteresis,

68

Lemma, Trigonometrical, 161 Lightning Protectors, 403 Light, Velocity of, 358 Line Integral of Magnetic Force, 25 Line of Magnetic Induction, 29, 43 Lin*'-; of Force cutting Conductor, 8 Lines of Magnetic Force, 7 Linkage of Circuits and Lines of Magnetic

Induction, 47

Linkages of Magnetic and Conducting Cir- cuits, 49

Lodge, Dr., on Lightning Conductors, 401 Logarithmic Curves, 130 Magnetic Axis, 13 Magnetic Circuit, 33 Magnetic Circuit, Closed, 38 Magnetic Circuit, Open, 38 Magnetic Cycle, 60

BB

610

INDEX.

Magnetic Force, 43

Magnetic Force and Magnetic Fields, 13

Magnetic Force at the Centre of a Circular

Current, 18 Magnetic Force in Interior of Circular

Solenoid, 23 Magnetic Force in the Interior of Straight

i Solenoid, 20 Magnetic Force near a Straight Conductor,

Magnetic Hysteresis, 59 Magnetic Induction, 24 Magnetic Induction, Line of, 43 Magnetic Induction, Tube of, 43, 44, 45 Magnetic Induction, Unit of, 30 Magnetic Leakage, 328, 572 Magnetic Leakage, Calculation of, 576 Magnetic Leakage, Causes of, 574 Magnetic Leakage, Rule for, 578 Magnetic Permeability. 27, 39 Magnetic Permeability, Determination of.

52

Magnetic Potential, 24 Magnetic Reluctance, 40 Magnetic Researches, 56 Magnetic Resistance, 33 Magnetic Resistance, Definition of, 34 Magnetic Resistance, Specific, 34 Magnetic Saturation, 57 Magnetic Screening, 255 Magnetic Screening, Faraday's Experi- ments, 257 Magnetic Screening, Henry's Experiments

on, 258

Magnetic Screening, Theory of, 270 Magnetisation. Curves of, 51 Magnetisation, Intensity of, 42 Magneto-electric Induction; 6 Magnetomotive Force, 24 Mathematical Sketch of Fourier's Theorem.

89

Maxwell's Mode of shewing Inductance, 116 Maxwell's Theory of the Electromagnetic

Medium, 344 Maxwell's Theory of Molecular Vortices.

339

Mean - Square and Maximum Value of , Periodic Current, Relation of, 137 Mean-Square Value of Ordinate to Curve. . 103 Mechanical and Electrical Quantities,

Analogies of, 126 Slechanical Illustrations of the Properties

of the Electromagnetic Medium, 342 Molecular Vortices, 339 Mordey's Alternator, Curve of E.M.F. of.

537 Multiple-valued Function, 82

Mutual and Self Induction, 207

Mutual Induction between Telephone Cir- cuits, 216

Mutual Induction of two Circuits, 173

Mutuvl Induction of two Circuits, Theory of, 232

Non-inductive Circuit, 110

Open Circuit Transformer, 515

Open Magnetic Circuit, 38

Oscillations of Leyden Jar, 375

Oscillatory and Non-oscillatory Discharge, 379

Oscillatory Discharge Experimentally In- vestigated, 381

Periodic Currents, 79

Periodic Currents, Graphic Representation of, 139

Permeability, Curves of. 55

Permeability Magnetic, 27, 39

Permeability, Magnetic, at Different Temperatures, 56

Permeability, Magnetic, corresponding to various Magnetic Forces, 54

Polar Diagram for Periodic Currents, 190

Polar Diagram, Important Use of, 193

Power Curves, 150

Power Curves for Induction and Non- inductive Circuits, 152

Power Factor, 206

Power Factor of Transformers, 566

Power Factor, Relation of, to Secondary Output, 571

Power Factors, Influence of the Form of Curve of Electromotive Force on, 568

Power Factors of Various Types of Trans- formers, 567

Power of Periodic Current, Mean Value of, 147

Poyutiiig's Theorem, 483

Propagation of Alternating Currents through Conductors, 292

Propagation of Currents along Conductors, 488

Propagation of Electromagnetic Energy, 481

Ratio of Electromagnetic to Electrostatic Units, 355

Rayleigh, Lord, Induction Bridge devised by, 298

Rayleigh, Lord, Investigations on Induc- tion by, 289

Reactance, 146

Reaction of Closed Secondary Circuit on the Primary. 271

Relation of Power Factor to Secondary Output, 571

Reluctance, Magnetic, 40

Repulsion, Electromagnetic, 307

INDEX.

611

Resistance, Magnetic, 33

Resonance Phenomenon, 428

Rise of Current in Circuit, 132

Roessler's Experiments on Transformers.

• 580

Rotating Coil, Electromotive Force of, 76

Rowland's Experiments on a Rotating Electrified Disc, 357

Screening, Magnetic, 266

Secondary Drop, 572

Secondary Drop Curves of Transformer, 573

Self Induction, 207

Shunted Condenser in Series with Induc- tive Resistance, 187

Shunted Condenser, Theory of, 396

Side Flashing, 407

Simple Periodic Currents and Electro- motive Forces, 93

Simple Periodic Currents, Theory of, 79

Simple Periodic Curve, 83

§ine Curve, 88

Sine Curve, Mean-Square Ordinate of, 101

Sine Curve, Value of Mean Ordinate of, 98

Single-varied Function, 82

Slow and Quick Cycle Hysteresis, 65

Solenoid, Magnetic Force in Interior of, 20

Sonometer, 276

Specific Inductive Capacity, 350

Specific Magnetic Resistance, 3^1

Straight Conductor, Magnetic Forces near a, 15

Strength of Magnetic Field, 15

Strength of Magnetic Pole, 14

Surface Integral of Induction, 28

Symmetry of Current and Induction, 329

Symmetry of Transformer Diagrams, 549

Table of Efficiencies of Various Trans- formers, 560

Tallies of Specific Inductive Capacity, 351

Telegraphing to Moving Trains, 217

Telephonic Induction, 216

Temperature Effect on Hysteresis, 68

Test of Swinbourne Hedgehog Trans- former, 565

Te^t of a Westinghouse Transformer, 564

Theory of Air- Core Induction Coil, 174

Theory of Experiments on Alternative Path, 413

Theory of Hertz's Experiments, 435

Theory of Induction Balance, 285

Theory of the Transformer, 586

Thomson-Houston Alternator, Curve of

K.M.F. of, 536 Time-Constant of Circuit, 131

Time-Con.jtant of Condenser, 184

Time of Quickest Discharge of Condenser,

386

Transformation Ratio of Transformer, 575 Transformer, Analytical Theory of, 586 Transformer Curves of Current and Elec- tromotive Force, 538, 539, 540 Transformer, Current Diagram of, 561 Transformer Diagrams, 543. Transformer Diagrams, Description of, 535 Transformer, General Action of, 514 Transformer, General Analytical Theory

of, 586

Transformers, Classification of, 517 Transformers, Efficiency Curves of, 557 Transformers, Efficiency of, 555 Transformers, Power Factor of, 566 Transmission of Alternating Currents

through Conductors, 281 Trigonometrical Lemma, 161 Trowbridge's Experiments on Electro- magnetic Waves, 500

Trowbridge's Experiments on the Propaga- tion of Electrical Oscillations, 431 True Power given to Inductive Circuit, 157 Tube of Magnetic Induction, 43 Unit of Inductance, 122 Unit of Magnetic Induction, 30 Value of Mean Ordinate Sine Curve, 98 Value of Mean-Square Ordinate of Sine

Curve, 101

Variable and Steady Flow, 79 Variation of Hysteresis with Temperature,

70

Vector Potential, Explanation of, 360 Vector Quantities, 27 Velocity of Light, 358 Velocity of Propagation of Electromagnetic

Disturbances, 354 Velocity of Propagation of Electromagnetic

Waves experimentally determined, 511 Velocity of Propagation of Vector Poten- tial, 368

Wattmeter, Correcting Factor of, 168 Wattmeter Measurement of Periodic Power,

166

Wattmeter, Theory of, 157 AVave Diagram for Periodic Currents, 192 Waves in Rectilinear Wires, 458 Westinghouse Transformer, Current Dia- gram of, 562 Westinghouse Transformer, Diagrams of,

544, 545

Willoughby Smith's Experiments on Magnetic Screening, 262

LONDON : PRINTED BY CEOKGE TUCKER, SALI-SBl'RY COURT, FLEET STREKT.

IK 2792 • F 62 a " 1896

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Los Angeles This book is DUE on the last date stamped below.

Physic* Library

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JAN 6 -

JUN1 9 1961 m 24 1962

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JUN 081988

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