The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 12 of 36

§33. Initial Oonditiona on starting Current Flow in a Oir- enit having Resistance and Inductance. — It has been shown in the foregoing sections that if an impressed electromotive force of simple periodic kind acts upon a circuit having in- ductance, the resulting current is a simple periodic current, but lags behind the impressed electromotive force in phase. These, however, are the conditions when the resulting current has become steady. At the instant of closing the circuit there are peculiar conditions of augmentation of the current which are called initial conditions, gjid which have very important

SIMFLE FERIODIC CUJtiRENTS. 105

eonseqnencea in practice. It will be advisable, therefore, to •examine a little more closely how these are produced, and what results may be expected at the instant of starting or stopping the current in such a circuit;. To do this we will| in the first place, examine more carefully the solution of the fcmdamental equations for current creation in an inductive circuit. It has been shown that the differential equation for current at any instant in the circuit of constant inductance L and resistance B under an impressed simple periodic electro- motive force v^Y siaptia

L^* + R* = t; = Vsini>«, . • . (61)

To solve this equation completely, we differentiate it twice, and eliminate thereby the term V sin ^ t, thus obtaining the equation

A differential equation of this type is called a linear differ- -ential equation of the third order. It is shown in treatises on differential equations that its solution depends on the solution of a cubic equation called the auxiliary equation. The auxiliary equation in this case is

L7n3 + Rm«+i)2Lm+^2E=0. . • . (63) This cubic equation can be split up into two factors and be written

(m2 4-p2)(L?w + R)-0,

and hence the roots of the cubic equation (68) are m= ± J -Ip

--i-

For a linear equation of the jfcype of (62), the roots of the auxiliary cubic being a and ± J -ip, the solution is known to be of the form

t = Ae**+Bsin/?« + B'co9)8t. The solution of the differential equation (62) is, then, given by

i^ke ^+Bsm^tH-B'co8/>t. . . (64)

o2

196 SIMPLE PERIODIC CURRENTS.

By the Trigonometrical Lemma on page 161 we can write^ instead of the second and third terms on the right hand sid* of (64), the single term JWTB^ sin (pt - <f>), where VB« + B'« is obviously the maximum value of the current-~call it I-^ when the initial state is over, and <^ is the angle by which the current, when steady, lags behind the electromotive force in phase. Hence (64) may be written,

i-Aa ^^ + l3in{pt "<!>). • • . (65) To find the constant quantity A, we note that at the instant when the circuit is closed the current has necessarily a value zero. Let this closing of the current happen at a time if reckoned from the instant when the electromotive force ia- zero. Then at the instant of closing the circuit we have

0-Ar^*' + Isin(j9t'-<^),

R

or A=-Isin(p«'-<^)c+^*.

Hence, substituting this value of A in equation (65), we obtain

»-Isin(pt-<^) + Isin(p«'-<^)<j'ii^*"^\ . . (66) and this is the complete solution of the differential equatioa (61) for the current in the circuit.

We note that the expression for the current at any instant in the inductive circuit is made up oi two terms ; the first term, I sin (^ t^ <^), is a simple sine function, and represents hf itself a simple periodic current having a maximum value L

The second term, I sin {p t' - <i>) e' ^^^'^\ is an exponential function having a maximum value I sin (/>«'- <^), when «'=t^ and this term represents, therefore, a logarithmic curve begin- ning with the value I sin (p t' - <^) and dying away gradually to zero.

Hence the resulting current curve consists of these two curves superimposed upon one another, a periodic curve and a logarithmic or diminishing curve.

In Fig. 77 are shown two such curves ; curve 1 being the sine curve, curve 2 the logarithmic curve, and the resultant curve 8, represented by the dotted line, which is obtained by adding together the ordlnates of the sine curve and the logarithmic curve* It will be seen that the effect of the

SIMPLE FERIODIC CURRENTS. 197

^nperpoEdtion is to make the resultant curve lopsided with lespect to the ourve axis for a certain period, but beyond that time it is sensibly symmetrically situated with respect to the time axis. Hence, during this initial period, the maximum Talues of the current in opposite directions are not the same. At the instant when t = if the value of the current is zero. It is obvious that, when p<' = 90 + <^, sin (;)«'-<^) = l, and that then the logarithmic curve begins with its greatest value ; but that, when ^^ = <^, then the logarithmic curve has no existence at all. When pif =^90 + 4>—th2kt is if the circuit

is closed at an instant t» — — ^ reckoning from the instant

P when the impressed electromotive force is zero— the disturbance of the uniformity of the periodic curve of current is the

Fio. 77.

greatest possible. At the same time the maximum value of

the current in the negative direction can never be greater

than 2 1 where I is the maximum value of the steadily periodic

43arrent. For the maximum value of the logarithmic curve at

the instant of closing the circuit is - 1, and at that instant

«=0 ; hence the value at which the periodic component of the

current must begin will be + 1. At the time when the periodic

part has reached a maximum of - 1, which happens after half

a period, the ordinate of the logarithmic curve will have fallen

to something less than I by an amount depending on the rate

R of decrease, which in turn depends upon the value of ^- or the

jj

xeciprocal of the time-constant of the circuit.

1»8 SIMPLE PERIODIC CUBRENT3.

Consider the case when the circuit is closed at a time f reckoned from the zero of electromotive force such that

t' = ?lt^, thenpf-<^=90.

T If T is the complete periodic time, then at a time - after

2

the instant of closing the value of the periodic part in the

expression for the current is

Isin jp(e'+|.)-<^|

= 1 sin {pt' -<f>-[-7r) = -I since /> /' - <^ = 90.

The value for the exponential part at this instant t'+^ ia

_ RT

• 1 sin (p «' - <^) ^ " t: a

and hence the total value of that current at that instant is

• 1 { 1 + sin (;> «' - <^) « " lt}

= -I {l+^"ll}' since ^«'—<^= 90.

The greatest value which the time constant :^ can have is

infinity. Hence, when _ = 0, or approxunately zero, e ^ > = 1^

and the quantity in the bracket in the last equation may approach to 2 but can never exceed it. This shows us that, with an inductive circuit of very large time constant, the value of the current after half a period from the instant of closing the circuit may be something a httle less than twice the value of its periodic steady maximum, provided that the circuit is dosed at the instant when the electromotive force has a value e such that

e=E sin (90 + <^),

or esEcosi^,

where tan <^ = --?, a

This amounts to saying that the circuit must be closed at

the instant of zero electromotive force.

SIMPLE PEBIODIC CURRENTS. 199

Hence we see that daring the initial period there may be a greatly increased mean-square value of the current, and thus the production of a current-rush on closing the circuit.

When therefore a circuit of constant inductance is switched on to a source of steadily periodic electromotive force at the instant when the electromotive force has a value corresponding to the maximum value of the current, when the variable state is over, we find that, before the current settles down into its steady swing, lagging behind the electromotive force in phase, there is a period of disturbance during which the current has greater maximum values in one direction than in the other, and the current virtually consists of a rapidly evanescent tmidirectional current superposed upon the normal periodic current which ultimately survives. If however the circuit is closed at the instant when the electromotive force is passing through a value which corresponds to that at which the current has its zero value, when the variable period is passed, then there is no period of disturbance, but the current begins at once in its normal manner lagging behind the electromotive force and having constant maximum values + 1 and - 1 alter- nately. This phenomenon of current rushes into inductive circuits such as transformers will be treated at length in a later chapter, and the attention of the reader is merely at this point directed to the general nature of the effects taking place in a circuit of constant inductance when suddenly switched on to a source of simple periodic electromotive force.

§ 34. Initial Oonditions in Circuits having Capacity, Indnctance and Resistance. — It is somewhat more difficult to discuss completely the conditions which arise at the instant of connecting to a source of periodic electromotive force a circuit having not only inductance and resistance but also capacity. Generally speaking, they may be described as consisting of a variable period and a succeeding steady period. 'W^e shall in outline indicate how these conditions respectively arise. Sup- pose, in the first place, that a condenser of capacity G is con- nected through an inductive resistance of inductance L and resbtance R to a source of periodic electromotive force of which the value at any instant is represented by v=' sinpt.

200 SIMPLE PERIODIC CURRENTS.

Let % be the instantaneous value of the current flowing into the circuit, and let v be the value at the same instant of the potential difference between the terminals of the condenser, and vi the fall of potential down the inductive resistance. Then we have the following fundamental equation connecting the current and potential : —

For the resistance L ^ + R t = I'l, (67)

at

for the condenser C -- = i, (68)

a t

and for the total fall of potential

v' = v + i\ (69)

Hence, from (67) and (68),

L'ii + Ri + lfirf« = t;' = V'8ini}«, (It cj

and eliminating i by the help of (68), we get

LC^+R0'l^ + t; = V'8ini>«. . . . (70) dt* dt

This is the differential equation defining the value of the condenser terminal potential in terms of the constants and the time. To solve (70) we must eliminate V sin^^ This is done by differentiating (70) twice and eliminating V sinj> t between the original equation (70) and the twice differentiated equation. As a result we reach the equation

LC^ + CR^V(l + CLp^)^ + CByl!+2,'r = 0.(71) tt r civ dt^ dt

The solution of this linear differential equation of the fourth order depends on the solution of the biquadratic

L C m* + C R wiH (I + C Ly>2) ,,^84.0 R/?* m+i?« = 0, and this last splits up into two factors

(w* +y ) (C L m'^ + C R wi + 1) = 0. Hence the roots of this biquadratic are,

2L- >/ V 4CL^

SIMPLE FEBIODIC CUBBENTS. 201

It is shown in treatises on differential equations that the solution of equation (71) is then

v^Amnpt+BooBpt + k' e «l sin qt + B' e'v^oos qt, . (72) where A, B, A', B' are constants and

V 4CL* The solution for v may obviously be written

«- VA2 + B2sin(|?t-<^)+ v'A'24-B'2«"iS«gin(^e-^'). (78)

This equation for the value of the condenser potential v shows us that the variation of v is made up of two parts. First a simple periodic part VA^ + B'^ sin {p t - <^), which may be written V sin (;? t - <^), and which indicates a simple periodic -variation of v differing in phase from the impressed electro* motive force v' by an angle <^. The other term of the solution indicates a superposed periodic variation, gradually decreasing in amplitude as time increases, and dying out as the exponent

~ t increases with time. If, for the sake of brevity, we write 2Ii

C for V^A'*+B'', we can put the solution for v in the form

v^Y sin {pt''<f>)'^Ce''^^ Bin (qt-if>'). . (74)

Two constants have therefore to be determined, viz., C and <f>\ in order that we may completely solve the problem.

Since the quantity of electricity in the condenser at any instant is numerically equal to the product of the capacity and potential, if we multiply (74) all through by C, the con* denser capacity, we have an expression for the charge x in the <sondenser at any instant. Since this charge is null at the instant of closing the circuit, if the circuit is closed at the instant t', reckoning time from the instant when the impressed electromotive force is zero, we have

0 = V sin (pf - <^) + C' «""i^ sin (yt'- <^') as an equation to determine the constant C.

Therefore C'= -I?i"i^1.<^)^. . . • (76) .

sin (q t' - </)')

202 SIMFLJH FERIODIG CURRENTS.

No sufficient advantage is to be obtained by. working oat the rather complicated algebraical expressions in terms of G, L, B, and p, for the constant <^\ but we can indicate gene- rally what the equations teach. They show us that if such a condenser in series with an inductive circuit is switched on to a source of periodic impressed electromotive force, before the oscillations of the condenser potential settle down into a regular state, there is a variable period in which a second set of oscillations of gradually diminishing amplitude are superimposed on the steady set, and the second set of oscil- lations have a quite different frequency and initial amplitude to the steady set which ultimately survive. In the initial period the superposition of the two sets of oscillations may increase the instantaneous value of the condenser terminal potential difference to a value greater, and perhaps much greater, than it would have if there were no such additional oscillations. If the circuit is closed at an instant t' such that j9t' = <^, viz., at an interval after the impressed electromotive force has passed its -zero value equal to the final permanent difference of phase of condenser and impressed electromotive force, then, since this value makes the constant Q' zero, we see that there are no superposed vibrations at all. On the other hand, if t' is so chosen that pt' = 90° + <^, then the disturbing effect is greatest. It is clear, therefore, that, in switching on a condenser to an alternating current circuit through an inductive resistance, initial effects of abnormal rise of condenser voltage may result. We shall see later on that these are practically very important matters in dealing with alternating current systems of supply, and that caution must always be used in switching on a con- denser to such a circuit.

§36. Oomplex Periodic Functions.— Before leaving the subject of periodic currents and electromotive forces it is desirable to explain some properties of the trigonometrical expressions or series by which such functions can be represented as the sum of a series of simple periodic con- stituents or terms. It has already been explained that the value of an ordinate y of any single valued function can be expressed by Fourier's method as follows :

y = Yo + Yi sin (/?« + <^i) + Ya sin (2jd« + <^,) + , &c. ; (76)

SIMPLE FERIODIC CURRENTS, 203

ir, taking advantage of the trigonometrical equality

A sin ^ + B cos 0=^ >/ A^ + B* sin (^ + </>), where ^tan <^ = j, we can write the above expression for // y = Yo + yi8in|t)« + 2iCos2?« + !/iSin 2;?« + 24Cos2;?t + itc. (77)

The coefficients Yi, Ya, &c., in the series in (76) are the amplitudes of the simple sine curves or harmonic constitutents whose added ordinates together build up the function y.

If the periodic quantity represented is a wave curve symmetrical with respect to the axis of time, and repeating the same form continually, then the constant term Yq is absent.

We can therefore express the instantaneous value e of any periodic electromotive force by the expression

<«Eisin/)« + FiCosp« + E2 8in2^« + FaCos22)t + ,&c (78)

and the instantaneous value i of any periodic current by the series

t = Iisinpt + JiCOSj?e+Iasin 2^^* + JaC0s2p« + , &c. (79)

In the series (78) the amptitude of the first harmonic is i^Bi* + Fi«and that of the second Ve7+F7, and so on.

We have already seen that if we take the mean or average value of sin* 0 over one half-period, or from 0 to ir, the value

of the mean is 1- ; but that the mean value of such a product

as sin d cos d or sin ^ sin 2 0 over half a period is zero.

Accordingly, if we square the series in (78) or (79) we find that the values for ^ and for t* partly involve terms like sin'pt, sin'2p<, &c., and partly terms hke sinjpt sin2/?e. Hence, if we integrate the value of ^dtot i^dt over half a period and divide the result by tt, or take the definite integrals

Z-f^^dt, ^j^i^dty which is equivalent to finding the mean- square values of e and i, we find by the above theorem that

204 SIMPLE PERIODIC CURRENTS.

and, similarly, by multiplying (78) and (79) and taking thd mean value of the products from ^ t = 0 to /? t = ir, we obtain

'jfj Q 2 2 2 2

The ordinary alternate current ammeter or dynamometer measures the Vmean' value of the current i, or the quantity

j^ -fi^dt; and the ordinary alternating voltmeter, such as

an electrostatic voltmeter, measures the Vmean* value of the

•electromotive force e, or the quantity a/ - r^ ^ *» whatever be

^ Wo ihe force of the curves of e and L A wattmeter in proper form

reads the true mean product of«, t or - {^ eidtyihea currents

Wo respectively proportional to « and t traverse its two coils. For

-shortness, let us write e' for \J -T ^^dt, and i' for the similar

^ Wo

function of i", and (^ i') for -T eidt; then

W 0

2«' = Ei« + F;- + , Sec,

2t' = V + Ji>+,&c.,

2(/i') = EiIi + FiJi + , &c.

We see therefore that twice the V mean''^ value of e or t is •equal to the sum of the squares of the coefficients of the sine and cosine terms, or to the sum of the squares of the amplitude of the harmonic constituents. Also that the ttcice mean product of two periodic functions taken over a half period at similar instants is equal to the sum of the products of the coefficients of similar sine or cosine terms taken in pairs from each expansion. Suppose that e and i represent the instantaneous values of the impressed electromotive force and current of any circuit, we may ask whether the product •of the Vmean* value of e and the v^mean^ value of i is equal to, greater, or less than the mean of the product of e and if

or

is V- re^dtxV- r^^dtoT- HeidU

SIMPLE PERIODIC CURRENTS. 20&

We see that the square of the expression for 2 (^'i') is made up of terms like E^^Ij^, and F^^J^^, and also of terms likfr

The product of the expressions for ^ and i' consists obviously of terms like Ej*!^, F^ J^*, and also of terms like F,«Ii«andEi«Ji2.

Hence, the question whether the product of e' and t' or 0* X i' is or is not greater than (e* t') will depend upon the relative collected magnitude of terms like Fj^Ij* + Ej* Jj* in the one series, and of terms hke 2 EiIiF^Ji. If FilEiUJilIj^

or if |^' = ^S then E^^ J,2 + F,2I2 = 2 EiFJiJi, and if the samfr El Ii

proportion holds good for the coefficients of the terms in

Bin2jpg, &c., then we see that e' x i' = (e'i'), or the product of

the ^mean* values e and i is equal to the mean product of e

and i. If the above proportionality does not hold good, then,

since generally d^ + h^ is greater than 2 a 6, it is not difficult

to see that c'Xi' is greater than («' %') or the product of

^y 0 ^y 0

is greater than the mean value- jetdL

ttJ 0

Hence it follows that, in this last case, the product of the^ amperes and volts as read on alternating current instruments is greater than the true value of the power as read on a watt*

meter. The mean value - je idt is called generally the

true fvatts or power given to the circuit, and the product

V^ f^€^dtx\ - fi^dt is called the apparent vfattf or

power given to the circuit. The apparent watts are equal under some conditions to the true watts. This is the case when the circuit is non-inductive, and when the different harmonic constituents of the current and electro- motive force are in step or in synchronism with each other.. Under these conditions the angle of lag of the electromotive force harmonics is equal to the lag of the current harmonics of the same degree, and this is expressed by relations of the

206 SIMPLE PERIODIC CURRENTS,

form Pi : El : : Ji : Ij holding good. Under other conditions, the apparent watts are greater than the true watts. The ratio of the true power or watts taken up in any oircuit to the apparent power or watts is called the power factor of the circuit, and the power £EM!tor (P.F.) is therefore given for any alternating current circuit by the ratio of the wattmeter reading to the product of the ammeter reading and voltmeter reading for that circuit.

CHAPTER IV.

MUTUAL AND SELF INDUCTION.

§1, Researches of Prof. Joseph Henry— We have already, in the first chapter, made a brief allusion to the share taken by Joseph Henry in the fundamental discovery of the induction of

Fig. 78.

electric currents. A full account of his labours in this field is to be found in the collected ** Scientific Writings of Joseph Henry/' republished by the Smithsonian Institution. It will be of advantage to consider at this stage some of his chief investigations.

The principal pieces of apparatus used by Henry in his experiments on the induction of electric currents consisted of

• See also the Pkilosophtcal Mofjazine, Vol. XVI., 3rd Sen, 1840, and TranmctUms of the American Philosophical Society, Vol. VI., 1838, pp. 303-337.

208 MUTUAL AND SELF INDUCTION.

a number of flat ooils of copper strip or band, which were designated by the names Coil No, i, Coil No., 2^ &c., also several long bobbins of wire, and these, to distinguish them from the ribands, were called Helix No, i, Helix No. 2, &c.

His description of these coils and helices is as follows : Coil No. 1 was formed of thirteen pounds of copper strip one inch and a-half wide and ninety-three feet long; it was well covered with two coatings of silk, and was generally used in the form represented in Fig. 78, which is that of a flat spiral sixteen inches in diameter. It was, however, sometimes formed into a ring of larger diameter, as is shown in Fig. 79.

Fio. 79.

Ooil No. 2 was also formed of copper strip of the same width' and thickness* as coil No. 1. It was, however, only sixty feet long. Its form is shown at 6 in Fig. 78. The opening at the centre was sufficient to admit helix No. 1. Coils No. 8, 4, 5, 6, were all about sixty feet long, and of copper strip of the same thickness, but of half the width of coil No. 1.

Helix No. 1 consisted of sixteen hundred and sixty yards of copper wire ^th of an inch in diameter; No. 2 of nine hundred and ninety yards, and No. 8 of three hundred and fifty yards of the same wire. These helices were wound on bobbins of such size as to fit into each other, thus forming one long helix of three thousand yards, or, by using them separately and in dififerent combinations, seven helices of different lengths. The wire was covered with cotton thread

MUTUAL AND SELF INDUCTION. 209

saturated with bees' wax, and between each stratum of spires a coating of sUk was interposed.

Helix No. 4, shown at a, Fig. 79, was formed of five hundred and forliy-six yards of wire ^^th of an inch in diameter, the several spires of which were insulated by a coating of cement.

Helix No. 6 consisted of fifteen hundred yards of silvered eopper wire, ^l^th of an inch in diameter, covered with cotton, and of the form of helix No. 4.

In addition, a long spool of copper wire covered with cotton, ^th of an inch in diameter and five miles long, was provided. It was wound on a small axis of iron, and formed a solid oylinder of wire eighteen inches long and thirteen in diameter.

For determining the direction of the induced currents Henry employed a magnetising spiral, which consisted of about thirty spires of copper wire in the form of a cylinder, and so small as just to admit a sewing needle into the axis.

Also a small iron horseshoe is frequently referred to, which was formed of a piece of soft iron about three inches long and f ths of an inch thick ; each leg was surrounded with about five feet of copper bell wire. This length was so small that only a current of considerable strength could develop sensible magnetism in the iron. This horseshoe was used for indicat- ing the existence of such a current. The battery which was used was a simple copper-zinc cylinder battery, having about If square feet of zinc surface. In some experiments a series of cells was used, but most experiments were performed with one or two cells of the above kind. For interrupting the circuit of the conductor Henry employed the simple device of scraping one end of the conductor along a rasp held in contact with the battery terminal.

Provided with this apparatus, Henry entered on a pre- liminary series of experiments on the induction of electric currents, and in 1888 published an account of his investiga- tions on the phenomenon which had been previously named by Faraday electro-dynamic induction. The fact which seems to have chiefly attracted the attention of the numerous investi- gators who rapidly entered the region of research opened out by Faraday's discovery of the mutual induction of electric circuits and the production of electric currents in conducting circuits by the variation of the magnetic inducti(Hi linked with

p

210 MUTUAL AND SELF INDUCTION.

them, and by Henry's discovery of the self-induction of electric circuits, seems to have been the possibility of obtaining from a single cell of a galvanic battery effects such as spark and shock. These effects connected what was then known as voltaic electricity with the then more familiar effects of electrification by friction.* Henry took up the train of investigation at this point, and proceeded to employ the above described helices and coils in an investigation of the facts of the self- and mutual-induction of electric circuits. His mode of operating was to dose the battery circuit by dipping the ends of a coil or helix into two mercury cups connected with the terminal plates, and then to break the circuit by lifting out one end from its mercury cup, the hands being at the same time in contact with the battery terminal and the end of the conductor which is being raised. In this way the extra current, or electro-magnetic discharge of the coil| passed through the operator's body.

When the electromotive force was small, as in the case of a thermopile or a large single cell, and the circuit taken was the fiat riband coil No. 1, ninety- three feet long, it was found to give brilliant snaps at the surface of the mercury when contact was broken, but the shocks were very feeble, and could only be felt in the fingers or through the tongue. The induced current in a short coil, which thus produced deflagration but not shocks, he called, for distinction, one of quantity.

When the length of the coil was increased, the battery being the same, the deflagrating power decreased, while the intensity of the shock continually increased. With five- riband coils in series, making an aggregate length of three hundred feet, and a small battery the deflagration was less than with coil No. 1, but the shocks were more intense.

There appeared to be, however, a limit to this increase of intensity of the shock, and this took place when the increased resistance or diminished conduction of the lengthened coil began to counteract the influence of the increasing length of the current. The following experiment illustrated this fact.

A coil of copper wire -^th of an inch in diameter was increased in length by successive additions of about thirty-

• For a more extended description of the historical order of disooveriet in connection with the induction coil the reader is referred to the First Chapter in the Second Volume of this treatise.

MUTUAL AND SELF INDUCTION. 211

two feet at a time. After the first two lengths, or sixty-four feet, the brilliancy of the spark began to decline, but the shocks continually increased in intensity until a length of five hundred and seventy feet was obtained, when the shocks also began to decline. This was, then, the proper length to produce the maximum effect with a single battery and a wire of the above diameter. With a battery of sixty cells (Cruick- shank's trough), having plates four inches square, scarcely any shock could be obtained when the coil formed a part of the circuit. If the length of the coil was increased, then the inductive effect became very apparent.

When the current from ten cells of the above-mentioned trough was passed through the large spool of copper wire, the induced shock was too severe to be taken through the body. Again, when a small battery of twenty-five cells having plates one inch square, which alone would give but a very feeble shock, was used with helix No. 1, an intense shock was received from the induction when the contact was broken. Also a slight shock in this arrangement was given when the ■contact was formed, but it was very feeble in comparison with the other. The spark, however, with the long wire and -compound battery was not as brilliant as with the single battery and short riband coil.

When the shock was produced from a long wire, as in the last experiments, the size of the plates of the battery might be very much reduced without a corresponding reduction in the intensity of the shock. A small battery was made, formed of six pieces of copper bell wire one inch and a* half long and an equal number of pieces of zinc of the same size. When the current from this was passed through a coil consisting of five miles of wire, the shock was given at once to twenty-six persons joining hands.

With the same coil, and the single battery used in the former experiments, no shock, or at most a very feeble one, could be obtained.

The induced current in these last experiments he called one of eonnderable intensity and small quantity.

§ 2. Mutual Induction. — Henry then passed on to consider the mutual induction of two circuits. Coil No. 1 (see c,

p2

212

MUTUAL AND 8ELF INDUCTION.

Fig. 80) was arranged to receive the current from a small battery of a single cell, and coil No. 2, 6, was placed over it with a plate of glass between to secure perfect insulation. As often as the current in No. 1 circuit was interrupted, a powerful secondary current was induced in No. 2. When the ends of the secondary were joined to a magnetising spiral, the enclosed needle became strongly magnetic. Also when the ends of the second coil were attached to a small water decomposing apparatus, a stream of gas was given off at each pole; and when the secondary current was passed through the wires of the iron horse-shoe, magnetism was developed. The shock, however, from the secondary coil was very feeble, and scarcely felt above the fingers. This secondary current had, therefore, the properties of one of moderate intensity but considerable quantity (to use the

Fio. 80.

terms then employed) when developed by the current in one flat riband coil acting on another flat riband coil.

Coil No. 1, remaining as before a longer coil, formed by uniting Nos. 3, 4, and 6, was substituted for No. 2. With this arrangement as a secondary circuit the magnetising power of the current and the brilliancy of the spark at breaking contact was less than before, but the shocks were more powerful — in other words, the intensity of the secondary induced current was increased, whilst its quantity was decreased;

A compound helix, formed by uniting Nos. 1 and 2 helices, and therefore containing two thousand six hundred and fifty yards of wire, was next placed on coil No. 1. The weight of this helix happened to be precisely the same as that of coll

MUTUAL AND SELF INDUCTION. 213

No. 2, and hence the different effects of the same quantity of metal (as secondary circuit) in the two forms of a long and short conductor could be compared. Wifch this arrangement the magnetising effects with the apparatus above-mentioned •disappeared. The sparks were much smaller and the decom- position less than with the short coil, but the shock was ahnost too intense to be received with impunity except through the fingers of one hand. The secondary current in this case was one of small quantity but of great intensity.

The following experiment is important in establishing the fact of a limit to the increase of the intensity of the shock as well as to the power of decomposition with a wire of given <liameter.

Helix No. 5, consisting of a wire y^^^h of an inch in diameter, was placed on coil No. 2, and its length increased to about seven hundred yards. With this extent of wire neither decomposition nor magnetism could be obtained, but shocks were given of a peculiarly pungent nature. The wire of the helix was further increased to about fifteen hundred yards ; the shock was now found to be scarcely perceptible in the fingers.

As a counterpart to the last experiment, coil No. 1 was formed into a ring of sufi&cient internal diameter to admit the great spool of wire, and, with the whole length of this (five miles), the shock was found so intense as to be felt at the shoulder when passed only through the forefinger and thumb. Sparks and decomposition were also produced, and needles rendered magnetic. The wire of this spool was yV^ of an inch in diameter ; and Henry noted therefore from this experiment that, by increasing the diameter of the wire, its length might also be increased with increased effect of shock.