The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 14 of 36

curreDt there is a certain size of electrode, above which gas is nxt visibly evolved, and for every given size of electrode there is fn current below which gas is not apparently liberated. When the conditions are suitable for the liberation of gas, the gaseiB collected at both electrodes have the same composition. If the quantities of electricity passing in each alternate and oppositely- directed flux are equal, then the electrodes are not sensibly polarised. If, however, the quantities are not equal, then there is, on the whole, a greater flow of current in one direction than in the other, and the electrodes exhibit the state known as polarisation, and yield a reverse current when connected with the galvanometer. Verdet, in his experiments, made use (tf flat spirals, the wires of which were insulated from each other ^th great care by silk and a layer of gum-lac varnish. Th& primary spiral was made of copper wire f chs of an inch in dia- meter and 92 feet in length, forming 24 spirals. The secondary circuit consisted of three spirals of wire -^th of an inch in dia- meter and 157 feet in length, making 95 turns. The inducing discharge was supplied from a Leyden jar battery of nine large jars. The induced discharge was sent through a voltameter having small platinum electrodes, and which could be connected with a delicate galvanometer for detecting polarisation of the electrodes immediately after the discharge. Yerdets experi- ments led him to recognise that when the induced circuit is continuous, and not interrupted anywhere except by the insertion of the voltameter, no traces of polarisation are obtained except by very powerful discharges. This indicates that the induced discbarge consists of a double current of two oppositely- directed and equal quantities of electricity In the case of very powerful discharges there was a slight galvano- metric deflection, indicating a preponderating secondary dis- charge in the same direction as the primary. If the induced or secondary circuit is interrupted at one point, so that the discharge has to pass as a spark at that place, then very per- ceptible polarisation of the electrodes presents itself, and the direction of this is such as to indicate a predominant induced current passing in the same direction as the primary.

To sum up. It follows from all the numerous researches on induced discharges that this is a very complex phenomenon, and is influenced by a large number of conditional circum-

MUTUAL AND SELF INDUCTION, 229

stances, and also by the very mode employed for determining it. It may be, however, taken as proved that an induced dis- charge, produced either as a secondary discharge by a transi- ^ tory primary, such as the discharge from a Ley den jar, or a • tertiary current produced by induction by a secondary current of very brief duration, is, in its simplest form, a wave of electric ■current, consisting of two short fluxes or currents in opposite Sections, and succeeding each other immediately. This Poggendorff * holds to be shown by the action of such tertiary or higher order currents on a galvanometer. If these currents Are led through a galvanometer of which the arrangement is £ruch that the magnetic axis of the needle is accurately at right Angles to the direction of the magnetic axis of the coil, then no deflection of the needle is observed, or at most a very slight one. If, however, the needle makes an angle with the plane of the coils, then these induction currents cause a deviation of the needle. This efiTect {die doppelsinnige Ahlenkung) arises from the fact that the magnetism of the needle is not rigid, ■and that the alternate twisting couples to which the needle is subjected are not equal, by reason of the fact that one of the hi^lves of the complete induced current — say the direct half — increases the magnetic moment of the needle, and hence increases slightly the deflecting couple in the direction tending to increase the deviation of the needle ; the other half — say the inverse part of the induced current — tends to reduce the moment of the needle, and hence to subject it to a smaller reverse couple. Hence it follows that, if discharges of equal quantity and opposite sign succeed each other through a galvanometer when the needle is accurately in the plane of the €oils, little or no deviation is observed ; but if the coils are turned so that the needle makes an angle with them, then these alternate currents will affect the needle and increase the angle of deflection.

This behaviour towards a galvanometer, and the action on a voltameter of liberating mixed gases of equal composition at «ach pole, prove that each induced current of the third and higher orders consists of two oppositely-directed discharges, produced by the operation of two successive electromotive impulses of opposite sign and very brief duration acting upon • PoggendorflF, Vogg, Ann,, Bd. XLV., 1838, p. 353. ^

230 MUTUAL ANV SELF INDUCTION.

the circuit-. The quantitus of these discharges are equal ; buv the durations are different, and hence the maximum value of the current strength during the opposite discharges may be^ very different.

This may be illustrated graphically thus : — . Let the curve a P 6 Q c (Fig. 84) be a current curve represent- ing two waves of current of opposite sign succeeding each, other. Let the horizontal line a c be a time line, and vertical ordinates represent instantaneous current strengths. Then th& sha.ded areas will represent the quantity in each discharge. Let these shaded areas be equal, then the diagram repre'tcnts two discharges of equal quantity succeeding each other in opposite directions, but having different maximum current

Fio. 84.

strengths I and I'. The duration of the first discharge is^^ represented by a 6, and that of the second by h c. This diagram represents the conditions in the simplest case of tertiary current. If the instantaneous value of the current at any time is called i, then the whole quantity of the discharge will

be represented by the shaded area and by the integral \i d t between proper limits.

We may classify the effects of induced discharges or currents in the following way : —

(1) Those effects dependent upon lidt, or upon the whole

quantity of the discharge. These are the galvanometric and the electro-chemical effects. If a discharge is passed through a^

MUTUAL AND SMLF INDUCTION. 231

galyanometer, the duration of which is very small compared with the time of free oscillation of the needle, the galvanometer needle experiences a ''throw" such that the sine of half the angle of deflection is proportional to the whole quantity of the discharge. Also in a voltameter, hy Faraday's law, the whole quantity of the electrolyte broken up is proportional to the quantity of electricity which has passed through it.

(2) Those eflfects dependent upon / i-rff, or upon the average

of the square of the strength of the current at every instant during the discharge. These are the heatinfj and the electro- dynamic effects. By Joule's law, at every instant the rate of dissipation of eitergy is proportional to the square of the current strength, and hence the whole heat generated by the discharge is proportional to the integral above. Similarly, if the dis- charge passes through a circuit, part of which is movable and can react upon a fixed part, so that attraction or repulsion may take place between them, the force is dependent at any instant on the square of the current strength, and hence the whole effect or average force upon the same integral.

(8) We have, lastly, effects dependent chiefly upon the maximum ordinate I, or upon the rate of change of the cur- rent— that is, upon the steepness of the slope of the current curve. These are the physiulofjical^ telephonic, luTninoiLs, and magnetic effects.

The physiological effect of a discharge in giving a shock appears to depend in great part upon the suddenness with which the maximum current strength is reached. Of two dis- charges which reached equal maxima, that which arrived at it in the shortest time would be the most effective in producing shocks. The value of the maximum current strength is also important. Two induced currents of equal quantity but different durations cause a greater shock in proportion to their lesser duration. The telephone in this respect resembles the animal body. It is affected more by the rate of change of the current strength thain by the absolute current strength at any instant.

The magnetic effect depends, as has been shown by Lord Bayleigh,* upon the maximum current strength during the

• See PhU. Mag,, Sen 4, Vol. XXXVUL, 1869, p. 8 : The Hon. J. W Sfcrutt (Lord lUyleigh), " On some Electromagnetic Phenomena." AUo PhU. Mag,, Ser. 4, VoL XXXIX., 1870, p. 431.

232 MUTUAL AND SELF INDUCTION.

discbarge, or upon the initial current strength, in those cases in which the current dies gradually away. In the two Papers referred to below it is shown by direct experiment that, since the time required for the permanent magnetisation of steel is small compared with the duration of induced currents generally^ the amount of acquired magnetism depends essentially on the initial or maximum current strength during a transitory current, without regard to the time for which it lasts. It is, then, not difficult to understand that the effort to settle by experiment with a magnetising coil the direction of induced discharges may lead to very conflicting results, and, in any case, it is hardly competent to do more than indicate the direction in which the maximum current flow takes place during the discharge.

The spark effects are also included in this category. The air or other dielectric is broken down when the difference of potentials between the two discharging points reaches a certain magnitude, and in the case of a varying electric pressure the question whether a spark will pass or not is evidently determined by the maximum magnitude of that quantity.*

It is evident from the above considerations that the complete analysis of the effects and phenomena of induced currents of the higher orders, and of those of secondary currents due to discbarges from condensers, requires a knowledge of the form of the current curve in each case. We proceed to consider the problem of the theory of induced currents in some of its simpler aspects.

§ 6. Elementary Theory of the Mutual Induction of Two Oircoits. — Aiming rather at the elucidation of principles than very copious treatment, we shall consider in the next place the problem of the mutual induction of two circuits in its simplest form. Let there be two bobbins of wire in suitable positions for producing mutual induction and without iron cores. Let the constant inductance of the first or primary coil be denoted by L and its resistance B, and the similar quantities for the

• Set Bertin, " Notes on Electrodynamic Induction," Ann. de Chimie, 4tb Ser., Vol. XXII., April, 1871, p. 486.

MUTUAL AND SELF INDUCTION. 233

seoond or secondary coil be N and S, and let M be the co- efiQcient of mutual induction.*

Let there be a source of constant electromotive force, E, 'which can be applied or withdrawn from the primary circuit. We shall denote by x the strength of the current in the primary at any time t after closing the primary circuit by applying the battery to it. Also we shall denote by y the current in the secondary circuit at any time reckoned from the same zero.

If, then, at any instant the currents are x and y, the follow- ing state of things exists in the circuits.

The electromotive force E is the impressed force on the primary circuit.

That part of the impressed electromotive force producing the current a; is B a;. That part employed in overcoming the

counter-electromotive force of self-induction is L — ?, and the

dt

counter-electromotive force of mutual induction due to the

current p at that instant in the secondary circuit is - M -Ji.

(It

Hence the relation which at any instant holds good between

these quantities is

L^+Miy+Ra:=E. dt dt

The above equation is an expression of the fact that the external impressed electromotive force at any instant is equal to the internal electromotive forces and the effective electro- motive force driving the current.

Similarly, for the secondary circuit we have an induced tiectromotive force due to the induction of the primary on

dx the secondary equal to M -- and a counter-electromotive force d t

of self-induction N -^. d t

Hence N llf+M^+Sy=0,

dt dt

since there is no external impressed electromotive force. The complete solution of the problem of finding the currents x

• Continental writers often call L and N the potentials of the bobbins on themselves, and M the potential of one bobbin on the other.

234 MUTUAL AND SELF INDUCTION.

and y at any instant is obtained by the solution of these- simultaneous differential equations —

L^+M^+Ra-=E, at at

dt dt^ ^

As our object is to illustrate principles rather than mathe- matical methods, we shall simpUfy the problem by supposing that the two circuits are similar in every respect. This makes B a S and L = N, and the equations become

^^+^^t+«^=^' • • • • («»)

L^ + M4^+Ry = 0 (84)

at (It

Bearing in mind that the inductance L is, in ordinary parlance, the "number of hues of force** which are Unked with the primary circuit when unit current flows in its own circuit, and that M signifies the number of lines of force which are common to both, or linked in with both circuits, when unit current flows in each, we see that M can never be greater than L, but that under all circumstances we must have

M <or = L, also M <or = N ;

hence M*<or«LN,

or LN — M^ always a positive quantity, and the maximum value which the co-efficient of mutual inductance M can have is >/LN, or the square root of the product of the self-induc- tances of the separate circuits.

In order to separate the differentials in (88) and (84) we differentiate each equation with respect to t, and obtain —

■ L^+m5| + B!L^-0. . . . (85) dt* dt^ dt ^ '

L^If + M'-^ + Rly-O. . . . (86) dt' de dt ^ '

MUTUAL AND SELF INDUCTION. 235

Multiply (86) by L, (86) by-M, and (88) by R, and then adding the three equations together we obtain —

(Px 2LB dx B" ER /q^v

and a similar elimination gives us

We have now separated the differentials in x and y, and the solution of these equations depends, as is well known,* upon the solution of an auxiliary quadratic equation —

the solution of which is —

R R

m= - - — — , or-;

L + M' L-M* Hence the general solution of (83) and (84) is —

Rf Rt E

a; = Ae"^+^+B«"^-*^ + j-^, . . . (89)

Rt Rt

and y = AV"^+-^I+BVi'-^ (90)

where A, B, A', B' are constants of integration to be determined from the circumstances of the flow. To do this, however, a preliminary discussion is necessary. Let us suppose that the primary current is fully established, and has a steady value I, and hence that M I lines of induction penetrate through the secondary circuit. This quantity is then the electromagnetic momentum of the secondary circuit, because when the current in the primary is steady there is no current in the secondary circuit.

Let us now suppose that the primary circuit is broken, and that the circumstances of the *' break " are such that all these MI lines of induction are removed at a uniform rate in a Bmall time 8 1 from the secondary circuit.

During this time 8 e an electromotive force wiU operate upon

MI the secondary circuit equal in magnitude to - -jj^ or to the rate

of decrease of the included lines of force. We have seen in

• See Boole's *' Differential Equations," p. 192, 2Qd Edition.

236 MUTUAL AND SELF INDUCTION.

Chap. III. that when an electromotive force E acts on a drcnit of inductance L and resistance B that the current i at any time after the commencement of the application of the electro- motive fdrce is given by the equation

In the case ' considered the inductance and resistance of the secondary circuit are L and B, and the impressed electro-

MI motive, force applied during a time Bt ia '^. Hence, at the

end of the interval of time St, the value of the secondary current is given by the equation

This gives us the value of the inverse induced current at the instant of breaking the primary. Expand the above expression by the exponential theorem, and it becomes

I""L"Ln-2'^L?l~2-3*

At th^ instant when the removal of lines of force or the ces- sation of the induction through the secondary takes place the impressed electromotive force ceases and the secondary current begins to die away. If we suppose the '' break " of the primary to be very sudden, 5 1 becomes practically zero, and we have

that is to say, the secondary current starts with a value equid

M

to -^ of that of the steady primary. h

' The state t)f things in the secondary circuit immediately (rfter the break of the primary is, then, this : The electromotive impulse due to stoppage of the primary has generated a current

of initial value--. I in the secondary, but there is no impressed L

electromotive force in the secondary circuit. If at any instant after the break the current in the secondary circuit is t, tha law of decay of this current is expressed by the eq[uation

Lli + Bi=0. (it

MUTUAL AND SELF INDUCTION.

237-

The solution of this is

--?.*

i^Ge ^

and the constant C is found from the condition that when

M t = 0 i = — - 1. Hence we have

1TJ.T

(91)

This gives us the value of the direct or "break" induced current in the secondary at any instant after the break of the primary. Graphically, this may be represented by a curve, such as that in Fig. 85. During the time 0 T in which the primary is being broken the induced electromotive force is

Fig. 85.

creating an induced current, the rising strength of which is represented by the rise 0 P. The time occupied by the break 8 t is 0 T. As 0 T is diminished in value, the magnitude of

M

the maximum ordinate P T approximates to —I, and this is

L

the initial value of the inverse secondary current when the

break is very sudden. After the break the current decays

away along a path represented by P Q, and becomes zero only

after an infinite time.

The whole quantity of the induced current is obtained by

integrating equation (91) with respect to the time from zero to -

infinity, thus :

/:-=/: 'l'

ir'rf«=^.

MI R

238 MUTUAL AND SELF INDUCTION.

We see, then, that both the maximum value and whole quantity of the direct secondary current are proportional ix) the coefficient of mutual induction and to the strength of the primary current, and, moreover, that the whole quantity of electricity set in motion in a secondary circuit of total resis- tance B by suddenly removing from it M I lines of force is equal to the quotient of number of lines removed by the total resistance of the secondary circuit.

If the induced current is sent through a galvanometer the

M I indications are proportional to the magnitude of -— -. K, how*

R

•ever, the induced current is employed to magnetise steel

needles, the magnetisation acquired is dependent upon the

M I magnitude of — -, and is therefore greater in proportion as

the coefficient of self-induction of the secondary circuit is less. Lord Rayleigh has pointed this out,* and shown by experi- ment that, within certain limits, the magnetising efifect of the break-induced current on steel needles is greater the smaller the number of turns of which the secondary consists, the opposite being, of course, true of the galvanometer. The galvanometer takes account of the total quantity of the induced current ; whilst the magnetising power depends mainly on the magnitude of the current at the first moment of its formation, without regard to the time which it takes to subside.

Returning to the equations (89) and (90), we can now find the constants of integration, counting the time from the instant of " make '* of the primary. It is obvious that when t = 0, y « 0 and x=^0, and that the whole quantity of the make-induced

• current, or | ydt, must be equal to the whole quantity of the

MI .break-induced current, which we have seen is equal to -^_.

R In (90) put t = 0, y = 0 ; we get

A'-fB' = 0, or B'=.-A'.

(Bt Rt \

e L+M - e L-M j

, r^ ,, 2A'M

and / ydt-— — - — .

Jo K

• Pha, Moif., Sen 4, Vol. XXXIX., 1870, p. 429.

MUTUAL AND SELF INDUCTION, 239

Hence the whole quantity of the " make ''-induced current is

2 A' M * M I

— — - — , and this must be equal to -— -, which is the whole

^quantity of the " break" current. Hence A' = - -.

Therefore we get for the instantaneous strength of the ** make " secondary current

y

= -^ftf--L+M_e""L^j . . . (92)

Again, in (89) put ^ = 0, a; = 0, and we get A+B+I = 0,

or B=-(I+A);

and by substitution in (89)

B R

x = Ae i'+M-(A+I)(j i—M+L From this equation we can find the value of A by substitut- ing the value of ^-^ derived from equation (92),andi--!? derived d t at

from the above in the original differential equation (83), and we find A= - — . Hence we arrive at the equation for the value of the primary current at any instant, and it is

a-'-le'—L+i+e L=MJ . . . (98)

This gives the law according to which the primary current grows up in its circuit. If M = 0, that is, if there is no secon- dary circuit ; then

a;=I f l-e L J,

which is the ordinary law of current growth. K M = L, which is the greatest possible value of M, then

Hence it is obvious that the presence of the secondary circuit hastens the rise of the primary current and operates on it to reduce its inductance.

On making the primary we get a " make " or inverse secondary current according to the law of growth expressed by the equation

I / Ri _ B<\

2^

MUTUAL AND SELF INDUCTION.

and we see that under the circumstances assumed the <' make ' secondary starts from an initial value zero, rises up to a maxi- mum, and then decays away again. To find the time of reach- ing maximum, equate — to zero, and we find

t'.

dt

-(^

2RM

and this function increases as M decreases. So that the more nearly M is equal to L the sooner does the secondary reach its maximum. It is not diflScult to show that when M » L the

L R-

above value for t' becomes zero, and when M = 0 t' =

M-L (nei^rry)

M«0 ( nearly j

Tine «9im Fig. 86.

Carvei representing roughly the cmreDt value of the make-indiiced oarrent for different and increaBlng Taluea of M.

If, then, we trace a series of curves (Fig. 86) representing the ^'ulues of y, or the make-induced current at each instant

for various and increasing values of ^, as the coils are moved

M

further apart, we find a series of curves with decreasing maxima, but the maxima happening later as M decreases.

Lastly, on breaking the primary current we have a break- induced current in the same direction as the primary, which at

MUTUAL AND SELF INDUCTION. 241

any instant after the ** break ** is decaying away according to the law

L If the break was absolutely instantaneous, the induced current would start with a finite value equal to of that of the primary,

but as no form of break entirely eliminates sparking, the rise of the direct secondary current is a gradual one. Also we have another element of disturbance which enters into the case. The self-induction of the primary creates direct electromotive force in its own circuit at the instant when the induction through the primary due to its own current vanishes. When the primary is broken either at a mercury cup or at a platinum point the fusion and volatilisation of metal which takes place keeps open for a little time a conductive path through which flows the extra current due to the self-induction of the primary. As will be explained later, the decay of the current on breaking a circuit may often be by a series of oscillations or diminishing periodic currents.

This direct extra current in the primary will have its effect in introducing a very short inverse-induced current, which will precede the main direct-induced current due to the decay of the primary current. In any event it will introduce an electrical oscillation tending to render the growth of the direct secondary current a gradual matter. It is an interesting case to examine the relative maximum values and duration of the two induced currents under an assumption very nearly realised when the primary and secondary are wound together on the same bobbin, viz., when M = L. In this case the values of y and z become

I -^* y^.e ..

«=Ig— L-

The maximum of the direct currents ("break") is I, and

that of the inverse (or "make") is -. If we wish to know at

the end of what times t and t' the strengths of the two induced

currents y and z are reduced to — of that of the primary we

m

242 MUTUAL AND SELF INDUCTION

obtain by substitution of — for y and ;: in the two above equa- m

tions the following : —

1 -^^

__ = e l' for the direct-induced current,

m

and — = ^ sL for the inverse-induced current,

m 2

and therefore i' = 2 (^1 - ]^^) .

t \ loff m/

log?

We see that t* is always greater than «, and that, in propor- tion as m increases, f tends towards a limit 2 t, or the inverse current has a duration about double that of the direct secondary. We shall now see how this theory is confirmed by experiment.

§ 7. Oompariflon of Theory and Experiment. — Masson and Bregaet carried out a series of experimental researches on induced currents which illustrate and confirm the foregoing theory. The principal part of their apparatus was a commutator keyed on a revolving shaft, which enabled them to separate the direct and inverse-induced currents. Two brass wheels were keyed on one shaft, but insulated &om it, and the wheels had depressions cut in their periphery which were filled up with ivory. These wheels could be shifted relatively to each other, and were insulated from each other and from the shaft (see Pig. 87). Two springs pressed against the edge of the wheels, and two against the hub of the wheel. The whole arrangement served as a means to break and make one circuit, and at the same time to control a second circuit so that it was broken at the time when the first was made, and made at the time when the first was broken, or vice versa. One of these wheels was inserted in the circuit of a primary coil and battery, and the other in the circuit of a secondary coil and galvanometer. On rotating the wheel at a certain fixed speed the series of "break" and ** make "-induced currents ai'e separated out ; all one set are stopped out and all the other are sent through the galvanometer. In this way it was shown that the quantities of the induced currents were equal, but very different in maximum magnitude, and hence in duration, the break-induced currents being greatly superior in making sparks.

MUTUAL AND SELF INDUCTION.

243

Lenz* wound a spiral of wire on the soft iron armature of a magnet and connected the ends of the wire to a ballistic gal- vanometer. He detached the armature suddenly, and observed the throw of the galvanometer. If 6 denotes the angle of deflection and x the number of windings, he found that the

1 0 product -.sin - was a constant quantity, which shows that,

eaeteris patibusy the quantity of electricity set in motion was in proportion to the number of lines of induction withdrawn from the circuit. He also established experimentally, in con- firmation of Faraday, that the electromotive force of induction was independent of the width, thickness or material of the

Fio. 87.

wire windings,! and by other experimentalists also the fia.ct has been established that the electromotive force is indepen- dent of everything except the form of the conductor and the nature of the change it experiences in relation to the magnetic induction through it. Felici| carried out an extensive series of experiments on induction, using a form of induction balance.

• Lenz, Poggeruiorjgrs Armalen, Bd. XXXI., 1835, p. 385.

t See Faraday, '' Exp. Researches," Ser. II., § 193, et »eq. ; also for Electrolytic Circuits, ue L. Hermann, Pogg. Ann,^ 1871, p. 686.

X Felici, Nwvo dmento. Vol IX., 1859, p. 345, also Ann, de Chimie [3], VoL XXXIV., 1852, p. 64.

Ii2

244

MUTUAL AND SELF INDUCTION.

In this apparatus a seoondary circuit, consisting of two coils, is arranged in series with a galvanometer. These coils are so far apart as not to influence one another. In contiguity to each secondary coil is a primary coil, and the primaries are wound in opposite directions. The primaries are in circuit with a hattery and a key. The circuits can be so arranged, by adjusting the distances of the coils, that the induction of the primaries on their respective secondaries balance each other, and the galvanometer indicates no current, however strong may be the primary current. If three pairs of coils (see Fig. 8^ are thus taken and balanced, two and two, so that the induction of A on a is equal to that of B on 6 and C on c, then, if we con- nect the primary A in series with B and C in parallel, so that

FiQ. 8a

the current divides between them in the ratio of their resis- tances, and connect the secondaries with a galvanometer, all in series, so that the current in a is opposed to that in b and in c, then no induced current is detected when the battery circuit is made and broken. This proves that the quantity of the induc- tion current is proportional to the strength of the primary current.

If a primary and secondary coil are taken in fixed positions and the ** throw " of a galvanometer observed when a definite steady electromotive force E is applied to the primary, then, if the position of battery and galvanometer are reversed, the application of the same electromotive force E to the secondai;y

MUTUAL AND SMLP INDUCTION, . 246

will give the same " throw" on the galvanometer now attached

to the primary circuit, provided that the galvanometer and

battery either have equal internal resistance or that their

resistance is negligible in comparison with that of the coils.

Hence we may assert that the induction of a circuit A upon B is

the same as that of B upon A. For, if the resistances are B and

8, then we have seen that the total quantity Q of the secondary

MI current is ---, where I is the steady value of the primary o

•  .  P'  M  P 
  

and'M is the mutual inductance; but I = --, hence Q = — ^-.

B SB

If, then, the positions of battery and galvanometer are reversed,

we get a quantity of induced current equal to -— — , which is

B S

the same as before. For any two coils it is possible to find a

number of relative positions in which the interruption of a

current in one produces no induced current in the other. In

such cases the ooils are said to be corrugate to each other. It

is manifest that when in these positions the lines of induction

produced by one coil do not pass through the other. It is

possible to use one coil in this way to explore the field of

another.

Let P be a primary coil and S be a small flat secondary coil, both being shown in section in Fig. 89. Then, if S is placed in a position conjugate to P, it will be found possible to move the coil S along a certain line ABO, maintaining the flat face of the coil always tangent to that line and so that in all these positions P and S are conjugate. It is evident that such a line is a line of induction of the coil P.

When one coil is in a conjugate position to the another, as far as regards inductive action they may be considered to be at an infinite distance apart. It follows, therefore, that if a coil is moved suddenly from a conjugate position to one not conju- gate in the field of a primary traversed by a steady current, and then the primary current is stopped at the instant of arriv- ing at the second position, a galvanometer in the second circuit will have its needle jerked from one position of rest to another of rest, because the interruption of the current takes out of the circuit of the second coil just as many lines of induction due to the first coil as the motion from one position to the

^4^

MUTUAL AI^D S^LP IUfDUGTIOHf.

other pat in. A series of well-devised experiments on Uie conjugate positions of two coils has been carried out by Mr. W. Grant.*

An elaborate investigation into the duration of induced currents was made by Blasema.f

A commutator was constructed which consisted of two insu- lating cylinders keyed on one shaft and having on part of their surface brass coverings cut into steps {see Fig. 90). These cylinders were capable of being set in any relative position to each other on the shaft. The shaft could be revolved at a high rate of speed, and its velocity ascertained by a siren plate attached to the axis. This siren plate consisted of a disc pierced with holes against which was directed a jet of air.

Fig. 89.

From the pitch of the musical note given out, when ascertained by comparison with standard tuning forks, the speed could be determined. Two springs pressed against the hubs of these cylinders and two against the surfaces of these cylinders, and a current entering by the hub was conducted to the brass coating and escaped by the other spring, if the cylinder was in such a position that this last spring was pressing on the metal part. The apparatus, therefore, formed a device by which each

• See Proc. Physical Soc, London, Vol. UL, p. 12i ; also Proo. Physical Soc. London, VoL IV., p. 361.

t Blasema, " Sul sviluppo e la durata delle Correnti d'induzione," QiomaU di SoUnce Naturali, Vol. VL (Palermo, 1870).

MUTUAL AND 8BLF INDUCTION.

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