The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 17 of 36

Next, let the interrupter be removed to the battery circuit, all the other arrangements remaining the same {see Fig. 105). It will be found that the balance is destroyed, and that no- mere change in the value of the resistance B will enable a per- fect balance to be obtained. The reason for this is that, on

• These experiments formed the subject of Prof. Hughes's Inaugui-al Discourse to the Society of Telegraph-Engineers on the occasion of hia- election to the office of President. See Journal of the Society of Telegraph.- Engineers^ January 28, 1886, " The Self-induction of an Electric Current in Relation to the Nature and Form of its Conductor."

MUTUAL AND SELF INDUCTION.

283

dosing the battery circuit, the inductance of P introduces a counter electromotive force into P and the potential rises at e faster than at d, and on breaking the circuit the potential at e dies down faster than at d ; and hence at each make and break the telephone is subjected to an alternate flux of current which causes it to emit a sound. Supposing that an attempt is made to get rid of this sound by shifting the point c so as to alter B, the steady balance will be destroyed, and the telephone will be traversed by a current during the time when all the currents have become steady ; but no such change in the value of B will prevent a variation of current taking place through

the telephone during the complete period from the first instant when the battery circuit is closed to the instant when it is opened again.

The only way in which a balance can be obtained in this last arrangement is by introducing into the telephone circuit an elec- tromotive force which shall be capable of being made to balance at every instant the inductive electromotive force due to the inductfimoe of P. Prof. Hughes does this very ingeniously by introducing a pair of mutually inductive coils into the battery

284

MUTUAL AND SELF INDUCTION.

and telephone circuits, and the final arrangement is as shown in Fig. 106. Mi and M, are a pair of coils, one of which, M» is in the battery circuit and is fixed, and the other, Mi, is in the telephone circuit, and can be placed so that, whilst its centre coincides with that of Ms, its axis makes any required angle with that of M^. In this way the mutual inductance between Mi and M, can be varied from zero when the coil axes are at right angles to a definite maximum value when they are co-linear.

It is found that, when the coils Mi Ma are in certain positions, the inductive electromotive force set up in the

telephone circuit by the induction of Mi on M9 can be made to neutralise the electromotive force of self-induction due to the inductance of P, when, in addition, a certain value is given to the resistance E. Under these circumstances the bridge can be balanced and the telephone completely silenced, both when the interrupter is in the battery circuit and also in the tele-* phone circuit ; in other words, the bridge can be balanced both for steady and for variable currents.

In the arrangement adopted by Prof. Hughes the resist- ances Q, B, and S, were sections of one and the same fine

MUTUAL AND SELF INDUCTION.

285

German silver, 1 metre long, and having a total reEOstance of 4 ohms {see Fig. 107). The ends of this wire were joined to

Fial07.

the conductor P under investigation, and the rest of the apparatus was arranged as described.

In order to investigate the relation between the resistances and inductances which holds good when the bridge is balanced for

Fio. 108.

steady and also for variable currents, a diagram must be drawi^ (Fig. 108) representing the network of conductors. Then call

286 MUTUAL AND SELF INDUCTION.

the current at any instant in the inductive branch P, x, that in the branch Q, y, and that in the telephone circuit «. The current in the branch battery is then x-'y. Let L be the inductance of P, and M the mutual inductance of the coils placed in the circuits B and T, and let all the other circuits, Q, B, and S| have no sensible inductance. Let e be the electromotive force of the battery at any instant U Then the currents in the various branches at that instant are as follows : — In the branch P the current is x „ „ B „ „ x-k-z » »» '^ >» >> 2/~*

»> >> ^c n »i y

   >f        -^        >»       n       x-hy 

T z

Applying Kiiclihoffs corollaries to each of the three meshes of the network, we have three equations, viz.,

^x + BJ^ + nJT'z^e^Jj T^-M j^ (94) BJVy-^S^z + Qy^e^U^ • • • (95)

B.^^ + T^-Si7^=-M^^^, . . (96)

and these three equations enable us to find at any time t the current in any branch.* If we suppose the bridge to be balanced for variable currents, then z is zero, and on making this limitation we find the above equations reduce to the two,

Qy-Pa:-L|-^^' = Bx-Sy, . . . (97)

dx dy and -M^-M^=Ba;-Sy. . . . (98)

Furthermore, let us assume that the currents vary according to a simple periodic law. In this case, if X is the maximum value of Xf then we can write

X = X Bin pt,

• The general method of finding the current equations for any network is given in MaxweU'e* " Treatise on Electricity," 2nd Edition, VoL II., § 765. Also see " Problems on Networks of Conductors," by J. A. Fleming, Pha, Mag,, September, ie85, Vol. XX., p. 221 ; or Proceedings Phya. Soc., Lond., 1885.

MUTUAL AND SELF INDUCTION. 287

where p as usual « 2m, n being the frequency of the altema.

dx tion. Hence jr ^pXcoapt,

and j^ = -2)2 X sin jpt

Adopting the fluxional notation, it is convenient to write x for 77 and X for xs* Hence, for simple periodic variation of a

(It CL t

current x, we always have the condition

If we differentiate with respect to t the two equations (97) and (98), and eliminate x by the help of the equation dPs ~p^, we obtain two other equations, which, together with the original two (97) and (98), give us the necessary four equations for elimi- nating the four variables x, y, x, y. We have thus,

Qy-Vx-hx ^Ex-Sy, . . . (99)

. -M;i-Mi/ ^JXx-Sy. . . . (100)

Qy-Px-hhp^-x^R'x-Sy. . . . (101)

Mp2a; + Mj/2y =Ri-Sy. . . . (102)

The student who has mastered the elements of determinant analysis will recognise that the variables a?, y, Xy y can be eliminated from these equations, and the relation which must always hold good between the constants can be found by equating to zero the determinant of these four equations. We have th^

-L, 0, -(P + R)> (Q + S)

-M, -M, R, S

-(P + R), (Q + S), lsp\ 0 . .

-R, S, lip\ Mp« =0.

This determinant writes oat into the sum of three terms, viz. : —

;>»L[L(M«j)a + S«)-M(S«+SQ + RQ + 8R)] + -(P + R)[MV(P + Q + R + S)-S(QR-SP)] +

• (Q + S)[MLp*(R + S) + R(SP-RQ)-MV(P + Q

• E + S)]-a

288 MUTUAL AND SELF INDUCTION.

This long equation reduces to the simpler form

[(pBL)«-(pM(P + Q + E + S))>] + [(MLp7

-(QE-SP)2]-a In order that the sum of the two left hand terms in the above equation may always be zero, each factor in the square brackets must be separately zero, and it will be seen that each of these factors equated to zero are oquivalent to the two equations : —

QE-SP = ML/, (103)

and M(P + Q + R + S) = SL (104)*

These equations express the relation which holds good between the resistances of the branches and the self and mutual induction coefficients of a Hughes bridge when the bridge is balanced for variable currents.

It will be seen that the ordinary relation of the resistances for steady balance, viz., P:Q » E: S is departed from, and that we have for the resistance of branch P, when traversed by variable currents, the value

P S y- -s". • • (105)

and for the inductance of branch P under these circumstances^ the value

L„M(P + Q + R+S)^ . . . (106) B In some of his experiments Prof. Hughes interpreted his

results on the assumption that P was always equal to -^, and

b

L was equal to M ; but the complete investigation shows that this is not the case. A very full theoretical and practical examination of the induction bridge has been given by Prof. H. P. Weber, for which the student is referred to the pages of the Electrical Beview.yol. XVIIL, p. 821, 1886, and Vol XIX., p. 80, 1886.t

• These equations were given by Lord Rayleigh in the discussion on Prof. Hughes's Paper. See also Lord Rayleigh " On the Self -Induction and Resistance of Compound Conductors,*' Phil. Mag,, Dec. 1886, p. 471. I^uivalent equations have been also arrived at by Prof. H. F. Weber and Mr. Oliver Heaviside.

t See alto Mr. Oliver Heaviside in the PhU. Mag,, August, 1886.

MUTUAL AND SELF INDUCTION. 289

The whole method of the construction and use of the induo- tion bridge has been the subject of elaborate examination by Lord Bayleigh in a Paper on the self-induction and resistance of compound conductors {Phil. Mag., December, 1886), from which we shall quote freely in what follows. Lord Bayleigh discarded the tooth- wheel interrupter, as it does not give a regular variation of current correspondmg in period to the passage of a tooth ; and he substituted a harmonium reed, the vibrating tongue of which made contact once during each period with the slightly-rounded end of a brass or iron wire advanced exactly to the required position by means of a screw cut upon it. Blown with a regulated wind, such reeds are capable of giving interruptions of current up to about 2,000 per second. The one usually employed had a frequency of 1,050 vibrations per second. The induction compensator consisted of two circular coils, one of which was fixed and the other movable round an axis, so placed that the fiat circular coils could be placed either with their planes coincident or at right angles. If the inner coil is very small compared with the other, and the coils are placed with centres coincident and axes inclined at any angle, 6, and if Mq be the maximum mutual inductance and M the inductance in any position, d, then

M = Mo cos 6.

This law is, however, not followed when the coils are sensibly of the same size. In this case Lord Bayleigh has shown that the mutual induction is very approximately proportional to the angle between the axes of the coils for a range between 40** and 140*. In the actual experiments the mutual inductance of the Goila was determined for each degree of angular displacement of the axes by comparing it with the calculated coefiScient between two wires, wound in measured grooves, out in a cylinder, and it was found that every degree of movement of the movable eoil, when the axes were not far removed from perpendicu* larity, was equal to 7768 centimetres of mutual induction, the maximum when d-0 being 56,100 centimetres. The first experiment described in the Paper referred to is one on the self-induetioA and resistance of a coil of copper wire. In the bridge used the resistances Q + B + S were together equal to 400 ohms. Besistanoes were, however, measured in scale

u

290 MUTUAL AND SELF INDUCTION.

divisions of the bridge wire, each one equal to 204 x 10 centimetres per second. The copper coil being balanced on the bridge, it was found that the readings of the three resistances and of M were as follows : —

Q = 610, R=190, 8 = 1,160, M = 86° = 86 X 776 centimetres,

and the frequency « of the vibrations » 1,050. Hence p^2ir X 1,050. Taking the equations (108) and (104) on page 288, and eliminating L, we have for the value of P, the equation

p /M'(Q + B + S)-| Q.R ^' S.Q.R

Substituting the values above, we find

Q R P = -876 ^ = 87-6 scale divisions.

This gives the value of the real resistance of P for the periodic currents used ; and we see that if we neglected the peculiarity of the bridge, and simply assumed the ordinary law, that the resis- tance of P was equal to Q R-^S, we should make an error of some 12 per cent. On actually balancing the bridge for steady currents the resistance of P was found to be 87*8 scale divisions, thus indicating that for this copper coil at the frequency employed the resistance to variable currents was the same as to steady ones.

On inserting a solid copper rod into the aperture of the coil and measuring again the resistance and self-induction, it was found that the values of the reading were Q«660, B«190, M = 295^ instead of as before, Q = 610, M » 86^. Hence the introduction of another closed secondary circuit (viz., the copper rod) increased the real resistance and diminished the real self-induction in accordance with the principles explained on page 180, at which place we demonstrated Maxwell's equations for the increased resistance and diminished self- induction of a primary circuit when in contiguity to a dosed secondary circuit.

The next example selected was that of a soft iron wire, 160 centimetres long and 8'8mm. dia. Here, with ihe variable

MUTUAL AND SELF INDUCTION. 291

currents from the reed interrupter of the same period as before, a balance was obtained for

<i « 178, B » 190, S - 1,592, M « 8 x 776 centimetres,

from which we find

P = -985 9^ = 20-98 scale divisions.

TFhe resistance of the same wire to steady currents was

p 100x190 ^Ij.gQ g^gjg divisions. ^ 1,670

Hence the efifective resistaoce to variable currents having a frequency of 1,050 was 1*84 times the resistance to steady currents. We have presented to us here the phenomena characteristic of the behaviour of conductors to electric currents rapidly intermittent or reversed. The real resistance of the conductor is increased. This is not to be confused with the fact that for intermittent currents the impedance {^/Br+p^Lfi) measured in ohms is greater than the ohmio resistance (B) ; but it is to be understood as a real increase in the rate at which energy is dissipated per unit of current. It is now well understood that such increase of resistance is due to the fact that the current density for rapidly periodic currents is not uniform over the cross-section of the wire, but is greatest along the outer layers of the wire. Hence, under rapidly periodic currents the inner portions of a conducting wire are never reached by the current, and, AS far as current carrying duty is concerned, might as well be away. This difference may be graphically represented thus : Let relative density of current or quantity passing per second through unit of cross-section of a conductor per unit of time be represented, like relative density of population, by degree of density of shading. Then the flow of a steady current through the section of a wire might be represented as in Fig. 109; and the flow of current over the cross- fiection when the current is rapidly periodic might be repre- sented as in Fig. 110.

We must consider that the current in beginning in a conductor starts its flow first on the outside, and soaks or penetrates inwards into the deeper layers by degrees. We see that, in consequence of this, if the current is reversed in sign,

u 2

292 MUTUAL AND SELF INDUCTION.

or rapidly intermitted, it will not have time to soak or diffose very far into the mass of the conductor before it is, so to speak, re-called, and its operations will be confined to the outer layers. This is a rather broad way of stating modem views on the viodus operandi of current flow* According to these views the current in a wire is not established by a process analogous to starting a flow of water in a pipe by a push applied one end, but it is put into the wire at all points of its surface by energy absorbed from the surrounding dielectric. Other things being equals the rate at which this equalisation of current across the cross- section of the conductor goes on will be a function of the magnetic permeability of the material. The current in flowing along a magnetisable circuit magnetises it circularly. This magnetisation involves work, and the impressed electro-

Fio. 109. Fio. 110.

motive force which is increasing the current has to do work,. not only against that which may be called the formal inductance of the circuit, or against that part of the counter electromotive force of induction which depends on the form of the circuit, but has to create this circular ma^etisation.

By keeping to the outer layers of the conductor the periodic current avoids magnetising the deeper layers of the material. Proof will be given later in describing the remarkable investi- gations of Hertz that this description of the mode of establish* ment of a current is one supported by experimental facts. We are thus able to offer a consistent theory of the real increase of resistance which we find for rapidly periodic currents. The inner core or central portion of the condu:tor is not used by the current, and, so far as conducting it goeSr

MUTUAL AND SELF INDUCTION. 5S93

might as well be absent ; hence the solid conductor does no more, or not much more, in the way of carrying the current than a hollow or tubular conductor would do : and, accord- ingly, the real or ohmio resistance of the conductor for such variable currents is greater than it is for steady currents.

Another way of regarding this inequality of current distribu- tion over the cross-section of a wire is as follows : — The counter -electromotive force arising from self-induction is greater at the axis or central portion of the wire than it is near the surface. If we consider the whole current flowing across any section of the conductor as made up of little streamlets of currents flowing parallel to each other, the central streamlets or filaments of current experience more opposition in reaching full magnitude than do the outer ones, because of the mutual induction with those surrounding them. The current there- fore arrives at its maximum value at the surface of the con- ductor before it does at the deeper or central portions. If the current is periodic or transitory the central streamlets or •current filaments are always greatly inferior in strength to those at the surface. There is reason, then, to believe that a sudden rush of current, very brief in duration, such as the dis- charge from a Leyden jar or condenser, moves chiefly along the surface of a discharging wire, and the same statement holds good for very rapid pulsatory or alternate currents. Although it may be said that the general principles governing the behaviour of alternating current flow as conductors were virtually given by Maxwell,* they have been subsequently <ihiefly developed mathematically by Mr. Oliver Heaviside and Lord Rayleigh, and were brought to the notice of practical electricians principally by the experiments of Prof. Hughes previously mentioned.

This increase of the resistance proper of a wire for rapidly periodic currents is one of the most striking of the results of Prof. Hughes's researches. The full mathematical develop- ment of the problem, even for comparatively simple cases, leads to some very complex mathematical expressions. Lord

♦ Maxwell's " Electricity," Vol. II., § 689-630. In this paragraph it 10 «hown that the counter electromotive force of self-induction at any point in a conductor is a function not only of the time but of the position of the point considered, and voriea over the cross-section of the conductor.

894 MUTUAL AND SELF INDUCTION.

Bayleigh has, however, treated with great fuhiess* one or two cases of practical importance. If B and L are the true ohmic resistance and inductance of a cylindrical straight wire of length I and magnetic permeability /^ to steady currents or currents of very slow alternations, and if an alternating current of simple periodic form and frequency n is sent through it, then the resistance is increased to E^ and the inductance diminished to L^ in such wise that if p~27rn, as usual y we have

L 1:^ B,^ 180 ii^ J

and L^:=/ rA+f.n^lPll^l+^1!^'^, . ^1 (108) L V2 48 R' 8640 R* / J ^ '

A being some constant depending on the position of the return wire.

These formulae express the fact that the resistance is in- creased and the inductance diminished in proportion as the frequency of alternation gradually increases from zero to* infinity.

At slow rates of alternation the chief opponent with which the impressed electromotive force has, so to speak, to contend is the ohmic resistance ; and the distribution of current across the cross-section of the conductor under these conditions is^ such as to make that resistance a minimum, and this is known to be so when the distribution is a uniform distribution. The- current is then taking the greatest advantage of the conductor, and the heat generated and dissipated per unit of time is less under these conditions than if the same total current were distributed in any other way over the cross-section of the conductor. This last statement can be easily proved. Let the cross-section of the conductor, supposed to be a cylindrical wire,, be divided into two equal zones by a circular line. Let the resistance per unit of length of the conductor be r for each portion corresponding to the outer and inner zone. Call the outer portion the sheath and the inner the core of the con- ductor for brevity. If a total quantity of current, x, fiowa

through the conductor, then the rate of dissipation of energy

• ..

• *'0d the Self-induction and Resiatance of Straight Conductura,r Pka. Mag,, May, 1886, p. 382.

MUTUAL AND SELF INDUCTION. 295

as heat is—- for each portion per unit of length, or ^ for the

whole conductor, on the assumption that the current is equally divided between the sheath and the core.

If, however, we suppose the total current, x, to be distributed so that a portion, y, travels by the sheath, and the remainder, z, travels by the core, then the heat generated per unit of length per unit of time is ny^ for the sheath and r^ for the core. Hence, for the equi-distribution of current, the energy

dissipation is !--=!1h^l^, and for the unequi-distribution it 2 2

is r{y^+s?). Which, then, is greater, ^Al^:t?loi r(y''+z^)? Consider the following inequalities : —

(y-2)« is greater than - (y-2)S

or y" + 2* — 2 y 2 is greater than - (y - tf \

but |(y-«)"-|(y+^)^-2i/e.

Hence, y"+^-2y«is greater than - (y + a)* - 2 // 2. Adding 2 ^ z to both sides, we have

y" + 2* is greater than 5 (y + ^)'. Accordingly it follows that

r y" + r 2" is greater than ^ (y + 2)2,

or r y' + r 2* is greater than - a?' ;

that is to say, the rate of energy dissipation is greater for the assumed unequal distribution than for the distribution in which the current is equal in density over the cross-section of the conductor. The same kind of proof may be extended to any other arbitrary distribution of current over the cross- section, and the reasoning will lead to the conclusion that the equi- dense distribution is that which causes the leaM rate of dissipation of energy per unit of current.

296 MUTUAL AND SELF INDUCTION.

For slew alternations, therefore, the current adopts that mode of distributing itself over the cross-section of the conductor which makes the rate of energy dissipation a minimum. On the other hand, for rapid alternations the current meets with its greatest obstacle from the counter eleotromotiye force of self-induction, and it accordingly distributes itself over the cross-section of the conductor, so as to get as much to the outside as possible, and thus avoids, in the case of magnetic conductors, magnetising the inner layers or portions of the conductor. The endeavour is to make the self-induction a minimum irrespective of resistance. This is only an instance of the broad, general principle that behaviour of current for very rapid pulsations, or alternations, is determined by the inductances rather than the resistances, whereas for steady or slowly periodic currents the behaviour is governed by resistance rather than by self-induction.

In order to see under what conditions the alteration of resis- tance and self-induction becomes sensible, we have to examine

the value of the term — i — -^ in the above-given series for 12 R'^ ®

B^ We will first take the case of an iron wire 04 centimetre, say, 0-16 inch diameter (No. 8 B.W.G.). The specific resis- tance of iron in C.G.S. measure is about 10 ; so that

B _ 10' _ 10°

jp*=47r*n', n being the frequency.

Let us take w=100, so that there are supposed to be 100 complete alternations per second. The value of /a is more difScult to assign. For small degrees of magnetisation and solid iron, we may, perhaps, take ft = 800;

then JL ^^!iV= 1 47r2nV«^^_5'2/;tgn« 12 R^ 12 B^ 10^°

n /x=800, n=100, /x«n2=:9xl0«, and 1 -2!^' =0-47

12 rC

■bO'6 nearly.

Accordingly, for this case B* = B (1+0-47) nearly, or the resistance is increased to about half as much again.

If n =« 1,000 we should find B^ = 48 B, or the resistance would be increased nearly fifty times.

I MUTUAL AND SELF INDUCTION. 297

Consider next the case of copper. The specific resistance is 1,6^0 C.G.S. units. If a be the radius of the wire in centi- metres, then we have

12 H^ 3 (1,640)« 10» '

If, as before, n^lOO, this fraction becomes equal to 0*12a^. This shows that for a diameter of one centimetre we should

have R^ = R (1+0-12);

and hence for diameters of one centimetre and upwards the resistance of round copper rods becomes very sensibly increased for alternating currents of a frequency about 100 and up- wards. The practical conclusions of importance in electrical engineering from the above investigation are these: — First, copper rods or conductors should be used, and not iron, for transmitting alternate or intermittent electric currents having a moderate frequency, say of 100 to 1,000 per second ; secondly, to avoid, as far as possible, the increase of resistance due to the current keeping to the outer portions of the conductor, the conductor should be in the form of a thin strip, or better, a tube having walls thin in proportion to the radius. It is to be noted that mere stranding of the conductor, or building it up of separate insulated conductors joined in parallel, will not prevent this augmentation of resistance, unless the stranding is of such a kind that portions of the cable which at one point of its length form the inner parts or heart of the cable at another part of its length form the outside.

The object to be achieved is to construct some kind of stranding by which all portions of the conductor are brought as near as possible to the dielectric, so that the energy arriving from the dielectric finds all parts of the mass of the cable, both surface and interior, equally accessible. In order to avoid external inductive disturbance, the proper form to give to a cable intended to convey rapidly intermittent or alternate currents is a couple of rather thin concentric tubes of copper well insulated from each other, and both insulated firom the ■ earth, of which one forms the lead and the other the return. By this device the metal will be most economically employed. An equivalent device used in practice is a concentric cable,

298 MUTUAL AND SELF INDUCTION.

which consists of a central core of stranded copper cable- covered with insulation, and then plaited over with a sheath of other copper wires which form the return conductor.

In a further experiment, Lord Bayleigh (loc. cit,) examined the resistance of an iron wire of hard Swedish iron 100^ metres long and 1-6 millimetre in diameter. In arbitrary units the resistance of the wire to steady currents was 10-4 units or 051 ohm, and to currents of 1,050 complete alternations per second its resistance was 12-1 units, or 0'59o ohm, which is an increase of about 20 per cent. In the case of a stouter wire, 1834 metres long and 33 millimatres in diameter, the

Fig. 111.

resistance to steady currents was 4-7 units, and the resistance- to the interrupted currents of the above-mentioned frequency was 8-9 units, or nearly double. This illustrates the fact that,. for a given frequency of alternation, the ratio in which the resistance is increased is greater the greater the diameter of the conductor, assuming it to be a round solid rod.

Lord Bayleigh found it more convenient in many researches to slightly alter the arrangement of the induction balance as described by Prof. Hughes, and to make it as follows (Fig. 111).

MUTUAL AND SELF INDUCTION, 29?

Two arms of a quadrilateral, B and S, consist of equal resist- ances of German-silver wire, wound double, so as to have negligible inductance. One arm, Q, consists of a coil having inductance and resistance greater than that of any conductor, P, to be placed in the fourth arm. B and I are a battery and an interrupter, T is a telephone in the <* bridge," and rr^ is a German-silver wire of appropriate resistance, along which slides the contact of the bridge. The arm P includes a pair of coils joined in series, and which act upon each other by mutual induction, so that the resulting self-induction of the two coils in series can be varied within certain limits by turning one coil round within the other. For the resulting self-induction of such a pair of coils used in this manner may be regarded as made up of the component self-inductions of each coil taken separately and of twice the positive or negative mutual self-induction, depending upon which faces of the coils are presented to each other. It is possible, then, within certain limits to vary the inductance of the branch PC, and to vary also the resistance of the branches Q and P G by shifting the contact of the telephone along rr^.

The condition for obtaining a true balance when the cnrrent is periodically interrupted is that the resistances and induc- tances of the branches Q and PC shall be separately equal. Suppose a balance has been obtained without the use of P, in which the resultant self-induction of C is made to balance the inductance of Q, and the resistance of C + r^ is made to be equal to that of Q + r. Let, now, any conductor, P, be inserted as in ihe figure ; the telephone contact will have to be shifted, and also the inductance of C will have to be changed to re-obtain a balance. The inductance of P is measured by the amount by which that of 0 has to be reduced on inserting P, and the resistance of P is measured by twice the resistance of that length of the German- si] ver wire ri^ by which the telephone contact point has to be shifted to regain the balance. This method of employing the induction balance separates out at once the real resistance of P from its effective induction.

With the aid of this balance an interesting experiment was made, showing the effect of a closed secondary circuit on the resistance and inductance of the primary. The frequency was again, as usual, 1,050 per second. A coil was prepared of

•300 MUTUAL AND SELF INDUCTION.

two copper wires, wound side by side on one bobbin. The diameter of each wire was about 0-08in., and the length of each wire 818in. There were 20 (double) turns, so that the mean diameter of the coil, wound as compactly as possible, was about Sin., and the resistance of each wire was 0*05 ohm.

The coefficient of mutual induction of the two wires was ■determined by comparison of the self-induction L of one wire with that of the two wires connected oppositely in series, viz., ^2 L - 2 M). In this way it appeared that

M = 43-P = 481 X 1,653 centimetres.

Observation showed that closing of the circuit of one wire reduced the self-induction of the other from 444° to 3'4°. The resistance to steady currents was 0-92 (arbitrary units). The resistance to the periodic currents was 097 with the secondary •circuit open, and 1*74 with the secondary circuit closed.

Hence, L = 44- i x 1,553 centimetres, and

R = 0-97 X 0-0492 x 10® centimetres per second. From Maxwell's formulaB, page 180, we get

P' M' ^ 10" X 1-951 ^

H^ + p^U 10^ X 0-023 + 10" X 2-071 Hence, L^ = L (1-0-932),

where L* is the decreased inductance. Hence,

Li = 0068L, or Li = 0068x44-4° = 8%

and the observed value is 3-4**, which is in very tolerable agreement.

Again, the steady resistance with secondary open is 0-92, and hence the resistance R^ with secondary closed is

Ri = 1-932x0-92 -1-77; and observation gives the value 1*74. We see, then, that observations with this bridge confirm, with a considerable degree of accuracy, the deductions from the theory of simple periodic currents, that the closing of a secondary circuit increases the resistance and diminishes both the inductance and the impedance of an adjacent primary circuit.

From a practical point of view the most important difference between the conduction of steady electric currents and rapidly

MUTUAL AND SELF imTUUTlON. 301

periodic onrrents is that of the locale of the currents in the conductor and the consequent rise in the ohmic resistance of the conductor as a whole when employed with such periodic currents. Prof. Hughes dalled attention in 1888 to this great difference in the resistance of an electrical conductor if mea- sured during the variable instead of the stahU condition of the current.*

In experiments with his induction bridge Prof. Hughes was- able to assure himself that the resistance of an iron telegraph wire of the usual size was more than three times greater for rapid periodic currents of about 100 per second than for steady currents. The full elucidation of the propagation of currents in conductors under periodic electromotive force is not to be attempted without following out some very elaborate mathe- matical analysis. The subject has received its most complete treatment perhaps in the published writings of Mr. Oliver Heavisidet and all that can be attempted here is to give a slight sketch of the views which are now very generally held on this subject.