The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 17 of 35

• 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. Equivalent equations have been also arrived at by Prof. H. F. Weber and Mi-. Oliver Heaviside.

t See also Mr. Oliver Heaviside in the Phil. Mag., August, 18B6.

MUTUAL AND SELF INDUCTION. 239

The whole method of the construction and use of the induc- tion bridge has been the subject of elaborate examination by Lord Kayleigh in a Paper on the self-induction and resistance of compound conductors (Phil. May., 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 corresponding 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 flat 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 M0 be the maximum mutual inductance and M the inductance in any position, 6, then

M = M0 cos 0.

This law is, however, not followed when the coils are sensibly of the same size. In this case Lord Eayleigh 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 coils was determined for each degree of angular displacement of the axes by comparing it with the calculated coefficient between two wires, Avound in measured grooves, cut in a cylinder, and it was found that every degree of movement of the movable coil, when the axes were not far removed from perpendicu- larity, was equal to 776*3 centimetres of mutual induction, the maximum when 0 = 0 being 56,100 centimetres. The first experiment described in the Paper referred to is one on the self-induction and resistance of a coil of copper wire. In the bridge used the resistances Q + R + S were together equal to 4'00 ohms. Resistances were, however, measured in scale

u

290 MUTUAL AND SELF INDUCTION.

divisions of the bridge wire, each one equal to 2-04 x 10s 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 = 36° = 36 x 776 centimetres,

and the frequency n of the vibrations = 1,050. Hence p = 2-7r x 1,050. Taking the equations (103) and (104) on page 288, and eliminating L, we have for the value of P, the equation

p /M2(Q + B + S)-) Q.E *- S.Q.B

•t| 13?

Substituting the values above, we find

O T? P = -876 -FT- = 87 -5 scale divisions.

b

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-i-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 87'3 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, R = 190, M = 295°, instead of as before, Q = 610, M = 36°. 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 closed secondary circuit.

The next example selected was that of a soft iron wire, 160 centimetres long and 3-3mm. dia. Here, with the variable

MUTUAL AND SELF INDUCTION. 291

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

Q = 178, R = 190, 8 = 1,592, M = 8 x 776 centimetres, from which we find

P = -985 %? = 20-93 scale divisions.

D

The resistance of the same wire to steady currents was

P0=100*190 = 11-88 scale divisions. 1,6/0

Hence the effective resistance 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 (/R--f2>2L2) measured in ohms is greater than the ohmic resistance (E) ; 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 he 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- section 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 diffuse 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 modern views on the modus 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 equal,, 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-

FIG. 109. FIG. 110.

motive force which is increasing the current has to do workr 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 magnetisation.

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 conductor is not used by the current, and, so far as conducting it goes,.

MUTUAL AND SELF INDUCTION. *^93

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 ohmic 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 •chiefly 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., § 689-690. In this paragraph it is shown 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 varies over the cross-section of the conductor.

294 MUTUAL AND SELF INDUCTION.

Eayleigh has, however, treated with great fulness* one or two cases of practical importance. If E and L are the true ohmic resistance and inductance of a cylindrical straight wire of length Z and magnetic permeability p 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 E1 and the inductance diminished to L1 in such wise that if p = 2?r n, as usual, we have

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

These formulas 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, flows through the conductor, then the rate of dissipation of energy

• "On the Self-induction and Eesistance of Straight Conductors,' Phil. May., May, 1886, p. 382.

MUTUAL AND SELF INDUCTION.

as heat is^- for each portion per unit of length, or If2 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 rf for the sheath and rz2 for the core. Hence, for the equi-distribution of current, the energy dissipation is 1^_ = r(y+z) t anc1 for the unequi-distribution it

is r^+z2). Which, then, is greater, r ^/+8)8or r(i/a+z8)? Consider the following inequalities : —

(y - z)2 is greater than 1 (y - z)2, or ?/2 + z2 - 2 t/z is greater than- (y-zf\

Hence, y* + z2 - 2 y z is greater than - (y + zf - 2 // z. Adding 2 >j z to both sides, we have

yz + z2 is greater than ~(y + zf. 2

Accordingly it follows that

r y* + r z2 is greater than - fa + zf,

or r y2 + r z2 is greater than - a;2 ;

2

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 least 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 electromotive 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 — ^ ^ in the above-given series for

E1. We will first take the case of an iron wire 0-4 centimetre, say, 0-16 inch diameter (No. 8 B.W.G.). The specific resis- tance of iron in C.G.S. measure is about 104 ; so that B_ 104 =106

I TTXO'04 477*

^2=4;r2w2, n being the frequency.

Let us take w=100, so that there are supposed to be 100 complete alternations per second. The value of //, is more difficult to assign. For small degrees of magnetisation and solid iron, we may, perhaps, take /A = 300;

then L P'Vf- 1 4-2nVZ2_

12 E2 12 E2 1010

If *=300, ?i = 100, ^^OxlO8, and — 1- =0-47

= 0'5 nearly.

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

If n = 1,000 we should find E1 = 48 E, or the resistance would be increased nearly fifty times.

MUTUAL AND SELF INDUCTION. 297

Consider next the case of copper. The specific resistance is 1,640 C.G.S. units. If a be the radius of the wire in centi- metres, then we have

_

12 ~~W~ IT (i,640)2 10*

If, as before, ?& = 100, this fraction becomes equal to 0'12a4. 'This shows that for a diameter of one centimetre we should

have B^K (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 from 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 Eayleigh (loc. cit.} examined the resistance of an iron wire of hard Swedish iron 10-03 metres long and 1-6 millimetre in diameter. In arbitrary units the resistance of the wire to steady currents was 10-4 units or 0-51 ohm, and to currents of 1,050 complete alternations per second its resistance was 12-1 units, or 0-595 ohm, which is an increase of about 20 per cent. In the case of a stouter wire, 18-34 metres long and 3-3 millimetres 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 Eayleigh 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. 292

Two arms of a quadrilateral, E 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 rr1 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 C by shifting the contact of the telephone along r ?J.

The condition for obtaining a true balance when the current is periodically interrupted is that the resistances and induc- tances of the branches Q and P C 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 + r1 is made to be equal to that of Q + r. Let, now, any conductor, P, be inserted as in the 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 C has to be reduced on inserting P, and the resistance of P is measured by twice the resistance of that length of the German- silver wire r?-1 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 AXD SELF INDUCTION.

two copper wires, wound side by side on one bobbin. The diameter of each wire was about 0-OSin., and the length of each wire 318in. There were 20 (double) turns, so that the mean diameter of the coil, wound as compactly as possible, •was about oin., 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 = 48-1° = 43-1 x 1,553 centimetres.

Observation showed that closing of the circuit of one wire reduced the self-induction of the other from 44-4° to 3-4°. The resistance to steady currents was 0-92 (arbitrary units). The resistance to the periodic currents was 0-97 with the secondary circuit open, and 1-74 with the secondary circuit closed. Hence, L = 44-4x 1,553 centimetres, and

E = 0-97 x 0-0492 x 109 centimetres per second.

From Maxwell's formulae, page 180, we get

*™ 10" x 1-951 _

B* +/>*La 107 x 0-023 + 1017 x 2-071 Hence, L^L (1-0-932),

•where L1 is the decreased inductance. Hence,

L! = 0-068 L, or L! = 0-068x44-4° = 3°,

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 R1 with secondary closed is

R! = 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 cf steady electric currents and rapidly

MUTUAL AND SELF INDUCTION. 301

periodic currents 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 called attention in 1883 to this great difference in the resistance of an electrical conductor if mea- sured during the variable instead of the stable 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 Heavisidef 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.

Consider a long level tank or canal full of liquid. There are, amongst others, two ways in which we might suppose this liquid to be set in motion. A paddle or the hand might be placed in the liquid, and by giving the liquid bodily a push it might be made to move forward ; or we might suppose some body floating on the surface, such as a plank of wood to be dragged along the surface. The friction between the plank and the layer of water beneath it would then cause the subjacent layer of liquid to move with the plank, and the motion of this layer would be gradually com- municated to the other and deeper-lying layers by reason of the viscosity of the fluid. Or take the case of a basin containing water. The liquid might be set in rotation by stirring it with a paddle or the hand, but it might also be set in rotation by twisting the basin rapidly. In this las: case the rotation of the basin would be communicated by friction to the water in contact with its sides, and then handed

• Discussion on a Paper by Mr. \V. H. Preece on " Electrical Conductors," Proceedings In&t. Civil Engineers, Vol. LXXY., 1883.

t "Electromagnetic Theory," by Oliver Heaviside in The Electrician Series, and " Electrical Papers," published by Messrs. Maemillan.

302 MUTUAL AND SELF INDUCTION.

on from layer to layer of the water by internal fluid friction. Thus the twist or spin of the basin would be gradually propa- gated inwards from the circumference to the centre. Imagine the whole mass of the liquid divided up into very thin con- centric shells, like the coats of an onion. If the liquid were a perfect fluid there would be no friction between these layers, but since every liquid possesses some degree of viscosity or internal fluid friction, the sliding of one layer of fluid over another gradually causes the second layer to partake of the motion of the first. Hence, when the rotation of the basin commences the friction between its sides and the first layer of fluid starts that gradually in motion ; this motion is then transmitted to the second layer, and so on, until the whole mass of the liquid possesses an equal angular velocity round the axis of rotation. The greater the fluid friction or viscosity the more rapid will be this equalisation of the angular velocity of all parts of the fluid, and so a rotating vessel full of tar would arrive at a stationary condition as regards angular velocity sooner than one filled with a limpid liquid as alcohol or ether. Just as the angular velocity diffuses inwards from the circum- ference to the centre in the case of such a revolving basin of liquid, so, according to modern views, does the current diffuse inwards from the circumference to the axis of the electric con- ductor, The student who has been accustomed to think of a current as produced in a conductor by a sort of push given to it in the conductor — such conception being based on a rough working hypothesis of a hydrodynamic nature — will perhaps have some difficulty in discarding this notion and realising that the current in a wire may perhaps be generated in it by an action taking place at all parts of the surface of the u-ire which gradually soaks or diffuses into the conductor out of the sur- rounding dielectric, but he will find that this new hypothesis serves to establish a mode of viewing the induction phenomena which makes various experimental results much more easily cor- related. It was well demonstrated by the experiments of Prof. Hughes and others that a flat sheet or strip of metal has a less self-induction than a round wire of equal cross-sectional area. On the present hypothesis, this is explained by saying that the flat strip offers a greater absorption surface to the dielectric ; the current therefore soaks in more quickly to the centre and arrives

MUTUAL AND SELF INDUCTION. 303

•at a uniform distribution over the cross section very soon— in other words, the variable state is sooner over, and we express this fact by saying that the self-induction is small. Again, if the electromotive force is oscillatory or rapidly periodic, we see at once that the current has not time to penetrate right into the core of the conductor before its sign or direction is reversed. It has hardly started on its journey inwards, soaking from sur- face to centre, before it is recalled ; hence the flow of a current when very rapidly periodic is confined to the surface of the conductor, the real or ohmic resistance is increased, and the self-induction is diminished.

Lord Kelvin has shown (Bath British Association Meeting, 1888) that for alternating currents of a frequency equal to about 150 complete alternations per second, the depth to which the currents penetrate into the substance of the copper is about three millimetres, so that portions of the conductor beyond this distance from the surface are almost useless for conduction. The practical moral of this is that the proper form for a con- ductor for alternate currents is either a flat sheet of copper or a copper tube, in which, for the above frequency, the thickness of material is not more than one-quarter of an inch. It is useful in this connection to note a few facts with regard to cables as used for alternating currents. A seven strand cable has an overall diameter of three times one strand. A nine- teen strand cable has an over-all diameter of five times one strand. A No. 12 wire (S.W.G.) has a diameter of 0-109 inch. Hence a 19/12 cable has a diameter of 0-5 inch, and a cross- sectional area of 0-1615 square inch. At a current density of 600 amperes per square inch this cable will carry 100 amperes, and it has a resistance of one- sixth of an ohm per 1,000 yards. For alternating currents, therefore, of about 100 frequency, a 19/12 stranded cable is about the largest size that should be employed. For alternating currents of 100 frequency, beyond about 100 amperes, a form of cable must be employed in which the thickness of the conductor is not at any part greater than about one quarter of an inch ; and this is only to be achieved by the employment of concentric tubes or concentric stranded cables in which the core or central strand is not of greater thickness than half an inch, and which con- dition necessitates, therefore, that when above a certain cross-

304 MUTUAL AND SELF INDUCTION'.

sectional area the central conductors should also be of tubular form. One of the advantages to be gained by the employment of alternating currents of low frequency is that the limiting diameter of the conductors is much larger for low than for high frequency. To return to our illustration of the twisting basin of fluid. Suppose the action on the vessel consists in rapidly twisting it through a small angle, first one way and then the other, the liquid in the interior would be subjected to a strain which would consist in the various concentric layers of the liquid sliding backwards and forwards over each other. The interior of the liquid would be thrown into stationary waves, in which the nature of the wave motion consisted hi each particle of water being displaced first one way and then another along an arc of a circle described on a horizontal plane, with its centre in the axis of rotation. The more rapid the motion the greater would be the rate of decrease in the amplitude of each wave in passing from the circumference to the centre of the vessel ; in other words, for very rapid oscilla- tions the bulk of the water in the centre of the basin would remain nearly at rest.

Every experiment as yet made on the self-induction or change of self-induction in conductors is consistent with the above hypothesis. It shows, for instance, why a conductor composed of thin insulated wires or thin insulated strips has a less self- induction than a solid conductor of equal cross-section. Prof. Hughes says* : — " We can reduce the self-induction of a current upon itself to a mere fraction of its previous value by simply separating the contiguous portions of a current from each other, the results proving that a comparatively small separation, such as is obtained by employing ribbon conductor? in place of a wire of the samo weight, reduces the self-induction 80 per cent, in iron and 85 per cent, in copper, and if we still divide the current by cutting the ribbon into several strips (separating the strips at least 1 centimetre from each other), then the combined but separated strips show a still greater reduction, being 94 per cent, in iron and 75 per cent, in copper."