The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 23 of 36

§ 7. The Function of the Condenser in an Induction Ooil. — Pizeau appears to have been the first* to suggest that the action of an induction coil employed for raising the electro- motive force of a current would be increased by the employ- ment of a condenser. Its mode of use is as follows : — Let P be a primary circuit which takes current from a few cells of a battery, and let I be an interrupter in the primary circuit, either automatically worked by the magnetisation and demagnetisation of the iron core or by any other means. Let S be a secondary circuit of many more turns and high resistance. Under these circumstances each break of the primary current is accompanied by the production of an electromotive force in the secondary cir- cuit capable of producing a discharge across an air space in the secondary circuit. This electromotive force in the secondary is increased by any action tending to increase the suddenness of the stoppage of the primary current, and decreased by anything promoting a spark at the points of rupture of the primary circuit. Fizeau found that if a condenser, formed of alternate sheets of tinfoil and mica or paraffined paper in such fashion as to form a Leyden jar, has its two opposite coatings connected with the two extremities between which the rupture of the primary circuit takes place, then the electromotive force in the secon- dary circuit under these circumstances is increased. In most current text-books this action is explained by saying that the extra current in the primary circuit, instead of being expended in making a spark at the contact points, darts into the condenser and hastens the decay of the primary current. This explanation as generally given is, however, very imperfect. A more complete examination of the nature of the condenser action has been given by Lord Rayleigh {Phil, Mag.^ Vol. XXXIX., 1870, p. 428, et seq.). In the experiments there detailed a sewing needle was submitted to the magnetising action of an induced secondary current produced by the ** break " of the current in a primary circuit. In some previous experiments • Comptes Jiendus, VoL XXXVL, p. 418, 1853.

DYNAMICAL THEORY OF INDUCTION. 391

by the same writer (PAtZ. Mag.^ July, 1869, p. 9) it had been shown that the magnetising effect of the secondary current was, cet. par,, proportional to the initial strength of the in- duced current, and that this initial strength was proportional to the quotient of M by N, or to the value of the ratio of the coefficient of mutual induction to the coefficient of self-induc- tion of the secondary circuit. It was then found that the mag- netising effect of the secondary current was greatly increased by connecting the plates of a condenser respectively to the two points between which the break of the primary circuit occurred. The complete investigation of the values of the induced and primary currents would under these conditions be a good deal more complicated than the investigation of the more simple case of the discharge of a condenser through a single inductive circuit. We are here, however, only concerned with the first part of the electrical motion, the manner in which the currents wear down under the action of the resistances being of subordinate importance. It appears that when the electrical motion is decidedly of the oscillatory type the first few oscil- lations will take place almost uninfluenced by resistance, and on this supposition the calculation (following Lord Bayleigh) becomes remarkably simple.

Let L, M and N be the primary, mutual and secondary in- ductance, and let i and i' be the primary and secondary current strengths at any instant, and q and g' the quantities of elec- tricity which have flowed through these circuits from the instant of beginning to reckon the time t,

then ^J-tand^'-r;

at at

and if we neglect resistance effects, as we can do at the instant after *^ breaking " the primary circuit, and call C the capacity of the condenser bridging across the *< break" of the primary circuit, the equations giving the values of the primary and secondary current i and i' at the instant after breaking the primary circuit are —

L'^-' + M^' + ^H-O. . . . (123) dt dt C ^

M^' + N^' -0. ... (124) dt dt

892 DYNAMICAL THEORY OF INDUCTION.

Eliminatiiig i' we have

(125) may be written

{"-"^IrH-o- ■ ■ ■ <'^»)

A differential equation of this type always indicates an

oscillatory motion. For, consider the simple periodic function

2ir d;== Asin p ^, where ^ = -=-9 T being the periodic time of the

motion, we have — ?=;? A cos /?«, and ^x= -p^ A sin pt; a t dr

hence, -r^ +p* « = 0, and therefore a? » A sin p t is a particular or

solution of this equation.

In the above differential equation p is seen to be 2ir times the frequency of the oscillation.

Accordingly, equation (126) indicates an oscillation of the primary current, of which the periodic time is equal to

-yc(L-f).

and this is the periodicity of the electric oscillation set up in the primary at the first instant after " break."

Equation (124) gives by integration the connection between i and i', and it is

Mt + N»' = constant, . . . (127)

which shows that the currents in the primary and secondary oscillate synchronously, the maximum of the one coinciding with the minimum of the other. Since t' is zero at the instant of ' break,' the constant in equation (127) must be equal to M I, where I is the current strength in the primary just before *' breaking " primary circuit. Accordingly, we have

so that when, after half an oscillation of the primary, i becomes equal to -I, we have

»' = 2|l (128)

DYNAMICAL THEORY OF TNBUCTION. 393

This equation gives ns the initial value of the secondary

onrrent i' in terms of the value of the primary current just

before the '' break *' when the condenser is used. Comparing

equation (128) with the results on page 286, where it is shown

that, if the condenser is not used across the '' break *' of the

primary, the initial value of the secondary current under the

M assumption of a perfectly sudden break is equal to ~-I,

N

we see that the value of the secondary current just inmie- diately after the break of the primary, is double that which is there deduced as the value when the primary is simply suddenly stopped without the intervention of the condenser. Stripped of symbolism, what the above amounts to is this : if a condenser is inserted across the '* break points " of a primary circuit, then on breaking the primary circuit, the primary current continues to run on into the condenser for a short time ; it then rebounds, and is reversed in sign, retaining initially its full strength. The electromotive force set up in the secondary circuit is then the result of a stoppage of a primary current and its immediate reversal in direction, and this is equivalent to the removal of a certain number of lines of induction from the secondary circuit, and their immediate insertion into it in the opposite direction. Hence, when a condenser is so employed, the inductive electromotive force in the secondary must be just double that which it would be if there were no such rebound of the primary. The condenser acts by setting up electrical oscillations, and it does away with the spark, or largely dimi- nishes it, in virtue of the fact that the condenser acts at the moment of '* break " as if it were a shunt circuit of negative self-induction, only with this difference — that instead of dissL pating energy like a conducting circuit it returns it again to the primary circuit in the form of a reversed current, and increases the total change of induction through the secondary circuit in the short interval of time immediately succeeding the ** break."

Since the sparking distance of the secondary current depends on the initial electromotive force in the secondary — that is, on the maximum of the electromotive force — we see that the con- denser so applied can greatly increase the sparking distance of the secondary discharge.

894 DYNAMICAL THEORY OF INDUCTION.

The action is essentially a phenomenon of resonance. The condenser causes an elastic recoil in the current and enables the electro-kinetic energy of the steady primary current to be utilised in producing secondary electromotive force rather than suffer dissipation in the form of a contact spark. In order to be efficient in quenching spark the capacity of the condenser must be great enough to take the full primary current, or to receive charge at a rate equal to the delivery of the full primary current for a time during which the contact or break points are separating to a distance too great to permit of much sparking jumping across. There is a certain capacity of condenser suitable for any given coil which produces the most beneficial result in quenching contact spark and lengthening secondary spark. The required capacity is best determined by trial, since the experimental data necessary to furnish the means to calculate it would be probably more difficult to obtain, owing to the fact that it will be determined by several variables, viz., the effective resistance and inductance of the primary circuit, the rate of breaking, and probably also by the ampli- tude of movement of the '* break points." If the primary coil of an induction coil is traversed by an alternating current then the condenser as ordinarily used becomes superfluous. It will be remembered that the late Mr. Spottiswoode obtained secondary sparks of great magnitude from his large coil by so using the alternating current of a De Meritens machine.

If a condenser is discharged through a circuit of which the resistance is so small that it may be neglected in numerical comparisons, then the equation of discharge is

L^ + i^O, dt^ G '

where the symbols have the same signification as before. As above explained, this indicates that the discharge is oscillatory, and that the time of a complete oscillation is 2w ^LC.

In describing the experiments of Blaserna we saw that the frequency of the electrical oscillation set up in circuits on starting and stopping currents in them could be reckoned by tens of thousands per second. In the case of Leyden jars dis- charged through very short circuits, the frequency may rise to numbers reckoned by millions per second. Since the frequency

DYNAMICAL THEORY OF INDUCTION. 396

of luminous vibrations falls between 400 and 700 billions per second, these condenser oscillations fall in frequency in the gap between the acoustic and luminous vibrations.

It is of interest here to note that since these electrical oscillations in a circuit are creating pulsatory electrical dis- turbances, which spread out from the wire laterally, the wire in which the electrical oscillations are going on is virtually emitting "light," although not such light as can affect our eyes. The ether waves in the case of these elec- trical disturbances are too long to be eye-affecting. If the velocity of a wave disturbance is V, and the wave length is A, and the frequency of the oscillations corresponding to this wave length is n, then V-nA, for the wave motion travels over the length of one wave in the time of one complete oscil- lation. In the case of ether disturbances we have seen that V is 8 X 10" centimetres per second, or 186,000 miles per second. Hence when the frequency of the electrical oscilla- tions is known, the wave length of the lateral disturbance emitted can be found. According to Dr. Lodge, a microfarad condenser discharging through a good conducting coil having an inductance of one henry gives a current alternating 160 times in a second, and emits ether waves about 1,200 miles long. A gallon Ley den jar (capacity about 0*008 microfarad) dis- charging through a stout wire suspended round an ordinary sized room emits ether waves between three and four hundred yards in length, its current alternating at the rate of about one million per second. A pint Leyden jar sparking through an ordinary pair of discharging tongs gives a current of 15 miUion alternations per second, with ether waves some 20 yards in length. An ordinary electrostatic charge on a sphere two feet in diameter, if disturbed in any way, will surge to and fro at the rate of 800 miUion vibrations per second, emitting ether waves a yard long. Electric charges on bodies of atomic dimensions, if able to oscillate at all, would vibrate thou- sands of billions of times a second, and produce ultra-violet light.

The ordinary use of a condenser with an induction coil shows how it can be employed to neutralise the effect of self -induction in a circuit. We have considered on page 187, § 81, of Chapter IV. I the case of a condenser having its terminals shunted by a

396 DYNAMICAL TffEORT OF INDUCTION,

resistance, and the combination placed in series vnth an inductive circuit and there shown that capacity can neutralize self-induction. We may also consider the case of a condenser in parallel with an inductive circuit as another similar problem. Let L B (Fig. 140) be an inductive circuit, and let the ter- minals a 6 be closed by a condenser 0 of capacity G. Let L be the inductance and B, the resistance of the coil. Let t be the value at any instant of a simple periodic current sent through the relay and condenser in parallel, and let i, i^ be the simul- taneous current strengths at that instant in the condenser circuit and the coil circuit. As the potential difference of the points a and b oscillates, an ebb and flow of current is produced in the condenser circuit ; the condenser, in fact, is charged and discharged by the periodic current ; also a periodic current is produced in the inductive circuit L E. The current in L B

lags in phase behind the impressed electromotive force or potential difference of the points a h, and the current flowing into the condenser lags 90deg. in phase behind the same im- pressed electromotive force. From this it results that the mean current through the inductive circuit may, under some circumstances, be greater when the condenser is joined up to its ends than when it is not so joined ; its effective self -induction is thereby lessened, and it acts as if it had experienced a diminution of self-induction. The condition most favourable for producing this result may be investigated as follows : —

Let V be the potential difference of the points a and h at the instant when the current in the undivided circuit is i and that in the branches is i^ and lij. We then have, by the principle of continuity,

i^h + h, (129)

,UNIV-: .--TTjr]

DYNAMICAL THEORY OF INDUCTION. 3»7

also »i = 0l^ (180)

and L^' + Bt,=t;, .... (181)

and we may take the original current before division to be simply periodic, and to be expressed by

t = Isinp« (182)

where I is its maximum value.

Then by elimination of v and I'l and i from the above four equations we arrive easily at the equation —

CL^ + CR^ + t,=Isinp«. . . (188) air at

Now, since t, must be a simple periodic current lagging in phase behind that of the undivided current i, we may take it to be of the form

i^^ J^ Bin (pt-0), (184)

J^ being the maximum value of t,, and 6 its phase lag be- hind Ii.

Hence, by differentiation of (184) and substitution in (188) we arrive at

{l-CIjp*)l^sm{pt'-e) + CBplL,coa{pt-ff)=l8inptf (135)

which by the lemma on page 161 may be written —

P J(l-QIjpy + G^B;'p' (sin;? t - ^ + <^) -I sinp t (186)

Both sides of this last equation are the expressions for the same thing, viz., the value of i, and hence, equating the coefficients, we have

('iy=(l-CL;;y + C^R>«. . . (187)

This gives us the value of the ratio of the maximum or mean values of the strengths of the undivided current and the current in the inductive circuit. If we differentiate the right-hand side of (137) with respect to C, and apply the usual criterion to ascertain whether we have a maximum or minimum value, we find that the expression on the right hand side of (187) has a minimum value when

393 DYNAMICAL THEORY OF INDUCTION.

In other words, if the capacity of the condenser is so chosen as to have a capacity equal numerically to the quotient of the inductance by the impedance of the coil, then, under these circumstances, the mean strength of the current in the coil circuit will be greater than the mean strength of the current before subdivision ; and it is easily seen, by substituting in equation (187) the value of C given by (138), which makes the ratio of current strength a minimum, that with this value of the capacity the strength of the current in the inductive coU is to the strength of the current before division in the ratio of the impedance to the resistance of the inductive circuit.

The expression (188) gives the value of the condenser capacity which will produce the required result of minimising the self- induction of a relay of resistance B and inductance L when applied to it. Another problem of a like kind, but not so practically useful, is the investigation of the behaviour of a condenser when joined in series with an inductive coil and traversed by a simple periodic current. Let a condenser of capacity G be joined in series with an inductive circuit of resistance B and inductance L, and let a simple periodic current of frequency n be sent through the two in series. It is not difficult to show that, if we take p for 2ir n, as usual, and if the capacity and inductance are so related to the

frequency of oscillation that j? = - , then, under these

vJJC circumstances, the condenser just annuls the self-induction of the coil, and the two together permit the passage of the same current which would traverse the coil in virtue of its resistance B, assuming it to have no inductance. This is easily proved as follows : — Lot L be the inductance and B the resistance of the inductive circuit, and G the capacity of the condenser in series with it. Let v » V sin j) t be the potential difference at the instant t, measured over the condenser and inductive resistance, and let Vi and v^ be the fall of potential down the inductive circuit and condenser respectively. Then t? = vi + v,; (189)

also, L^+Et = r„ (UO)

at

and

ljidt = v„ (141)

DYNAMICAL THEORY OF INDUCTION. 899

where t is the value of the current flowing in the circuit at the instant when the potential difference between the ends of the whole circuit is v.

Hence by substitution we have

L^'+Rt + J,/"tei«=Vsin;>«. . . (142)

The current being also simply periodic must have a value t expressed by the equation,

i=^lBin(pt-e), .... (148)

since it will differ in phase by an angle d from the potential difference r.

Accordingly, we find that by differentiating (148) and substituting the values in (142) we arrive at the equation

The maximum value of the current, viz., I, is therefore given by the expression

I cvp

^ >/(l-CLp^)=' + R*CV If then 1 « C hp^ or p = , we see that the above equation

reduces to ^ — s»

and the whole circuit of coil and condenser is equivalent to a simple non-inductive circuit of resistance B, in other words the inductance is annuled. Hence a certain relation between the inductance, capacity, and frequency, causes the inductance to be neutralised by the capacity, and the whole circuit to be effectively non-inductive.

§8. Impulsive Discharges and Relation of Inductance thereto. — If between the ends of a conductor a difference of potential is created which is brought about slowly, the result shows itself in a current in the conductor, and the resulting current is determined as to strength by the mode of variation of the potential and by the capacity as well as by the induct-

400 DYNAMICAL THEORY OF INDUCTION.

ance and ohmio resistance of the conductor. If, however, the difference of potential is created with great suddenness, the resulting electric flow is less determined by the true resist- ance, and more by the inductance of the conductor. In this case we have the phenomena of impulsive discharges. We have a mechanical analogy in the case of impulses or sudden blows given to heavy bodies, which well illustrates how strikingly force phenomena may be altered when for steady or slowly varying forces we substitute exceedingly brief impulses or blows. If an explosive, such as gun-cotton, is laid on a stone slab in open air, and simply ignited, it burns away with com- parative slowness ; the slab is uninjured, and the evolved gases simply push the air away to make room for themselves. But it is well known that by means of detonators the same explo- sive can be fired with enormously greater rapidity, and in this case the blow or impulse given to the air is so sudden that it has not time to be pushed away, and in virtue of its inertia its incapacity of receiving a finite velocity in an infinitely small time bestows on it an inertia resistance, which causes nearly the whole of the effect of the explosion to take effect downwards on the slab, and this last is shattered. The inductance oi conductors introduces a series of phenomena which are the electrical analogues of the above mechanical experiment. We have seen that the counter electromotive force of self-induc- tion is proportional to the rate of change of another quantity, called the electro-kinetic momentum, and this quantity phy- sically interpreted is the total flux of induction or number of lines of induction enclosed by the conducting circuit at that instant. A conductor of sensible inductance can no more have a current of finite magnitude created in it instantaneously than a body of sensible mass can have a finite velocity in- stantaneously given to it. In both cases there is an immense resistance to very sudden change of condition. A very loose plug of snow or earth stuffed into the muzzle of a loaded gun will cause it to burst when fired, since the inertia resistance of the plug to very sudden motion is exceedingly large, though the frictional resistance may be small. Accordingly, the study of the behaviour of conductors under exceedingly sudden electric blows or electromotive impulses leads us to consider some very interesting effects. We shall best eluci-

DYNAMICAL THEORY OF INDUCTION.

401

date these effects by describing some interesting and suggestive experiments due to Dr. Oliver Lodge.*

His first experiment is called the experiment of the alternative path. The two ternunal knobs of a Voss or Wimshurst electrical machine (see Fig. 141) are connected to the two inside coatings of a pair of Leyden jars. The two outside coatings are con- nected to the balls of another discharger, B, and the terminals of this discharger are short-circuited by a metal wire, indicated by the dotted line. The Leyden jars stand on a badly insu- lating wooden base. On turning the handle of the electrical machine the inside coatings receive equal and opposite elec- trical charges, and there is an induced charge on the outer

L

Pio. 141.

coating of each, which, in the language of the old school of electricians, was called the '' bound " charge. When the differ- ence of potentials of the inner coatings reaches a certain value the air space at A is cracked, and a spark passes, discharging the inner coatings of the jars. At that instant the charges of the outer coatings are set '' free," or, in modem language, the potential of one rises and that of the other falls. The effect of this is that whereas before the spark passed at A the balls at B

• The acoouDt of these ezperimentB ia taken from the report of Dr. Lodge's Mann Lectures before the Society of Arts. These suggestiTe lectures were reprinted in The Eleetrioia/n, entitled " Protection of Build- ings from Lightning," Vol. XXL, pp. 234, 273, 302.

D O

402 DYNAMICAL THEORY OF INDUCTION.

were at equal potentiali on a spark at A happening the balls at B are instantaneously brought to a very great difference of potential. It might be thought that since the balls are short-circuited by a metallic wire this difference of potential will expend itself on making a current in the wire. On the contrary, very little of the discharge may take place through the wire. A spark passes at B, or, in other words, the discharge passes in great part across the exceedingly highly resisting air space at B, rather than take the circuit of the metallic wire of very low resistance, so that although there is a divided circuit open to the discharge, one branch of which measures hundreds of thousands of ohms or megohms and the other only a small fraction of an ohm, it nearly all goes by the route of higher resistance. The explanation of this is that when the balls at B are thrown with great suddenness into

FiQ. 142.

opposite electrical states the counter electromotive force of self-induction of the circuit of metal L makes it virtually non-conducting. The electromotive impulse meets with such resistance owing to the electro-magnetic inertia of the circuit that it rebounds and cracks through the air. In order that it shall do this, however, the distance of this air gap at B has to be less than a certain amount. There is a certain critical distance of the knobs B for less than which the dis- charge always jumps across B, and for greater than which the discharge keeps mainly to the metallic circuit. Even if the short-circuiting metal is a thick rod, still when B is not great the discharge chooses the air-gap path. The phenomenon here presented has had fresh interest and attention called to it by Dr. Lodge, but it has really been long a familiar one, though its explanation has not stood out hitherto so sharply as it does now.

DYNAMICAL THEORY OF INDVCTION.

403

Faraday was acquainted with it, and showed that if a charged Leyden jar is discharged by means of a wire crossed or bent so that there was a loop (Fig. 142), the wires at a nearly but not quite touching, then when the spark happened at 6, a spark took place also at a, showing that some at least of the discharge jumped across a instead of pursuing the course of the metal loop.

The same fact lies at the base of the action of all lightning arresters placed on telegraph or telephone instruments. Mr. C. F. Varley, we believe, first suggested that the coils of the single-needle instrument might be protected from damage by lightning by twisting together the earth and line wires where they leave the case, the theory being that although ordinary

Fia. 143.

currents were not short-circuited by reason of the cotton covering of the wires, yet lightning discharges would meet with such resistance in the inductive coils that they would jump across the knot from wire to wire rather than pass round and damage the coils. In the same way the ordinary comb protector is supposed to act. Between the line wire L {see Fig. 148) and the electromagnetic instrument, relay, telephone, &c., is placed a metal comb, which has its points in opposition to another comb in connection with the earth, and the other terminal of the electromagnetic instrument is also " to earth." An incoming current has then two paths open to it to get to earth, one of comparatively low resistance through the instrument, and one of enormously high resistance across the

DD 2

404 DYNAMICAL THEOBT OF INDUCTION.

air-gap between the comb points. Ordinary currents, steady or periodio, pass entirely through the metallic circuit. Very violent electric impulses, such as a lightning discharge, meet with an enormous inductive opposition in the electromagnetic instru- ment owing to the inability of an inductive circuit to respond to an electromotive impulse instantaneously. Hence the air- gap is cracked, and the discharge passes across the combs and the instrument may be saved. Evidence exists, however, pointing to the fact that the protection afforded by these con- trivances is very far from complete. We are not here concerned with their efficiency as practical devices, but only with them as illustrating the principle of the alternative path and the behaviour of inductive circuits to impulsive discharges. That these devices are insufficient has been fully demonstrated by many experimentalists.*

In these experiments of the alternative path it was found by Dr. Lodge that the critical distance at which the discharge just prefers to jump the air-gap was greater for a thick copper rod 40 feet long (No. 1 B.W.G.) than for an iron wire (No. 27 B.W.G.) of 88*8 ohms resistance, indicating a less inductive inertia on the part of the iron ; but this fact is only true for the particular circumstances of the experiment. A very clear difference was established between copper rod and tape, using conductors of the same length and weight. The tape has an advantage in permitting more easily the passage of sudden electric discharges. A controversy on the relative suitability of rod and tape for lightning conductors dates from the time of Faraday and Sir W. Snow Harris, and a possible explanation of the reasons for preferring one rather than the other presents itself when we consider the matter in the light of those con- siderations which induce us to think that an electric current begins always at the surface of conductor, and takes a certain time to diffuse or soak into the mass of the metal. It is not cross-section but surface which is here concerned ; and, other things being equal, the conductor which offers the greatest surface to the dielectric is able to drain the energy out of the

• For an account of Bome interesting experiments by Ptof. Hughes and Prof. Quillemin on ** Lightning Protectors " see Ths EUeirieian, YoL XXI., p. 304, July 13, 1888. It was found that a protects coouBting ci two opposed flat plates waa better than a comb or opposed Doint

DYNAMICAL THEOEY OF INDUCTION. 405

dieleotrio most quickly and dissipate it as heat in the oon- duotor. We have referred to this on a previous page (see ante^ p. 252) I and it will be mentioned again in connection with some views of Prof. Poynting.* With respeob to the apparent supe- riority of iron, it would naturally have been supposed that, since the magnetic permeability of iron bestows upon it greater inductance, it would form a less suitable conductor for dis- charging electric energy with great suddenness. Owing to the fact that the current only penetrates just into the skin of the conductor, there is but little of the mass of the iron magnetised, even if these instantaneous discharges are capable of magne- tising iron. This last fact has been thought to be due to an actual time lag of magnetisation, viz., that magnetising force required to endure for a sensible time in order to produce magnetisation, but recent views tend in the direction of con- sidering the apparent lag as a consequence of the fact that the eddy currents produced in the surface layers of the metal by the discharge shield the inner and deeper layers from inductive influence, as described under the head of Magnetic Screening. In any event the final result is the same ; the electromotive impulses, or sudden rushes of electricity, do not magnetise the iron, and hence do not find in it any greater self-inductive oppo- sition than they would find in a non-magnetic but otherwise

• For lome special remarks on the sell-induotion of wires of various cross- seotioDs see Mr. Oliver Heaviside in the PhiL Mag,, January, 1887| p. 11 : — " The magnetic energy per unit of length of a circuit is } L t^, where i is the current in the wire and L the inductance per unit of length. As regards the diminution of L in general by spreading out the current in a strip instead of concentrating it in a wire, that is a matter of elementary reason- ing founded on the general structure of L. If we draw apart currents, keeping the currents constant, thus doing work against their mutual attraction, we diminish their energy at the same time by the amount of work done against their attraction. Thus the quantity ) L r* of a circuit is the amount of work that must be done to take the current to pieces, so to speak — ^that is, to separate all its filamentary elements of currents to an infinite distance. If wires are taken, each of a unit of length and of the same total cross-sectional area, but of different forms of croes-section, round, square, elliptical, equilateral triangle, narrow rectangle, kc, the ratio of their inductances is the same as the ratio of their torsional rigidities. Thus the narrow strip has the least torsional rigidity, and the circular- sectioned wire the greatest, and this is true also for their relative self- inductions."

406 DYNAMICAL THEORY OF INDUCTION.