The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 22 of 35

by a uni-directional discharge. It is highly probable that Prof. Joseph Henry, as far back as 1842, was the first to recognise that the discharge of a condenser might be of an oscillatory character. It is remarked by him* that "The discharge, whatever may be its nature, is not correctly represented by a single transfer of imponderable fluid from one side of the jar to the other ; the phenomena require us to admit the existence of a principal discharge in one direction and then several reflex actions backward and forward, each more feeble than the preceding, until equilibrium is attained. All the facts are shown to be in accordance with this hypothesis, and a ready explanation is afforded by it of a number of phenomena which are to be found described in the older works on electricity, but which have until this time remained unexplained." A little later on in the Paper he gives an explanation of the reversal of polarity of the needles by the oscillatory discharge. In his celebrated Essay, " Erhaltung der Kraft " (Berlin, 1847), Helmholtz alluded also to such a possible form of electric discharge in the following words : " We assume that the dis- charge (of a jar) is not a simple motion of the electricity in one direction, but a backward and forward motion between the coatings in oscillation, which become continually smaller until the entire vis viva is destroyed by the sum of the resistances." He adds : " The notion that the discharge consists of alter- nately opposed currents is also favoured by the phenomena observed by Wollaston while attempting to decompose water by electric shocks, that both descriptions of gases are evolved at both electrodes." The investigation which, however, marks an epoch in this subject is the Paper by Lord Kelvin (then Sir William Thomson) in the June number of the Philosophical Magazine for 1853, on " Transient Electric Currents." In this Paper the author discusses, first, the equations which determine these currents at any instant when a condenser or Leyden jar is discharged through a conductor. The dis- charging conductor is supposed to have self-induction, or as

• " The Scientific Writings of the late Prof. Joseph Henry." Washing- ton : 1886. Vol. I. This statement of Prof. Henry had attention directed to it by Mr. A. D. Raine in 2 he Electrician of November 2, 1888, p. 831. It had been previously mentioned, however, in the sketch of the life of Prof. Joseph Henry, given in the Encyclopedia Brittanica, Ninth Edition.

376 DYNAMICAL THEORY OF INDUCTION.

Lord Kelvin then called it, " electro-dynamic capacity," and also to have ohmic resistance, which is constant, and indepen- dent of the rate of discharge. On these two assumptions he builds up an equation which mathematically contains the whole theory, as follows : —

If C is the electrostatic capacity of the jar or condenser, and E the ohmic resistance, and L the constant inductance of the discharging conductor ; and if q is the electric quantity hi the jar, and v the potential difference of its coatings at any instant t, then by the definition of electric capacity we have

q = C v ,

and _? = i = the current at that instant in the conductor, which di

is equal by Ohm's law to ^. By the principle of conservation of energy the rate at which electro-magnetic energy is being

taken up by the conductor, viz., — (A L i2), together with the a t

rate at which energy is being dissipated as heat in the con- ductor, viz., Ei2 (by Joule's law), must be equal to the rate of decay of the energy contained in the jar, or to

-£<»•)--;£(£)•

tit a t \ \jj

__ (I f , Q \ f?

Hence - — ( £ L 1 » _

Cdt~ J Jt but t = '-^, or the current is the rate of loss of charge, there-

The value of q, or the charge in the jar at any instant, is given by the solution of this equation. Let us write the equation in the form

In order to solve this equation we may proceed as follows : The charge g in the jar begins by possessing a certain initial

DYNAMICAL THEORY OF INDUCTION. 377

value, and ends by being zero. Let us assume that q can be expressed as a function of the time t in the form q=A.emt, where A is a constant and e is the base of the Naperian logarithms, and m is also a certain function determined by the capacity, resistance, and inductance of the system. For it is clear that by a suitable value for A and m the func- tion A emt may be made to express the mode in which the charge q dies away with increase of the time t. The problem is reduced, then, to finding A and m. The solution of nearly every differential equation is by a process of happy guessing ; there is generally no systematic or direct method of obtaining the required result. Take, then, the expression g^Ae"", form the first and second differential coefficients, and sub- stitute these results in the original equation, and we arrive at the expression

Hence, the value A emt assumed for q will satisfy the equa- tion (115) ; that is, when substituted for q in the original expression, render it zero, provided that m is such a quantity that ma+ a ?/i-|- 6 = 0. The two roots of this last quadratic equation are obtained by a simple solution, and they are

Two cases then arise, first, when — is greater than b — that

is, when J?l- is greater than JL , or -— greater than _ . In

4 J_r Li O 4 Jj

this case the roots of the quadratic are real, and if we call them m^ and m2 we can say that the solution of the dif- ferential equation is

(116)

where A and B are constants determined by the initial circum- stances of the discharge, and w: and m.2 are equal respectively

to-" + A? -6 and--- /--b. This solution for the

2V 4 'J V 4

value of q is called an exponential solution, and it indicates that under these circumstances when the inductance, resistance

378 DYNAMICAL THEOEY OF INDUCTION.

and capacity are of such magnitudes that E is greater than / __ , the quantity q dies away regularly, diminishing with

the time in a continuous manner. In this case the discharge of the jar is always in one direction, and the current or rate of

decay ( - — ) of the charge is also always in one direction. \ d t/

If, however, E is less than / — , then (^.-6) is a nega- f\J C \4 /

tive quantity, and the square root of it is an imaginary one, and the roots of the quadratic m'2 + am + b = Q are unreal. It is shown in treatises on algebra that a quadratic equation has either two real or two imaginary roots, and when this last is the case the roots of the quadratic can always be expressed in the form a + /3 V - 1.

Accordingly, the solution of the original equation (115) under these circumstances is of the form

a-^i*. . (117)

By a simple transformation, based on the employment of the exponential values of the sine and cosine, as given on page 106, this solution can be thrown into the form

q = eat(Pcoapt + Plsmpt) . . . (118) where P and P1 are constants, and

a = -«=--?., and/?- lb-?L= /I 2 2L V 4 YLC

The general result is then that the equation

L(V

has two solutions — one, called the dead beat case which applies

when E is greater than . /. — , and is of an exponential form,

C 

and indicates that the charge q dies away regularly with lapse of time, and the discharge current is uni-directional ; the other, called the oscillatory case, which applies when E is less

than */ — > contains sine and cosine terms, and indicates a periodically changing discharge decreasing by a series of

DYNAMICAL THEORY OF INDUCTION. 379

oscillations, in which case the charge on each plate of the condenser is first positive and then negative, but at the same time always decreasing; or, in other words, is a periodic variation superimposed on a steadily decreasing variation, the currents or rates of discharge following the same distinction. These two modes of discharge, or

Time

Curve representing the Discharge of a Condenser through a large .Resistance. Discharge Uni-directional and Continuous.

Curve representing the Discharge of a Condenser through a Small Resistance.

Discharge is Periodic and Alternate. Maxima gradually diminishing in Geometric

Progression.

FIG. 137.

solutions of the differential equation, are best indicated graphically by the two curves in Fig. 137, in which the upper curve represents the gradual decrease, according to an exponential law, which is indicated as the proper solution of the equation, when the value of B or the resistance of the

resistance E is less than */ -£. When B has such a

380 DYNAMICAL THEORY OF INDUCTION.

discharging circuit is greater than */ — r, and the lower one

V G

the oscillatory discharge, which is indicated by the trigono- metrical solution of the differential equation, when the

'4L __ C"

value that R=/-— » the discharge is just non-oscillatory.

' G

We find, then, that according to Lord Kelvin, analysis indicates that for a certain relation between the resistance and inductance of the discharge circuit and of the capacity of the jar the discharge is a simple current in one direction or an oscillatory but decreasing current, according as E

is greater or less than */ — . If the discharge is oscillatory,

then the electrical oscillations are isochronous, and the periodic time of a complete oscillation is

rp_ 2?T

V LC~H7 for in the second solution (118),

q = ea *(P cos (3 1 + Q sin £ t), we see that at intervals of time equal to ^ the sine and cosine

terms have the same values, since sin£« = sm/3u + 5Y and

the same for the cosine. Hence, the trigonometrical factor in the value for q periodically repeats itself in value at intervals

of time equal to - , and is zero at times when tan y8 t = - -.

ft Q

Hence the complete periodic time of the oscillation is

ft and the frequency of the oscillations, or number in one second,

18 t£I.\J f-y.-* T¥V

DYNAMICAL THEORY OF INDUCTION. 381

Accordingly, when E= /y/_ there are no oscillations in one

second, or the motion is just non-oscillatory, or dead beat. In the case of the uni-directional discharge the values of the in- stantaneous current in the discharge circuit can be represented as we have seen by the ordinates of an exponential curve, and in the case of the oscillatory discharge by those of a periodic curve whose successive maxima descend in geometric progression as the time increases in arithmetic progression. During equal intervals of time the whole quantities which pass decrease also in geometric progression, and the zero points, or instants of reversals of sign of current, are uniformly separated.

The foregoing predictions of analysis have been confirmed by the experiments of Feddersen, Paalzow, Bernstein, Blaserna, Helmholtz, Schiller and Eood. Lord Kelvin in his original Paper pointed out and suggested the application of Wheat- stone's mirror in the examination of the discharge. In Feddersen's experiments the spark from a Leyden jar battery was taken between two brass balls placed in front of a revolv- ing mirror. The discharge was passed through a high resist- ance. The image of the spark was viewed by a telescope. Under these circumstances the image of the spark was drawn out when the mirror revolved into a continuous band of light in a direction perpendicular to that of the discharge.* When the resistance was gradually reduced a point was reached at which the image was broken up into a series of separated strips, each strip corresponding to a discharge. This showed that the discharge was intermittent.

In Paalzow's experiments a similar discharge from a Leyden battery was passed through a resistance coil and through a vacuum tube, and the image of the discharge in the vacuum tube viewed in a revolving mirror. As before, with a small resistance the image consisted of a number of separate images, each of which corresponded to a discharge, and a bluish light showed itself at both poles of the vacuum tube. When the

• An experimental research of a very complete character on the duration and nature of the discharge of a Leyden jar is described by Prof. Ogden Rood in the American Journal of Science and Arts for September, 1869 ; January, 1871; September, 1871 ; October, 1872; November, 1872; March.

382 DYNAMICAL THEORY OF INDUCTION.

resistance was increased the bluish light showed itself only at one pole. In the former case a magnet held outside the tube split the discharge into tico lines of light, showing that it con- sisted of currents travelling in both directions ; but in the last case the magnet did not divide the discharge. This sufficiently indicated that with a low resistance the discharge was oscillatory and alternate, and not uniform or uni- directional.

Feddersen found that the critical resistance at which the discharge just becomes oscillatory varies inversely as the square root of the capacity of the battery, which is in agreement with the predictions of theory.

A good account of the researches of these experimentalists is given in Wiedemann's Galvanismus, Part II, § 800, et seq*

We can cast the expressions for the charge at any instant left in the condenser into more convenient forms. First, consider the dead-beat case (equation 116) is

where ml and m.2 are the real roots of the quadratic equation

• For the sake of readers wishing to pursue the subject we give here a few references, to original Papers, in which are included some collected by Mr. Tunzelraann in a series of articles on Electrical Oscillations in The Electrician of September 14, 1 888, and succeeding numbers.

Feddersen, PoggendorfFs Annalen, Vol. CIIL, p. 69, 1858 ; Vol. CVIIL, p. 497, 1859; Vol. CXIL, p. 452, 1861 ; Vol. CXIIL, p. 437, 1861 ; Vol. CXV., p. 336, 1862 ; Vol. CXVI., p. 132, 1862.

Paalzow, Pogg. Ann., Vol. CXIL, p. 537, 1861 ; Vol. CXVIIL, p. 178, 1863.

Bernstein, Pogg. Ann., Vol. CXLII., p. 54, 1871.

Heltuholtz, Monatsberichte der Berl. Akad., 1874.

Kirchoff Gesammdte Abhandlungen, p. 168, containing remarks and criticisms of Feddersen's results.

Von Oettingen, Pogg. Ann., Vol. CXV., p. 115, 1862; also Jubelbaud, p. 269, 1874.

L. Lorenz, Wiedemann's Annalen, Vol. VII., p. 161, 1879.

Schiller, Pogg. Ann. Vol. CLIL, p. 535, 1872.

Mouton, These, Paris, 1876, Journal de Physique, Vol. VI., pp. 5 and 46, 1876.

Kolacek, Beiblatter en Wiedemann's Annalen, VoL VII., p. 541, 1883.

Oleareky, Verhandlungen der Academic von Kralcau, Vol. VII., p. 141, 1882.

Oberbeck, Wiedemann's Annalen, Vol. XVII., pp. 816—1,040, 1882; Vol. XIX., pp. 213 and 265, 1883.

Bichat et Blondlot, Comptes Rcndus, VoL XCIV., p. 1,590, 1882.

DYNAMICAL THEORY OF INDUCTION. 383

and as a = - and 6 = __, we have m, = - _ + /_?! _. l

L CL' 2L V4L2 CL'

which we will write as - a + ft, and similarly, m2 is -a- f3.

The constants A and B are determined by the condition that when t = 0 the charge q is the original charge Q ;

hence Q = A + B, (119)

and since the current i at any instant is the rate of loss of charge, or - _?, we have i = - — - = - A m^ e mi i - B m2 emz t,

when t = 0, i = Q.

Hence A.m1 + 'Bmt = 0 (120)

From these two equations (119) and (120) A and B are deter- mined in terms of ml and m2, or of a and (3, and we find

Let the quantity be called T1 and let . be called

a — p a + p

T2, then it is easily seen that A = TI Q, and B = - T^ Q,

and the equation for q may be written

The ratio of the potential v of the condenser at any instant to its original potential V is the same as that of q to Q.

The two quantities Tx and T2 are such that their sum is equal to C E and their product to C L — statements easily verified by taking the values of T! and T2 in terms of a and /?,

T? / T?2 i

and recollecting that a stands for — — , and ft for * / __ - _.--.

A Lt v 4 LJ (J Jj

Hence also the current * at any instant is given by the equation

(122)

These two equations (121) and (122) contain the complete solution of the discharge in the dead-beat case, giving the current, potential and quantity at any instant reckoned from the moment of closing the circuit of the condenser.

384 DYNAMICAL THEORY OF INDUCTION.

Suppose that the discharging circuit possesses no inductance, then L = 0, and the equation reduces to

In the above expression the product E C, or the product of the resistance of the discharging circuit and the capacity of conden- ser, is a quantity of the dimensions of a time, and is called the time constant of the condenser. It represents the time in which

the charge of the condenser falls to -th part of its original value (« being 2-71828). Let E C be denoted by T. Then if we begin with a charge Q, in a time T the charge left is — . In a time

2 T it is 5, and in a time n T it is 9. Now, since (? = (2-71828)3,

6 6

or nearly 20, and e4 is nearly 54, it follows that in time 7 T only one-thousandth of the original charge remains, and in a time 21 T only one thousand millionth ; so that in a period of time equal to 5 or 6 times the length of the time constant the condenser is practically discharged. If the discharging circuit possesses inductance then in the dead-beat case there are two time constants of unequal importance. These are the quantities we have called Tx and T2 above. T: is the larger of the two. The rapidity of decay of the charge with an inductive dis- charger depends chiefly on T:. For if we refer again to equa- tion 121, we see that q will become zero when the quantity

-L -t

in the bracket, viz., the function {T^ Ti-T2e 1's}, becomes

zero.

Starting with given values of Tj and T2 depending on the values of L, C, and E, and knowing that T! is greater than T2, the function starts with a value equal to Tx - T2 when t = 0, and as t increases without limit both exponentials tail away down to zero ; but since T: is greater than T2, the first expo-

_ t nential,viz., e TJ, is longer getting down to practical zero

_^

than the other. Hence, the evanescence of e TI practically determines the time of discharge of the condenser, and we may call T: the principal time constant of the system.

DYNAMICAL THEORY OF INDUCTION.

385

If we call the expression __ 2 A, then bearing in mind that

1

^

where a = — anc

/E^J V 4r,a ni.'

we can express Tt and T2 in terms of A and C E or T, and we have by simple substitution

T,.. 2TA

and

1- Vx-4A' 2TA

l+Vl^4A' and the product Tx T2= T2 A.

Hence, if a horizontal line is taken, on which the values of A are set off (see Fig. 138), and values for Tt and Ta plotted

FIG. 138.

off vertically, the locus of the extremities of these ordinates is a parabola. In the figure, lengths along 0 1 represent values of A, and the corresponding values of T: and T2 define a parabola P M 0, such that 0 P = T = C E, and the ordinates of the upper portion P M of the curve are the values of Tv and those of 0 M are those of T2. The value of A = £ is the abscissa

0 A, for which T1 = T2, for when

then (3 = 0, and in

cc

386 DYNAMICAL THEOEY OF INDUCTION.

this case T1 = T2, and Tl has its minimum value. For this particular value of A, which is just the value when the dis- charge ceases to be dead-beat, and becomes oscillatory — that

is, when - = TTT or -TT^> = \ — ^ne ^me constants have equal 4 L2 C L C K

values, and TJ. becomes a minimum. Hence, for this particular value of the inductance the time of discharge of the condenser is a minimum, and less, therefore, than the time of discharge when the discharge circuit has no inductance.*

Turning next to the case when the inductance of the dig-

charge circuit is such that A is greater than £, or when _±L. is

^ 4L2

less than - , we have to consider the periodic function which CL

then applies.

Referring to equation 118 for the value of q in terms of t we have = e

where a = - _^ as before, but /3 now stands for / __ - J*L From the conditions that q = Q when t = 0, and that when

t = 0, t = ^f = 0, we find that P = Q and P: = Q |. Hence, q = Qe~>*< |cos/3« +

— ^- On the convention that y is such an angle that

we can write the above expression

sin 7

and t = l?= Q e

dt /2LC

Hence, we see that the expression for the currents and for the remanent quantity of electricity at any time t consists of a periodic part, which is a sine function, and a decreasing part,

• This appears to have been first noticed by Dr. W. E. Sumpner (Phil. Mag., June, 1&77), and discussed by Prof. Oliver Lodge in an interesting paper in The Electrician for May 18, 1888, p. 59, from which article some portion of the above paragraph and figures have been taken.

DYNAMICAL THEORY OF INDUCTION. 387

which is an exponential function, and that the rate of decay of the maxima of the waves is determined by the value of

•p Q T

__; in other words, _ is the time constant for the oscillatory form of discharge.

This is expressible as 2 T A. in our notation, and is, hence, simply proportional to A. In Fig. 138 the variation of the time constant T3, or _ , for oscillatory discharge is represented

by the straight line M Q.

The really important part of the time constant curve is the part P M Q, consisting of a bit of a parabola and a straight line, and having a minimum ordinate corresponding to A = ^.

The current at different times for the two cases A = 0 and X = % are plotted in Figs. 139 and 140.

For A = I r we have T3 = T, since T3 = 2 T A. In other words,

the time of discharge of the condenser when 2 = | is the

C B

same as when L = 0, and just double that when A = ^ ; and in this last case the rate of discharge is a maximum. Hence, so far from reducing the rate of discharge, a little self-induction in the discharge circuit is a positive help to the condenser in getting rid of its charge. Dr. Sumpner* has pointed out that since a lightning discharge resembles that of a condenser, a little inductance in a lightning rod may assist matters instead of blocking the way of the discharge.

A pendulum swinging in treacle was long ago suggested by Lord Eayleigh as a mechanical analogue to the Leyden jar dis- charge. Dr. Lodgef has pointed out that we may make the analogy exact by considering a loaded spring bent aside or compressed in a resisting medium in such way that gravity is not concerned in the motion and then let go.

The pliability of the spring corresponds to the capacity of the condenser, its displacement to the electric charge. The load or inertia corresponds to the self-induction of the circuit ; the viscosity of the fluid to its resistance. If the viscosity friction be supposed to vary accurately as the speed, then the equation of motion is

m^-Rv = Rx,

dt '

» Loc. cit. t See The Electrician, May 18, 1888, p. 41.

Od2

388 DYNAMICAL THEORY OF INDUCTION.

where x is the displacement and v the velocity = - —. Writing

i ilt

L for m, and _ for C, and x for Q, we have the condenser

\

FIG. 139.

Curve I. represents the strength of the discharge current of a condenser in a circuit of no self-induction. T0=S R. This curve corresponds to the point P in Fig. 138.

Curve II. represents the strength of the discharge current ot the same condenser in a circuit of the same resistance, but with self-induction enough just to bring the discharge to the verge of oscillation, this being the condition which effects complete discharge in the shortest time possible. This curve corresponds to the point M in

Charge in Jar

o ±TO TO i zero 2 TO Time

FIG. 140.

Curve I. shows the charge remaining in the jar at any time, the circuit being practically devoid of self-induction.

Curve II. shows the same thing for L=J S R2 — that is, for the quickest discharge possible. At first Curve I has the advantage, but at a time 1-26 K 8 the second curve overtakes it and discharges the jar more rapidly.

equation (115); the two are seen to be the same, and every- thing we have said of the electrical problem applies to the mechanical one.

DYNAMICAL THEORY OF INDUCTION. 389

It is obvious mechanically that if the resistance is moderate and the mass considerable, the recoil of the spring will be accompanied by oscillations, and that with great resistance and small inertia the motion will be a slow sliding back without oscillation ; and there must exist between the strength of the spring, the mass of its load, and the viscosity resistance of the medium some definite relation which shall constrain the recoil to be dead beat, just returning to the original position of equilibrium without overshooting the mark. This relation is now seen to be

and under these circumstances the recovery of the spring is effected in the shortest possible time.

In addition to the experimental researches of Blaserna, to which reference has been made at page 246 et seq., very extensive experiments have been made by Bernstein* and by Moutonf on the subject of electrical oscillations in the case of induced currents. Bernstein's experiments were made with a revolving wheel interrupter, which closed a primary circuit, and for a very short time, at a determinable period after the closure of the primary, put the secondary circuit in series with a delicate ballistic galvanometer. In this way the state of the secondary circuit could be investigated at various instants of time after closing or opening the primary circuit, and the general results of Blaserna were confirmed. In Mouton's experiments a rather different form of commutator (see Jamin's " Cours de Physique," Vol. IV., p. 201, third edition) was employed to break a, primary circuit and to examine with a quadrant electrometer the electrical state of the terminals of an open secondary circuit at various instants afterwards. Mouton found that a potential difference declared itself at less than one four-mil- lionth of a second after rupture of the primary, and that this potential difference died away with decreasing amplitude by rapidly reversing sign, thus indicating the existence of electrical oscillations set up in the open secondary circuit. The duration of the first semi-oscillation was greater than that of succeeding ones. In the case of a secondary circuit of 13,860 turns he

• Pogg. Ann., Vol. CXLIL, p. 54, 1871.

t "Etude Experimental BUT lea Phdnom&nea d'Induction Electrodyna- mique." The'se de Doctorat, 1876.

390 DYNAMICAL THEORY OF INDUCTION.

found that the first semi-oscillation had a duration of 110 millionths of a second, and the succeeding ones about 77 millionths of a second, and he was able to count about 30 complete oscillations.

§ 7. The Function of the Condenser in an Induction Coil. —

Fizeau 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 Ley den 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 Kayleigh (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 Rendus, Vol. XXXVL, p. 418, 1853.

DYNAMICAL THEORY OF INDUCTION. 391

by the same writer (PJiil. 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 Eayleigh) 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 q' the quantities of elec- tricity which have flowed through these circuits from the instant of beginning to reckon the time t,

then <*_«-,• and ^-f;

dt dt

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^'+^O. . . . (123) dt dt (j

M^ + N^l' =0. ... (124)

dt dt

392 DYNAMICAL THEORY OF INDUCTION.

Eliminating i' we have

(125) may be written

A differential equation of this type always indicates an oscillatory motion. For, consider the simple periodic function

x = A sin p t, where p = _^, T being the periodic time of the motion, we have _5=^ A cos p t, and — * = - p* A sin p t ;

(It (i t

hence, — 3^ + pzx = Q, and therefore x = A sin pt is a particular

solution of this equation.

In the above differential equation p is seen to be £TT times the frequency of the oscillation.

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

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

Mi + N t" = 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 i' 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

./ JV1 /T «\

.=-(!-*),

so that when, after half an oscillation of the primary, i becomes equal to - 1, we have

i' = 2I (128)

DYNAMICAL THEORY OF INDUCTION. 393

This equation gives us the initial value of the secondary current 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 236, 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

assumption of a perfectly sudden break is equal to — I,