The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 22 of 36

S72 l)YirAMlCAL THEORY OP INDUCTION.

of these values in equation (111) above, we see that the square of the velocity of propagation of the vector potential is

^., _ rf t^ K 1

dx

or

V^' ("^)

that is, the velocity of propagation of the magnetic force is the square root of the reciprocal of the product of the magnetic and electrostatic inductive constants of the medium. We have above proved that the ratio of the electro-magnetic to the electrostatic units of electric current is expressed by the same quantity, and indicated that accurate experiment shows this ratio to be numerically the same as the velocity of light.

Hence, the velocity of an electro-magnetic disturbance or magnetic force is the same as the velocity of light, and the conclusion is urged upon us with great force that the medium concerned in both phenomena is the same.

§ 6. Electrical Oscillations. — A survey of the phenomena of electric current induction would be very incomplete if it did not contain some reference to the subject of electrical oscilla- tions. Becent researches have endowed this department of electrical investigation with fresh interest. We proceed to consider the manner in which electrical oscillations may arise. If a material body is subjected to elastic constraint, and is dis- turbed from a position of equilibrium, it returns when set free to its original position. If that body is endowed with masSf and hence possesses the quality of inertia, its motion of return to its position of equilibrium will, under certain circumstances, carry it beyond that point and set up oscillations^ which decay graduaUy away. Two illustrations of this readily present themselves, one a mechanical and the other a pneumatical example. The first case is that of a pendulum or straight spring. Let this pendulum or spring be deflected from its position or condition of equilibrium and held in constr&int. Next let it be set free — the elastic or restoring forces urge it back again to its first position. In virtue of its mass it will acquire

DYNAMICAL THEOBY OF INDUCTION, 073

a certain momentum, and on reaching the position of equili- brium this momentum may carry it past this point, and the acquired kinetic energy will then be expended in making a displacement against the elastic forces. If there is nothing of the nature of friction present to fritter away the work expended on the body in making the first displacement, then the energy would remain associated with it for ever, being alternately potential and kinetic, and the oscillations continue with undi- minished amplitude. If the spring or pendulum vibrates in a viscous fluid, then a frictional retardation will be experienced, and in so far as this is present the energy is gradually dissi- pated, and the oscillations decay away, becoming gradually less and less in amplitude. It may so happen that the work done against frictional resistance during the first quarter of a com- plete oscillation in starting to return from the position of greatest displacement is just equal to the work done in origin- ally making the displacement. When this is the case the whole energy is dissipated by the time the deflected or dis- placed body reaches its original position of rest, and there are then no oscillations. Accordingly a pendulum or spring may be set in a viscous fluid of such a kind that the frictional resistance is just sufficient to secure that when the body is disturbed and then set free it returns to its original position without ever passing it ; in other words, there are no oscillations. Another illustration of oscillatory and non- oscillatory establishment of equilibrium is as follows : Sup- pose there be two large vessels, or reservoirs, connected by a pipe, closed or able to be closed in the middle by a stop- cock. Let one of these vessels, A, be exhausted of its air, and let the other, B, have air in it at the atmospheric, or a greater than the atmospheric pressure. First, let the connecting pipe be supposed to be long and narrow ; on opening the stopcock air will rush over from B into A, and the flow of air will con- tinue uniformly in the pipe in one direction until the pressure in A and B is equalised. Second, let the connecting pipe be very short and large, so that little tubular friction is offered to the flow of air. Under these circumstances the result of open- ing the tap would be that a rush of air would take place, which would be succeeded by a series of oscillations of the air in the tube. The air, in i&ct, rebounds from side to side, and the

374 DYNAMICAL THEORY OF INDUCTION.

equilibrium is only finally established after a series of graduallj diminishing oscillations or backward or forward currents of air in the tube. This establishment of equilibrium or pressure by oscillatory movement takes place when the resistance to the flow is small. That this is no fanciful description is proved by the experience of MM. Clement and Disomies in their experiments to determine the ratio of the specific heats of gases. In these experiments a large glass vessel had a partial vacuum made in it. A stopcock was then quickly opened and closed, and the pressure of the air determined after a short time. These experiments were repeated by MM. Gay Lussac and Welter. See Journal de Physique, LXXXIX., 1819, 428, and Ann. de Oh. et de Fhija. [1], XIX., 1821, 436.

M. Cazin {Ann. de Ch. et de Phya. [3] LXVI., 1862, 206) first pointed out a source of error which resulted from these air oscillations, and showed that the final pressure depended upon the phase of the oscillation at which the stopcock is closed.

These examples are sufficient to indicate that when a material system of bodies having inertia is displaced against elastic forces which compel it to return, if free, to a definite position, whilst at the same time its motion is resisted by actions of the nature of frictional resistance which dissipate its energy, we have a resulting motion which may be oscillatory or non-oscillatory, according to the relation of the constants of the system. Under certain conditions as to mass, or inertia and friction, we have oscillations dying gradually away. Under other conditions we have a gradual return to the original position without ever passing it. The motion is then said to be perfectly dead-beat. We shall investigate presently the conditions which must hold good, and the relation between the inertia factory in virtue of which the moving system possesses kinetic energy, and the resistance factor, in virtue of which the energy bestowed upon the system at its first displacement is frittered away into heat, in order that the motion may be vibratory or dead-beat.

When a condenser or Leyden jar is discharged through a conductor, the potential energy runs down in the form of an electric current. In this case we have a similar state of things to that existing when a bent spring is released. This trans- formation of the potential energy may take place either by a vibratory current, that is, by a series of electrical oscillations— or

DYNAMICAL THEORY OF INDUCTION. 375

by a uni-directional discharge. It is highly probable that Prof. Joseph Henry, as far back as 1842, was the first to recogoise 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 &cts are shown to be in accordance with this hypothesis, and a ready explanation is a£forded 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 WoUaston 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 Sk Pac j William Thomson) in the June number of the Philosophical Magazine for 1863, 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 2he EUetrician 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 Encydopcsdia BriUanica, Ninth Edition.

37C DYNAMICAL THEORY OF INDUCTION.

Lord Eelvin then called it, ** electro-dynamio 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 B the ohmic resistance, and L the constant inductance of the discharging conductor ; and if q is the electric quantity in the jar, and v the potential difference of its coatings at any instant t, then by the definition of electric capacity we haye

3 = Ci;,

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

is equal by Ohm's law to --. By the principle of conservation K

of energy the rate at which elecbro-magnetic energy is being

taken up.by the conductor, viz., — (J L i^), together with the

a t

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

or -P'l^-Liii + Rt-^;

Cdt dt

but i = '~^, 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

dt^ dt ^

In order to solve this equation we may proceed as follows : The charge q 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 9= A 6*^, 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«*"'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 q=Ae^j 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 e^ 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 m^+am+l=0. The two roots of this last quadratic equation are obtained by a simple solution, and they are

Two cases then sxise, first, when — is greater than 6— that

4 T>4 1 R* 1

is, when -fL-. is greater than -—• , or -— greater than -. In 4 Li Jj C 4 Jj O

this case the roots of the quadratic are realy and if we call them till and m^ we can say that the solution of the dif- ferential equation is ;>t t- >*''^ ^

j= A 6'«i* + B «"»=»' (116)

where A and B are constants determined by the initial circum- stances of the discharge, and m^ and m, are equal respectively

to--+ /^-6and--- /l'-6. This solution for the

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

378 DYNAMICAL THEORY OF INDUCTION.

and capacity are of such magnitudes that B 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 (-;t^) of the charge is also always in one direction.

If, however, B is less than /-p-' ^®^ (t"") ^^ * ^®8*"

tive quantity, and the square root of it is an imaginary one, and the roots of the quadratic m' + am + 6 = 0 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 + p V^,

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

j«A,(a+^>A=T)t + Bf(«-^^^«. . (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^^' (T COS p t + T^ sin P t) . . . (118) where P and P^ are constants, and

2 2L' ^ V 4 VI'C 4L« The general rcsaU is then that the equation

dt^ hdt LC^ has two solutions — one, called the dead beat case which applies

when B is greater than / ~rr* ^^^ ^^ ^^ ^^ exponential form,

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 B is less

than A / — -., contains sine and cosine terms, and indicates a

periodically changing discharge decreasing by a series of

DYNAMICAL THEORY OF INDUCTION.

879

osoiUationSy 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

dure repretentinsr the Discharge of n Condenier throagh a Large Discharge Uni-directional and Continaoaa.

Carre representing the Discharge of a Condenser throngh a SmaU Resistance.

Discharge is Periodic and Alternate. Maxima gradually dlminiihlng in Geometric

ProgressloD.

FiQ. 137.

solutions of the differential equation, are best indicated graphically by the two curyes in Fig. 187, 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 tho

380 DYNAMICAL THEORY OF INDUCTION.

discharging circuit is greater than tj -^^ and the lower one

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

resistance B is less than /-^* When B has such a

value that B= w -tt' ^^ discharge is just non-oscillatory.

We find, then, that according to Lord Kelvin, analysis

indicates that for a certain relation between the resistsjice

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 B

/4l7 is greater or less than \j ~^- If the discharge is oscillatory,

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

27r

T =

for in the second solution (118),

2 = 6'*'^(Pcos^t + Qsin^«), we see that at intervals of time equal to ^ the sine and cosine

terms have the same values, since sin)8« = sin^(« + ^],and

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

tr P

of time equal to -^ , and is zero at times when tan ^ t = - - . Hence the complete periodic time of the oscillation is

2- or ^2^

/J_ Jl Vlc"4L'

and the frequency of the oscillations, or number in one second, is 1 /-\—W

DYNAMICAL THEOJiT OP INDUCTION. 381

/4L Accordingly, when B«= ,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 he 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, Blasema, Helmholtz, Schiller and Bood. Lord Kelvin in his original Paper pointed out and suggested the application of Wheat- stone's mirror in the examination of the discharge. In Feddersens 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 hght 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 Hood 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 two lines of light, showing that it con- sisted of currents travelling in both directions ; bat in the last case the magnet did not divide the discharge. This sufficientlj 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 fonus. Fint, consider the dead-beat case (equation 116) is

where m^ and m^ are the real roots of the quadratic equation m^ + a7n + b=0;

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

Feddersen, PoggendorflTs Annalen, Vol. CIIL, p. 69, 1868 ; VoL CVIII., p. 497, 1869; Vol. CXIL, p. 462, 1861 ; VoL CXIII., p. 437, 1861 ; Vol. CXV., p. 336, 1862 ; Vol. CXVI., p. 132, 1862.

Paalzow, Pogg. Ann., Vol. CXII., p. 537, 1861 ; Vol. CXVm., p. 178, 1863.

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

Helmholtz, Monatiherichte der Berl. Akad,, 1874.

Kirchoff Oetammelte Ahhandlungai, 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, Th^se, Paris, 1876, Journal de Physique, Vol. VI., pp. 5 and 46, 1876.

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

Olearsky, Verhandlungen der Academic von KraJcau, VoL VII., p. 141, 1882.

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

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

DYNAMICAL TREOttY OF INDUCTION. 383

andasa = =- and 6 = -—, we have wi,- - — + . /— - - —

which we will write as -a + p, and similarly, vi^ is - a - )8.

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 t at any instant is the rate of loss of

charge, or--^, we have t--— ?« - Am, «*"!<- Bw,«»»»2<|» dt dt

^en£»0, t-=0.

Hence Awi + Bw^ = 0 (120)

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

B=-^^ Q. 2)8 ^

Let the quantity ^ be called T, and let be called

T T

T2, then it is easily seen that A = - — L-- Q, and B = - "^2 Q,

Tj— Tj -li-l-j

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 ^ to Q.

The two quantities Ti and T, are such that their sum is equal to 0 R and their product to CL — statements easily verified by taking the values of T^ and T, in terms of a and 0,

and recollecting that a stands for ---:-, and fi for a / — —^ - 7^ ^.

2 L ^ 4 L* C L

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

^"t^J^"^'"''""*^} • • • • (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 R 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

{e being 2-71828). Let R 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 % and in a time n T it is 9. Now, since ^ = (2-71828)',

or nearly 20, and ^ 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 T^ and T, above. Ti is the larger of the two. The rapidity of decay of the charge with an inductive dis- charger depends chiefly on Tj. For if we refer again to equa- tion 121, we see that q will become zero when the quantity

-i - *

in the bracket, viz., the function {Tic 'A-Ta^'Ta}, becomes

zero.

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

^ t nential,viz., e '% is longer getting down to practical zero

than the other. Hence, the evanescence of e '^i practically determines the time of discharge of the condenser, and we may call Ti the principal time constant of the system.

DYNAMICAL THEORY OF INDUCTION. L

385

If we call the expression --— X, then bearing in mind that T1--I- and T> = -i-; where a = -^ and /?= . /^ - Jl.,

we can express Tj and Ta in terms of A and C R or T, and we have by simple substitution

and

T,'

1- >/i-4A' 2TA.

1 + n/I ^^TT and the product Ti Ta= T* A..

Hence, if a horizontal line is taken, on which the values of A are set off {see Fig. 188), and values for T^ and T^ plotted

X-Snr,

Fio. 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 T, define a parabola P M 0, such that 0 P = T = C R, and the ordinates of the upper portion P M of the curve are the values of Tj, and those of 0 M are those of T^. The value of A == ^ is the abscissa

O A, for which T^^T,, for when ^» = i then )8 = 0, and in

0B«'

00

886 DYNAMICAL THEORY OF INDUCTION.

this case T^aT,, and T^ 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

"p2 1 T

is, when _- — -. or <j^-= J — the time constants have equal

values, and T^ 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 dis-

charge circuit is such that X is greater than ^, or when — ~. is

1 ^^

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

then applies. Referring to equation 118 for the value of ^ in terms of t we

have ^ = 6**(Pcosi8t + P^sin)8t),

R / 1 -pa

where a = - -4- as before, but B now stands for / — - Jz^. 2L '^ VOL 41?

From ihe conditions that q^Q when t=0, and that when t=0, t = ^ = 0, we find that P = Q and F=Q g.

Hence, q^Qe »^ -(cos)8t + --=!- sin)8^|.

On the convention that y is such an angle that

^ -g-»

we can write the above expression

-»^|8in(^M:r)|

^ ^ V smy J

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. £. Sumpner {Phil, Mag.f June, 1&77), and discussed by Prof. Oliver Lodge in an intereatiDg paper in The EUctrician for May 18, 1888, p. 39, from which artioto aoma portion of the above paragFaph and figurea have been takeo.

DYNAMICAL THEORY OF INDUGTION. 887

which is an exponential function, and that the rate of decay of the maxima of the waves is determined hy the valae of

-— ; 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. 188 the variation of the

time constant Tg, 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 k^O and A = i are plotted in Figs. 189 and 140.

For A = J we have Tg = T, smce Tg = 2 T A. In other words,

the time of discharge of the condenser when --^3 = J is the

C R 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 Bayleigh as a mechanical analogue to the Leyden jar dis- charge. Dr. Lodget 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 pliabihty 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

wi — — R V = R a?,

dt, [

• Xoo. eU. t See 2%« Electrieian, May 18, 1888, p. 41.

002

888 DYNAMICAL THEOET OF INDUCTION.

dx

where a; is the displaoement and v the velocity ^ -—. Writing

•t dt

L for fill and ^ for 0, and z for Q, we have the condenser

Cunre I. represents the strength of the discharge current of a condenser in a dnmlt of no self-iiidactlon. Ta=S &. Iliis cunre corresponds to the point P In Kig. 18S.

Cunre II. represents the strength of the discharge current oc the same condenser in a circuit of the same resistnnce, but with self-induction enough )n«t to bring ifae discharge to the verge of oscillation, this being the condition which eftecta complete discharge In the shortest time possible. This curve corresponds to the point H in

Charge in Jar

tTo

Fio. 140.

Tim%

Curve I. shows the charge remaining in the Jar at any time, the drcnlt being practically devoid of selMiiductlon.

Curve II. shows the same thing for Ls J 8 RS— that Is, for the quickest discharge possible. At first Curve I has the advantage, but at a time 1*26118 the eeooud curve overtalccs 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. 889

It is obvious mechanioally 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

E2 = 4Rw,

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

In addition to the experimental researches of Blasema, 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 baUistic 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 Blasema 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-milr Month 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 18,860 turns he

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

t "£tude Experimental rar les Phdnom&nes d'Induotion Electrodyna- mique." Th^sa de Doctorat, 1876.

390 DYNAMICAL THEORY OF INDUCTION.

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