The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 7 of 35

The converse of the above proposition is also true. Let there be any periodic current-generating machine producing in a circuit an electromotive force, and therefore a current varying periodically according to any law. This kind or form of current could be exactly imitated by removing the given machine and substituting a series of sine inductors coupled in series and arranged so as to each produce a simple sine varying E.M.F., the respective sine currents having different phases and amplitudes, but being superimposed upon one another. That is to say, however complicated may be the nature of the periodic current which traverses a circuit, provided the same electric motions are repeated at regular intervals, we may build up this current by suitably super- imposing in the same circuit a number of simple periodic currents of certain amplitude and wave-lengths and fixed difference of phase.

The above remarks may be taken as an outline of the analysis of any single-valued continuous function into a series of simple harmonic functions. To simplify language, we shall in future speak of a curve whose equation is of the form y = A sin z as a simple periodic curve, and if such curve graphically represents the continuous variation of the flow of electricity past any section of a conductor, or the fluctuation of electromotive force in any circuit, we shall speak of such as a simple periodic current or a simple periodic E.M.F.

Any other mode of variation of these quantities which, graphically represented, would be a single-valued curve repeating the same form, will be spoken of as a complex periodic curve, current, or E.M.F., and, by the foregoing analysis, a complex periodic function can be analysed into a sum of simple periodic functions.

§ 7. Description of a Simple Periodic Curve.— The follow- ing method affords a very easy means of drawing a simple periodic curve. Take a cylinder or tube of pasteboard (see Fig. 39) and cut it through obliquely with a sharp knife, taking care to make the cut in one plane. The section of this cylindrical tube by an oblique plane will be an ellipse. Slit

SIMPLE PERIODIC CURRENTS.

97

the tube open along the line A B and unfold it. Lay it down on another sheet of paper and draw a pencil line guided by the curved edge A Y Y'. Draw a dotted line, X X', so that its vertical distance below the highest point Y on the curve is equal to its vertical distance above the points A and Y',

FIG. 39.

or make OY equal to AX. Then move this cardboard template forward through a distance equal to its own width, and draw another piece of curve, repeating the first, and similarly placed (see Fig. 40).

The resulting curve is a simple periodic or simple sine curve.* The distance X X', equal to the circumference of the tube or to the width of the template, is the wave length. The distance 0 Y of the highest point above the mean line

• " Elements of Dynamics " (Clifford), p. 22.

98

SIMPLE PERIODIC CURRENTS.

is the amplitude. If the bottom edge of the template is divided into 360 parts or units, then the distance O'M, measured in such units of the foot of the perpendicular, let fall from any point P on 0' M, is the phase of the point P, measured in degrees.

It is, perhaps, more convenient to reckon the phase of the point P by the magnitude of the line A N, or the distance of the foot of the perpendicular, let fall from P on X X' from the point A, where the curve crosses the mean line. The phase of the maximum ordinate 0 Y is then 90deg.

§ 8. The Value of the Mean Ordinate of a Sine Curve. — Let Fig. 41 represent the semi-wave of a simple periodic curve ; we shall proceed to prove some geometric proper- ties of such a curve. Considering this curve as bounding an area of which the other including line is the datum line

FIG. 41.

X X', we shall first find the value of the mean or average ordinate. Let XX' be divided into equal and very small intervals, such as N N', of which the length is d x, and let X N be called x. Assume as a unit of length the radius of the cylinder of which XX' is the semi-circumference. At each of these small elements raise ordinates, such as P N, to touch the curve. We require to find the mean value of all these equi-spaced ordinates when they are infinitely close. The arithmetic mean value of a number of things is the sum of them divided by their number. If y denote the length of one such ordinate P N, and 2 y the sum of all such ordinates when ruled at n equal and exceedingly small intervals, each of length dx, then the average value of these infinitely numerous ordinates is

ndx

ndx

SIMPLE PERIODIC CURRENTS. 99

but the sum of all such quantities as ydx, or PN . N N' is the sum of all the areas of the little rectangular slips into which these infinitely numerous ordinates divide the area bounded by the curve and X X' , and n d x is the length X X' ; hence we have

mean ordinate^™, length X X'

The area X Y X' is obtained by integrating the equation to the curve. Calling the maximum ordinate 0 Y, A, and the distance X N, x, the unit being the radius of the cylinder of which X X' is the semi-circumference, we have as the equa- tion to the curve

y = A sin x,

and therefore I ydx, or A / sin x d x,

between the limits 0 and TT, is the value of the area of the curve. But

A.lsmxdx= -A cos x,

and this, between the limits x — 0 and x = TT, is equal to 2 A. On the same scale, the length

hence, the average value of the infinitely numerous and equi-

O A

spaced ordinates is — , or the average ordinate of a simple

a

sine curve is equal to - times the maximum ordinate. The value of

• = 0-6369.

—

Therefore, the average value of the equi-spaced ordinates of a simple periodic curve, or the true mean ordinate, is 0-6869 of the maximum ordinate ; and, if the current or electromotive force varies according to a simple periodic law, the true mean current or the true mean E.M.F. is 0-6369 of the maximum cun-ent or E.M.F. during the phase.

We have here made use of one simple integration, and it is generally easier to master the elements of the infinitesimal calculus than to construct or follow proofs which aim at

H2

100 SIMPLE PERIODIC CURRENTS.

avoiding its use. We shall, however, indicate how the value- of this mean ordinate may be found from first principles. If we call the length of the base line X X' I, and divide it into n equal and very small parts of length 8 x, then n8x=L Erect at each interval an ordinate whose height is y, then the equa- tion to the curve is y = A sin 'j x, where x is, as before, the

distance X N. The mean value, M, of the ordinate is the sum of all the values of the ordinates divided by their number, or

is equal to i (yl + */2 + y3 + &e.). • M=1

The sum of the sine terms in the bracket is known by trigonometry to be equal to

Hence

. TT bx smT2

which may be otherwise written — ~1&X \

,, A I 1 /n8x irSx\ .

'"aKsnv* (-IT-IT) "

When n becomes infinite and 8 x becomes zero, n 8 x remains- still equal to I ; hence the above expression in this case- reduces to the following : —

for the value of £ is 2 when h becomes zero.

sin-

SIMPLE PERIODIC CURRENTS. 101

Accordingly the mean value of the ordinates, when they are infinite in number and equi-spaced, is - times the magnitude

of the maximum ordinate.

§ 9. The Value of the Mean of the Square of the Ordinates of a Simple Periodic Curve.— We require in the next place to find the value of the mean of the square of the ordinates to the same curve, assuming them to be equi-distant and infinite in number. If yv ?/2, &c., are the ordinates, and n the number, "we require to find the value of

the value of any ordinate being, as above, y = A sin y x.

If X X' or I is divided into n intervals, each equal to B x, so that n 8 x = I, we have to find the value of

sin2 0 + sin2 ^ 8 x + sin2 -

but, sinco sin2# = -fl -cos20Y

the series in the bracket can be replaced by

A ALL A A i

for n terms. Hence the mean value M is

M = — -- -VcosO+ cos- n 2 2;t\ /

for n terms.

The cosine series forms a progression of terms which begins with unity, since cosO° = l, and passes down through zero to — 1, and then up from — 1 through zero to unity again,

for e

"when n8x — l and S x becomes infinitely small.

102 SIMPLE PERIODIC CURRENTS.

Since the angles are in arithmetic progression, we can pick out from this series pairs of cosine terms such that they are equal in magnitude but opposite in sign, and, when taken pair and pair, cancel each other out. The sum of the cosine series in the bracket is thus equal to zero, and, therefore,

(22)

that is, the mean of the values of all the ordinates squared, taken equi-distant and infinite in number, is half the square of the maximum value.

We have, therefore, this result : If the current in a linear conductor varies in strength and direction in a manner which geometrically would be represented by the ordinate of a simple

sine curve, the true mean value of the current strength is -

or 0-637 of its maximum value, and the mean value of the square of the current strength, taken at equal and very small intervals, is half the value of the square of the maximum value.

Since 1 = 0-637 and — = = 0-707, and since the difference V2

= 0-07, the true mean current is less than the square root of the mean of the squares at each instant by an amount which, is very nearly 10 per cent, of the latter.

If we proceed by the ordinary rules of the integral calculus, we can find the value of the mean of the squares of the ordi- nates of the sine curve as follows : —

Let y = A sin x

be the curve ; then

y2 = A2 sin2 x

The mean value of the square of the ordinate between tha- limits 0 and 7r-that is, during the half -wave length — is

If-

SIMPLE PERIODIC CURRENTS. 103

Therefore, M^^T (I~cos2n)dx,

Hence we reach the same result as above. In order to avoid repeating constantly the clumsy phrase the square root of the mean of the squares of all the equi-spaced ordinates of a curve, we may call this, in speaking, the mean-square value of the ordinate, and express it by the symbol ^mean2. Hence,

^/mean2 y stands for the above particular kind of mean of y.

In practice, in alternating-current work, we hardly ever require to concern ourselves with the true mean of the ordinates of a simple or complex periodic curve. Chiefly we require to know or find the square root of the mean of the squares of the ordinates of a periodic curve taken at equi-distant positions throughout the period. Hence the

V mean2 value of the ordinate of a simple periodic curve is equal to the quotient of the maximum ordinate by the Jfi for the mean of the squares of the equi-distant ordinate is

equal to the value of — , as shown above, and hence the

_ A

•J mean2 value is -T=. Since N/2 = 1-414 nearly, we see that

the maximum ordinate of a simple sine curve is VI times the -J mean2 ordinate. In the practical measurement of alter- nating currents, the value given by the instruments is nearly always the Vmean2 value of the instantaneous values throughout the period.

§ 10. Derived Curves. — Let the curve in Fig. 42 represent the complete period of a simple periodic curve of which the

equation is y = A sin — x. Let P be any point on the curve.

ThenPN = i/, OY = A, XX' = J. At P draw a tangent P T to the curve, and let it meet the datum line at T.

104

SIMPLE PERIODIC CURRENTS.

We shall call the trigonometrical tangent of the angle P T N,

P N the slope of the tangent at the point P, hence -r™=. = the slope.

If two points, PP' (Fig. 43), are taken on the curve very near together, and a secant, P' P T, is drawn through them,

FIG. 42.

this secant will become a tangent when the points P P' move

~p' ivr up into contact. The ratio of f will then, in the limit,

Jt JVJL

become the slope of the tangent. If now X N = x - — , and

X N' = x + — , and P M = N N' is 8 x, we have the equations

and

The quantity in the bracket is identically the same as 2 cos ^ x sin 5 — ;

SIMPLE PERIODIC CURRENTS.

105

vxrt,«)f±m

L I "2 -"

When 8 x is made infinitely small, the quantity in the square •brackets is unity, and we have

If we plot a curve whose ordinates at any point are the slope -of the primal curve at the corresponding points, the above

FIG. 44.

equation shows us three things — first, that it is a sine curve or simple periodic curve of the same type as the curve from which it is derived ; second, that its maximum value is

• times the maximum value of the original ; and third, that its

zero ordinate corresponds to the maximum one of the original, .and vice versa.

In Fig. 44 the firm line curve is a curve of sines

y = A sin ^ x ;

the dotted line is a curve of sines, whose ordinate QN at any point represents the slope of the tangent at P on the original curve. Accordingly, at Y, where the original curve is -at its maximum, and the slope of its tangent is zero, the

106 SIMPLE PERIODIC CURRENTS.

derived curve cuts the datum line, or has its phase shifted 90deg. backwards relatively to the original- curve. In the language of the differential calculus, the firm line curve is

the plotting of the curve y = A sin — x, and the dotted curve is

the plotting of -^ as ordinates for the same abscissae.

We may regard it from another point of view. Let the simple sine curve be supposed to be generated or marked out by a tracing point, P, which moves to and fro along a line P N P' with a simple harmonic motion, whilst the point N moves uniformly along a straight line X X'. (See Fig. 45.)

FIG. 45.

Draw as before the dotted curve whose ordinate Q N at any point represents the slope of the firm curve at the correspond- ing point P. Then the magnitude of N Q will represent the rate at which the ordinate P N is increasing or decreasing. For, in this case, distances such as XN, measured along the mean line, are proportional to time, and hence N makes a small movement forward in a small time d t ; there is a corresponding decrease in the ordinate PN, which we may

denote by dy, and accordingly c-li represents the rate of

d t

decrease of P N. If the small forward movement of N causes N to advance through a space d x, d x is proportional to d t,

as the motion is uniform, and accordingly (-V- is proportional

d x

to — ^; hence --— is at any instant graphically represented by d t at

the slope of the tangent at P — that is, by the ordinate Q N. The dotted curve represents, therefore, the ruie of change of the

SIMPLE PERIODIC CURRENTS. 107

ordinates of the firm curve at that same instant. We shall call the dotted curve the derived curve.

If the ordinates of the original curve represent the instan- taneous values of a simple periodic current flowing in a con- ductor, then the ordinates of the curve called above the derived curve will represent the rate of change of that current at the corresponding instants. The derived curve is a similar- curve, but shifted backwards by one quarter of a wave length.

§ 11. Inductance and Inductive Circuits.— Before we can proceed to discuss the laws of periodic current flow in circuits of various kinds, we must call attention to some of the funda- mental properties of electric circuits. Every electric circuit in, which a flow of electricity, whether continuous or periodic, can take place possesses three primary qualities, viz., Resistance, Inductance, and Capacity. The resistance of the circuit is a. quality of it, in virtue of which a dissipation of energy takes place when an electric current flows through it. This specific quality is affected by change of temperature and by other alterations of physical condition. In the case of pure metals it has been shown* that, if the metal could be reduced to the absolute zero of temperature, its electrical resistance would vanish.

It is generally assumed that, apart from the change due to temperature or other altered physical conditions, the electrical resistance of a body is a constant quantity, which is independent of the current flowing through it. It is evident from experience that this is approximately, even if not accurately, the case. It would require very careful and extensive experiments before we should be entitled to say that the resistance of any circuit of any metal, when all corrections have been made for change of volume and temperature, is exactly the same when a thousand amperes are flowing through it as when one-thousandth of an ampere is flowing through it. Still less can we generalise and lay it down as absolutely and universally true. Careful experi- ments made by Prof. Chrystal at the Cavendish Laboratory (B.A. Eeport, 1876) showed that the resistance of a metallic circuit of one ohm is not different for currents of one ampere and for infinitely small currents by as much as 10" 12 part.

• Dewar and Fleming, fhil. Mag., Sept., 1?93.

103 SIMPLE PERIODIC CURRENTS.

There is, therefore, a strong prohability that the specific electrical resistance of a body is a quality which is not dependent upon the current flowing through it, but is only aftected by the temperature and physical condition of the body. According to Joule's law the rate of dissipation of •energy when a current flows through a conductor is propor- tional to the square of the strength of the current. The total resistance of a circuit may, therefore, be numerically defined by the rate at which energy is dissipated by it when unit current flows in that circuit. In the practical units a circuit which, when traversed by one ampere of current, dissipates energy at the rate of one joule per second, or has a dissipation rate of one watt, is said to have a resistance of one ohni. The energy required to heat one gramme of water one degree centigrade in the neighbourhood of its maximum density is 4-2 joules. Since the rate at which energy is being dissipated at any instant in a circuit is measured by the numerical value of the product of the strength of the current flowing in it and the fall in potential down that conductor, it follows that the resistance of the circuit, or of any part of it, is also measured by the ratio between the numerical values of the fall of potential down the circuit or down that part of it and the current strength in that circuit, provided that the inductance of that circuit is negligible. The resistance of a circuit is, therefore, the energy-dissipating quality of it, and the specific resistance of any material is the resistance of one cubic unit of it between opposed faces of the cube.

In addition to the quality of resistance every circuit possesses also inductance. This quality of a circuit is one in virtue -of which a current of finite value cannot be instantaneously produced even in a circuit of negligible resistance by a finite electromotive force, and when produced cannot be instanta- neously destroyed. On account of the fact that all bodies possess mass, and therefore inertia, a finite force cannot generate a finite velocity in any material body in an in- finitely small time. We see this fact exemplified in every falling body or starting train. A time element due to inertia comes into play which causes the motion of the mass to be acquired gradually, even under the action of a constant finite forc3. Experience shows that in all electric circuits

SIMPLE PERIODIC CURRENTS. tofr

there is a physical quality present which is related to current-, and electromotive force, just as the mass of a material body is- related to velocity and dynamical force. In virtue of the mass of a body time is required for a finite moving force to generate a finite velocity, and in virtue of inductance of a circuit time is required for a finite electromotive force to generate a finite current. The inductance of the circuit bestows on it a quality which may be called its electrical mass or electrical inertia. The mass of a material body enables it in some way to become the vehicle of energy when in motion, and this energy of motion is called its kinetic energy. This kinetic energy is capable of being removed from the moving body, and the moving body can be brought to rest again only by taking away from it the kinetic energy it possesses as a whole, and transferring that energy to some other body or bodies, or to the molecules of the body itself. In like manner the inductance of a cir- cuit may be said to cause it to be capable of being the vehicle of electrical energy when traversed by an electric current. A current cannot be instantaneously produced in finite value by any electromotive force, and when produced cannot be destroyed except by transforming that energy into some other form. Hence we have a very complete dynamical analogy between material bodies set in motion by what may be called materio-motive force and the flow of electric currents in circuits which possess inductance under the action of electromotive force.

These qualities may be compared as follows : — Motion in matter corresponds to Electric flow in circuits,

Mass=m ,, Inductance=L,

Velocity =v ,, Current strength =i,

Momentum=m, „ {^^^f^ m°men"

Kinetic energy oH f Electromagnetic energy

motion = iw v2 . . . f " j = |L i2,

Rate of change oA ( Bate of change of electro-

momentum =m^ \ » 1 a**11?1?0 momentum dt) ( = Lrf*.

dt

110 SIMPLE PERIODIC CURRENTS.

The force acting on a body which is being expended in 'making change of momentum is numerically measured at any instant by the rate of change of its momentum existing ;at that instant. So also the electromotive force which is being exerted to produce change of current strength or change of electro -magnetic momentum in a circuit is measured .at any instant by the rate of change of electro-magnetic momentum.

There is an exact analogy between a heavy body being set in motion against inertia and friction and between an electric current being generated against inductance and resistance. For in the first case one part of the impressed force is being expended to overcome friction and the remainder to accelerate "the mass against inertia, and in the second case one part of 4he impressed electromotive force is expended to overcome resistance and the remainder to increase the current against •electrical inductance.

A circuit possessing inductance is called an inductive circuit, and a circuit whose inductance is negligible is called a non- inductive circuit. A truly non-inductive circuit can no more be realised in practice than a mass-less material body. The •clear recognition that an electric circuit possessed a quality in virtue of which kinetic energy is associated with it when a current is flowing through it was first reached by Joseph Henry. In 1832 Henry made the observation that if the poles of a single galvanic cell are united by a short thick wire, then on breaking the circuit there is little or no spark ; but if the uniting wire is a very long one, and, better, if it is coiled into a spiral, then there is a considerable spark at the contact on opening the circuit. In 1835 he expanded and continued these observations,* and noticed that if the wire is coiled round an iron core, and thus forms an electro-magnet, the spark and shock at breaking circuit are still more marked. Henry still further elaborated these observations in 1835. t Later still Faraday attacked the same problem, and devoted to its consideration the Ninth Series (§1048) of his "Electrical Kesearches."

• Journal of Franklin Tnstitute, March, 1835, Vol. XV., pp. 169-170. t Phil. May., 1840; see alto Scientific Writings of Joseph Henry, rpp. 87-97.

SIMPLE PERIODIC CURRENTS. Ill

The chain of experiments which led to this inquiry was apparently started by a question addressed to Faraday by a Mr. Jenkin, one Friday evening, at the Eoyal Institution, as to the reason why a shock was experienced when a circuit containing an electromagnet was broken, the observer retaining in his two hands the ends of the circuit, but no shock was felt if the circuit contained neither magnet nor long coils of wire. Faraday seems speedily to have arranged an organised attack on the subject, and to have returned from his investigation burdened with the spoils of victory in the shape of the following facts : —

1. If a battery circuit is closed by a short thick wire, then, although there may be a very strong current existing in this wire, on breaking contact at any point little or no spark is seen, and if the two ends of the circuit are grasped in the two hands, and the interruption takes place between the hands, then little or no shock is experienced.

2. If a very long wire is used instead, then, although the absolute strength of the current may be less, yet the spark and shock at interruption are more manifest.

3. If this length of insulated wire is coiled up into a helix on a pasteboard tube, then, although the length of wire and strength of current are the same, yet the spark and shock are still more marked.

4. If the above helix has an iron core placed in it, both these effects are yet more exalted.

5. If the same length of wire is doubled upon itself, being, however, insulated, then the effects nearly vanish, and, whether straight or coiled, this doubled wire with current going up one side and down the other is no better in respect of spark and shock on interruption than a very short wire.

The first observation which Faraday makes upon the above results is that electricity would seem to circulate with some- thing like momentum or inertia in the wire, and that the greater the length and strength of the current, so much the more power is there to run on and jump over the obstacle presented by the first thin layer of air which is introduced between the contacts as they are separated, giving rise to a spark. He saw, however, at once that, since the form of this circuit is an important factor, the idea of inertia in tfte current

112 SIMPLE PERIODIC CURRENTS.

itself was fallacious, or else the mere doubling the wire could not nullify all the effects. He did not at that time see that the idea of momentum was exceedingly appropriate, but its- allocation in the electric current itself was wrong.

The observation, however, which led him to a consistent theory was as follows. A bobbin was prepared, having wound on it two insulated wires, 1 and 2. The ends of 2 being left unconnected, the wire 1 was used to complete a circuit, and gave a spark on interrupting a current traversing it. As we have seen (Chap. I.), Faraday* had three years previously established the fact that the commencement and cessation of a current in one circuit would produce in another circuit, if closed, an inverse or a direct induced electric wave or transi- tory current. Now, on closing the second circuit through a galvanometer or loose contact, and interrupting a steady current flowing in the first circuit, he found that when circuit 2 was completed, so that an induced or secondary current could be generated in it, little or no spark happened at thu place of interruption in 1 ; but, if circuit 2 was opened, then the interruption of circuit 1 gave rise to a bright spark at the contact. Faraday therefore inferred that when circuit 2 was closed adjacent to circuit 1, the current in 1 exerted its- full inductive effect in generating secondary currents in 2 -r but that, if circuit 2 was open, then, there being no adjacent conductors, the current in 1 expended its inductive effect in producing induced currents in its own circuit, and this self- vnduction manifested itself by temporarily diminishing the strength of the current at starting and assisting or increasing it momentarily at the interruption. He was thus able, from this point of view, to picture to himself the circuit of 1 as occupied by a steady current, superimposed on which was another current he called the inverse extra current, lasting but a very short time at starting the steady current ; and a direct extra current which flowed on and produced the effects of the spark or shock at the interruption of the circuit. These extra currents, or currents of self-induction, he found could be removed from the circuit itself and exhibited in a neighbouring circuit when that adjacent circuit was closed, and so fitted to be the seat of induced currents due to the mutual induction of

• Faraday's " Exp. Res.," § 1,090.

SIMPLE PERIODIC CURRENT*. 113

the primary on this secondary circuit. Faraday then placed this theory under test by requiring it to furnish an explanation of the following experiment : —

M and N (Fig. 46) were two mercury cups* which formed the terminals of three circuits — a battery circuit, B, a galvano- meter circuit, G, and a circuit consisting of an electromagnet or helix, C. The needle of the galvanometer was blocked in such a way that the tendency to deflect under the steady current was prevented and the needle kept at zero ; but it was free to deflect in the opposite direction under an oppositely directed current. This being the case, the raising of the battery wires out of the mercury cups was accompanied by a violent " kick " or deflection of the needle in the free direction.

6

JL

' rr\

N I G

I

\mmsmj

FIG. 46.

The action could clearly be explained by supposing that after the electromotive force of the battery is removed from the coil C, the current in it does not at once stop dead, but runs on like a heavy body and makes a backwash of current through the galvanometer in the direction from M to N. An illustration of the electromagnetic inertia of a coil on inter- rupting the current may be shown in a more modern form, thus : Let E (Fig. 47) be an electromagnet, and let L be an incandescent lamp of which the resistance is very large com- pared with that of E. Let S be a few cells of a storage battery supplying current, and let K be a key. On depressing the key the current flows both in the magnet and in the lamp

• Faraday s " Exp. Res.," Vol. L, § 1,079.

114

SIMPLE PERIODIC CURRENTS.

arranged as a shunt on the magnet. This current, however, is, by assumption, not strong enough to illuminate the lamp. On raising the key and stopping the steady current through the lamp the electric inertia of the coil sends a momentary powerful current through the lamp, which causes it to flash up. Again, if a small shunt-wound dynamo be occupied in supplying current to a few incandescent lamps, and the two hands be employed to raise simultaneously the brushes from the armature, the momentary rush of current from the field- magnet due to this extra current will disagreeably impress the phenomenon upon the mind of the observer if the experiment

is made with any but a very small dynamo. With a large dynamo this experiment is very dangerous to perform.

Neither of these experiments is well fitted to illustrate the extra current at the closing of the circuit or the effect of electric inertia on starting the current in a helix. The arrangement most suited to exhibit the whole effect is that of the differential galvanometer as used by Edlund, or that employing Wheatstone's bridge, due to Maxwell.

In Edlund's arrangement* a differential galvanometer is employed, of which the two coils GX G2 are so placed and

• See Poggendorff's Annalen, 1849.

SIMPLE PERIODIC CURRENTS.

115

n-ound that when equal and oppositely -directed currents are sent through them the needle is unaffected. The coils are then connected, as shown in Fig. 48, to a battery, B, an electromagnet or helical coil, L, and a wire, E, of equal v resistance to L, but wound double. The galvanometer coils are so connected to the circuits L and E that when the steady •current from the battery flows through the divided circuit the needle remains at zero. On closing the circuit it is then found that the needle makes a sudden deflection in a direction indicating a brief current passing in coil G.2, and on breaking the circuit it makes another deflection, indicating a transitory -current passing through Gj. In other words, the balance is

C,

FIG. 48.

destroyed at the instant of breaking and making, but restores itself again when the currents become steady. This experi- ment, therefore, most clearly shows that the electromagnetic helix L, although of exactly the same electrical resistance as the coil E, differs from it in possessing a peculiar quality, which it has in virtue of being in the form of a coil or helix, and to which the name self-induction or inductance has been given. We are able to define this term as follows : — The self-induction or inductance of a circuit is, speaking generally, a quality of it in virtue of which a finite and steady electromotive force applied to it cannot at once generate in it the full current due to its resistance, and when the electromotive force is with- drawn time is required for the current strength to fall to zero.

116

SIMPLE PERIODIC CURRENTS.

It miist, however, be noticed that not only does the inductance of a circuit depend upon tb.3 geometrical form of the circuit, but it depends upon the magnetic permeability of the region which surrounds the circuit and on the magnetic permeability of the conducting circuit itself. If, in the arrangement with the dif- ferential galvanometer, the steady balance is obtained by using a copper wire helix wound on a cardboard tube and balanced against a non-inductive but equal resistance, it is found that the insertion of a soft iron core into the helix greatly increases the "kick" on making contact, indicating the passage of a greater quantity of electricity through the opposite galvano- meter coil, and therefore a greater delay in the time of establishing the steady balance.

Maxwell's method of exhibiting the effect of inductance is a preferable arrangement.