Thus c was positive or negative according as Irw was greater or less than ?e, and was zero when Irw = ?e. Thus the electromotive force of the disk was opposed by a back electromotive force ?e due to the chemical action in the voltameter or battery, to which the wires from the disk were connected.

The conclusion arrived at therefore was that the electromotive force (or, as it was then termed, the intensity) of the electrochemical action was equal to the dynamical value of the whole chemical change effected by a current of unit strength in unit of time.

From this result Thomson proceeded to calculate the electromotive forces required to effect chemical changes of different kinds, and those of various types of voltaic cell. Supposing a unit of electricity to be carried by the current through the cell, he considered the chemical changes which accompanied its pa.s.sage, and from the known values of heats of combination calculated their energy values. In some parts the change was one of chemical combination, in others one of decomposition of the materials, and regard had to be paid to the sign of the heat-equivalent. By properly summing up the whole heat-equivalents a net total was obtained which, according to Thomson, was the energy consumed in the pa.s.sage of unit current, and was therefore the electromotive force. The theory was incomplete, and required to be supplemented by thermodynamic theory, which shows that besides the electromotive force there must be included in the quant.i.ty set against the sum of heats a term represented by the product of the absolute temperature multiplied by the rate of variation of electromotive force with alteration of temperature. Thus the theory is only applicable when the electromotive force is not affected by variation of temperature. The necessary addition here indicated was made by Helmholtz.

In the next paper, which appeared in the same number (December 1851) of the _Philosophical Magazine_, the principle of work is applied to the measurement of electromotive forces and resistances in absolute units.

The advantages of such units are obvious. Nearly the whole of the quant.i.tative work of the older experimenters was useless except for those who had actually made the observations: it was hardly possible for one man to advance his researches by employing data obtained by others.

For the results were expressed by reference to apparatus and materials in the possession of the observers, and to these others could obtain access only with great difficulty and at great expense--to say nothing of the uncertainty of comparisons made to enable the results of one man to be linked on to those made elsewhere, and with other apparatus, by another. It was imperative, therefore, to obtain absolute units--units independent of accidents of place and apparatus--for the expression of currents, electromotive forces, and resistances, so as to enable the results of the work of experiments all over the world to be made available to every one who read the published record. (See Chap. XIII.)

The magneto-electric machine imagined in the former paper gave a means of estimating the electromotive force of a cell or battery in absolute units. The same kind of machine is used here, in the simpler form of a sliding conductor connecting a pair of insulated rails laid with their plane perpendicular to the lines of force of a uniform magnetic field.

If the rails be connected by a wire, and the slider be moved so as to cut across the lines of force, a current will be produced in the circuit. The current can be measured in terms of the already known unit of current, that current which flowing in a circle of radius unity produces a magnetic field at the centre of 2p units. This current, c, say, in strength, flowing in the circuit, renders a dynamical force cIl necessary to move the slider of length l across the lines of force of the field of intensity I, and if the speed of the slider required for the current c be v, the rate at which work is done in moving the slider is cIlv. This must be the rate at which work is done in the circuit by the current, and if the only work done be in the heating of the conductor, we have cIlv = Rc, or Ilv = Rc, so that Ilv is the electromotive force. Any electromotive force otherwise produced, which gave rise to the same current, must obviously be equal to Ilv, so that the unit of electromotive force can thus be properly defined.

Thomson used a foot-grain-second system of units; but from this arrangement are now obtained the C.G.S. units of electromotive force and resistance. If I is one C.G.S. unit, l one centimetre, and v one centimetre per second, we have unit electromotive force in the C.G.S.

system. Also in one C.G.S. unit of resistance if c be unity as well as Ilv.

The idea of the determination of a resistance in absolute units on correct principles was due to W. Weber, who also gave methods of carrying out the measurement; and the first determination was made by Kirchhoff in 1849. Thomson appears, however, to have been the first to discuss the subject of units from the point of view of energy. This mode of regarding the matter is important, as the absolute units are so chosen as to enable work done by electric and magnetic forces to be reckoned in the ordinary dynamical units. A vast amount of experimental resource and skill has been spent since that time on the determination of resistance, though not more than the importance of the subject warranted. We shall have to return to the subject in dealing with the work of the British a.s.sociation on Electrical Standards, of which Thomson was for long an active member.

ELECTRICAL OSCILLATIONS

In his famous tract on the conservation of energy, published in 1847, von Helmholtz discussed some puzzling results obtained by Riess in the magnetisation of iron wires by the current of a Leyden jar discharge flowing in a coil surrounding them, and by the fact, observed by Wollaston, that when water was decomposed by Leyden jar discharges a mixture of oxygen and hydrogen appeared at each electrode, and suggested that possibly the discharge was oscillatory in character.

In 1853 the subject was discussed mathematically by Thomson, in a paper which was to prove fruitful in our own time in a manner then little antic.i.p.ated. The jar is given, let us say, with the interior coating charged positively, and the exterior coating charged negatively. A coil or helix of wire has its ends connected to the two coatings, and a current immediately begins in the wire, and gradually (not slowly) increases in strength. Accompanying the creation of the current is the production of a magnetic field, that is, the surrounding s.p.a.ce is made the seat of magnetic action. The magnetic field, as we shall see from another investigation of Thomson's, almost certainly involves motion in or of a medium--the ether--filling the s.p.a.ce where the magnetic action is found to exist. The charge of the jar consists of a state of intense and peculiar strain in the gla.s.s plate between the coatings. When the plates are connected by the coil, this state of strain breaks down and motion in the medium ensues, not merely between the plates, but also in the surrounding s.p.a.ce--in fact, in the whole field. This motion--which is not to be confused with bodily displacement of finite parts of the medium--is opposed by something akin to inertia of the medium (the property that confers energy on matter when in motion), so that when the motion is started it persists, until it is finally wiped out by resistance of the nature of friction. The inertia here referred to depends on the mode in which the coil is wound, or whether it contains or not an iron core.

If the work done in charging a Leyden jar or electric condenser, by bringing the charge to the condenser in successive small portions, is considered, it is at once clear that it must be proportional to the square of the whole quant.i.ty of electricity brought up. For whatever the charge may be, let it be brought up from a great distance in a large number N of equal instalments. The larger the whole amount the larger must each instalment be, and therefore the greater the amount acc.u.mulated on the condenser when any given number of instalments have been deposited. But the greater any charge that is being brought up, and also the greater the charge that has already arrived, the greater is the repulsion that must be overcome in bringing up that instalment, in simple proportion in each case, and therefore the greater the work done.

Thus the whole work done in bringing up the charge must be proportional to Q. We suppose it to be Q?C, where C is a constant depending on the condenser and called its capacity.

The idea of the charge as a quant.i.ty of some kind of matter, brought up and placed on the insulated plate of the condenser, has only a correspondence to the fact, which is that the medium between the plates is the seat, when the condenser is charged, of a store of energy, which can only be made available by connecting the plates of the condenser by a wire or other conductor. The charge is only a surface aspect of the state of the medium, apparently a state of strain, to which the energy belongs.

When a wire is used to connect the plates the state of strain disappears; the energy comes out from the medium between the plates by motion sideways of the tubes of strain (so that the insulating medium is under longitudinal tension and lateral pressure) which, according to Faraday's conception of lines of electric force connecting the charge on a body with the opposite charges on other bodies, run from plate to plate, when the condenser is in equilibrium in the changed state. These tubes move out with their ends on the wire, carrying the energy with them, and the ends run towards one another along the wire; the tube shortens in the process, and energy is lost in the wire. The ends of a tube thus moving represent portions of the charges which were on the plates, and the oppositely-directed motions of the opposite charges represent a current along the wire from one conductor to the other. The motion of the tubes is accompanied by the development of a magnetic field, the lines of force of which are endless, and the direction of which at every point is perpendicular at once to the length of the tube and to the direction in which it is there moving. In certain circ.u.mstances the tube, by the time its ends have met, will have wholly disappeared in the wire, and the whole energy will have gone to heat the wire: in other circ.u.mstances the ends will meet before the tube has disappeared, the ends will cross, and the tube will be carried back to the condenser and reinserted in the opposite direction. At a certain time this will have happened to all the tubes, though they will have lost some of their energy in the process; and the condenser will again be charged, though in the opposite way to that in which it was at first.

Then the tubes will move out again, and the same process will be repeated: once more the condenser will be charged, but in the same direction as at first, and once more with a certain loss of energy.

Again the process of discharge and charge will take place, and so on, again and again, until the whole energy has disappeared. This process represents, according to the modern theory of the flow of energy in the electromagnetic field, with more or less accuracy, what takes place in the oscillatory discharge of a condenser.

The motion of the tubes with their ends on the wire represents a certain amount of energy, commonly regarded as kinetic, and styled electrokinetic energy. If c denote the current, that is, the rate, -dQ?dt, at which the charge of the condenser is being changed, and L a quant.i.ty called self-inductance, depending mainly on the arrangement of the connecting wire--whether it is wound in a coil or helix, with or without an iron core, or not--the electrokinetic energy will be Lc.

This is a.n.a.logous to the kinetic energy mv of a body (say a pendulum bob) of ma.s.s m and velocity v, so that L represents a quant.i.ty for the conducting arrangement a.n.a.logous to inertia, and c is the a.n.a.logue of the velocity of the body. The whole energy at any instant is thus

Q?C + Lc, or Q?C + L(dQ?dt).

The loss of energy due to heating of the conducting connection is not completely understood, though its quant.i.tative laws have been quite fully ascertained and expressed in terms of magnitudes that are capable of measurement. It was found by Joule to be proportional to the second power, or square, of the current, and to a quant.i.ty R depending on the conductor, and called its resistance. The generation of heat in the conductor seems to be due to some kind of frictional action of particles of the conductor set up by the penetration of the Faraday tubes into it.

A conductor is unable to bear any tangential action exerted upon it by Faraday tubes, which, however, when they exist, begin and end at material particles, except when they are endless, as they may be in the radiation of energy. When the Faraday tubes are moving with any ordinary speed they are not at their ends perpendicular to the conducting surface from which they start or at which they terminate, but are there more or less inclined to the surface, and consequently there is tangential action which appears to displace the particles (not merely at the surface, unless the alternation is very rapid) relatively to one another and so cause frictional generation of heat.

The time rate of generation of heat is thus Rc, or R(dQ?dt), when the units in which R and c are expressed are such as to make this quant.i.ty a rate of doing work in the true dynamical sense. This is the rate at which the sum of energy already found is being diminished, and so the equation

d/dt{(Q?C) + L(dQ?dt)} = -R(dQ?dt)

holds, or leaving out the common factor dQ?dt, the equation

L(dQ?dt) + R(dQ?dt) + Q?C = 0

This last equation was established by Thomson, and is precisely that which would be obtained for a pendulum bob of ma.s.s L, pulled back towards the position of equilibrium with a force Q?C, where Q is the displacement from the middle position, and having its motion damped out by resisting force of amount R per unit of the velocity.

It is more instructive perhaps to take the oscillatory motion of a spiral spring hung vertically with a weight on its lower end, as that which has a differential equation equivalent to the equation just found.

When the stretch is of a certain amount, there is equilibrium--the action of the spring just balances the weight,--and if the spring be stretched further there will be a balance of pull developed tending to bring the system back towards the equilibrium position. If left to itself the system gets into motion, which, if the resistance is not too great, is added to until the equilibrium position is reached; and the motion, which is continued by the inertia of the ma.s.s, only begins to fall off as that position is pa.s.sed, and the pull of the spring becomes insufficient to balance the weight. Thus the ma.s.s oscillates about the position of equilibrium, and the oscillations are successively smaller and smaller in extent, and die out as their energy is expended finally in doing work against friction.

If the resisting force for finite motion is very great, as for example when the vibrating ma.s.s of the pendulum or spring is immersed in a very viscous fluid, like treacle, oscillation will not take place at all.

After displacement the ma.s.s will move at first fairly quickly, then more and more slowly back to the position of equilibrium, which it will, strictly speaking, only exactly reach after an infinite time. The resisting force is here indefinitely small for an indefinitely small speed, but it becomes so great when any motion ensues, that as the restoring force falls off with the displacement, no work is finally done by it, except to move the body through the resisting medium.

The differential equation is applicable to the spring if Q is again taken as displacement from the equilibrium position, L as the inertia of the vibrating body, 1?C as the pull exerted by the spring per unit of its extension (that is, the stiffness of the spring), and R has the same meaning as before.

In this case of motion, as well as in that of the pendulum, energy is carried off by the production of waves in the medium in which the vibrator is immersed. These are propagated out from the vibrator as their source, but no account of them is taken in the differential equation, which in that respect is imperfect. There is no difficulty, only the addition of a little complication, in supplying the omission.

The formation of such waves by the spiral spring vibrator can be well shown by immersing the vibrating body in a trough of water, and the much greater rate of damping out of the motion in that case can then be compared with the rate of damping in air.

It has been indicated that the differential equation does not represent oscillatory motion if the value of R is too great. The exact condition depends on the roots of the quadratic equation Lx + Rx + 1?C = 0, obtained by writing 1 for Q, and x for d?dt, and then treating x as a quant.i.ty. These roots are -R?2L v(R?4L - 1?CL), and are therefore real or imaginary according as 4L?C is less or greater than R. If the roots are real, that is, if R be greater than 4L?C, the discharge will not be oscillatory; the Faraday tubes referred to above will be absorbed in the wire without any return to the condenser. The corresponding result happens with the vibrator when R is sufficiently great, or L?C sufficiently small (a weak spring and a small ma.s.s, or both), to enable the condition to be fulfilled.

If, however, the roots of the quadratic are imaginary, that is, if 4L?C be greater than R (a condition which will be fulfilled in the spring a.n.a.logue, by making the spring sufficiently stiff and the ma.s.s large enough to prevent the friction from controlling the motion) the motion is one in which Q disappears by oscillations about zero, of continually diminishing amplitude. A complete discussion gives for the period of oscillation 4pL?v(4L?C - R), or if R be comparatively small, 2pv(LC).

The charge Q falls off by the fraction e^{-RT?2L} (where e is the number 2.71828...) in each period T, and so gradually disappears.

Thus electric oscillations are produced, that is to say, the charged state of the condenser subsides by oscillations, in which the charged state undergoes successive reversals, with dissipation of energy in the wire; and both the period and the rate of dissipation can be calculated if L, C, and R are known, or can be found, for the system. These quant.i.ties can be calculated and adjusted in certain definite cases, and as the electric oscillations can be experimentally observed, the theory can be verified. This has been done by various experimenters.

Returning to the pendulum ill.u.s.tration, it will be seen that the pendulum held deflected is a.n.a.logous to the charged jar, letting the pendulum go corresponds to connecting the discharging coil to the coatings, the motion of the pendulum is the a.n.a.logue of that motion of the medium in which consists the magnetic field, the friction of the air answers to the resistance of the wire which finally damps out the current. The inertia or ma.s.s of the bob is the a.n.a.logue of what Thomson called the electromagnetic inertia of the coil and connections; what is now generally called the self-inductance of the conducting system. The component of gravity along the path towards the lowest point, answers to the reciprocal, 1?C, of the capacity of the condenser.

It appears from the a.n.a.logy that just as the oscillations of a pendulum can be prevented by immersing the bob in a more resisting medium, such as treacle or oil, so that when released the pendulum slips down to the vertical without pa.s.sing it, so by properly proportioning the resistance in the circuit to the electromagnetic inertia of the coil, oscillatory discharge of the Leyden jar may also be rendered impossible.

All this was worked out in an exceedingly instructive manner in Thomson's paper; the account of the matter by the motion of Faraday tubes is more recent, and is valuable as suggesting how the inertia effect of the coil arises. The a.n.a.logy of the pendulum is a true one, and enables the facts to be described; but it is to be remembered that it becomes evident only as a consequence of the mathematical treatment of the electrical problem. The paper was of great importance for the investigation of the electric waves used in wireless telegraphy in our own time. It enabled the period of oscillation of different systems to be calculated, and so the rates of exciters and receivers of electric waves to be found. For such vibrators are really Leyden jars, or condensers, caused to discharge in an oscillatory manner.

This application was not foreseen by Thomson, and, indeed, could hardly be, as the idea of electric waves in an insulating medium came a good deal later in the work of Maxwell. Yet the a.n.a.logy of the pendulum, if it had then been examined, might have suggested such waves. As the bob oscillates backwards and forwards the air in which it is immersed is periodically disturbed, and waves radiate outwards from it through the surrounding atmosphere. The energy of these waves is exceedingly small, otherwise, as pointed out above, a term would have to be included in the theory of the resisted motion of the pendulum to account for this energy of radiation. So likewise when the electric vibrations proceed, and the insulating medium is the seat of a periodically varying magnetic field, electromagnetic waves are propagated outwards through the surrounding medium--the ether--and the energy carried away by the waves is derived from the initial energy of the charged condenser. In strictness also Thomson's theory of electric oscillations requires an addition to account for the energy lost by radiation. This is wanting, and the whole decay of the amount of energy present at the oscillator is put down to the action of resistance--that is, to something of the nature of frictional r.e.t.a.r.dation. Notwithstanding this defect of the theory, which is after all not so serious as certain difficulties of exact calculation of the self-inductance of the discharging conductor, the periods of vibrators can be very accurately found. When these are known it is only necessary to measure the length of an electrical wave to find its velocity of propagation. When electromagnetic waves were discovered experimentally in 1888 by Heinrich Hertz, it was thus that he was able to demonstrate that they travelled with the velocity of light.

Thomson suggested that double, triple and quadruple flashes of lightning might be successive flashes of an oscillatory discharge. He also pointed out that if a spark-gap were included in a properly arranged condenser and discharging wire, it might be possible, by means of Wheatstone's revolving mirror, to see the sparks produced in the successive oscillations, as "points or short lines of light separated by dark intervals, instead of a single point of light, or of an unbroken line of light, as it would be if the discharge were instantaneous, or were continuous, or of appreciable duration."

This antic.i.p.ation was verified by experiments made by Feddersen, and published in 1859 (_Pogg. Ann._, 108, 1859). The subject was also investigated in Helmholtz's laboratory at Berlin, by N. Schiller, who, determining the period for condensers with different substances between the plates, was able to deduce the inductive capacities of these substances (_Pogg. Ann._, 152, 1874). [The specific inductive capacity of an insulator is the ratio of the capacity of a condenser with the substance between the plates to the capacity of an exactly similar condenser with air between the plates.]

The particular case of non-oscillatory discharge obtained by supposing C and Q both infinitely great and to have a finite ratio V (which will be the potential, p. 34, of the charged plate), is considered in the paper.

The discharging conductor is thus subjected to a difference of potential suddenly applied and maintained at one end, while the other end is kept at potential zero. The solution of the differential equation for this case will show how the current rises from zero in the wire to its final steady value. If c be put as before for the current -dQ?dt, and the constant value V for Q?C, the equation is

L(dc?dt) + Rc = V

which gives, since c = 0 when t = 0,

c = (V?R)[1 - e^{-(R?L)t}].

Thus, when an infinite time has elapsed the current has become V?R, the steady value.

Thomson concludes by showing how, by measuring the non-oscillatory discharge of a condenser (the capacity of which can be calculated) by means of an electrodynamometer and an ordinary galvanometer arranged in series, what W. Weber called the duration of the discharging current may be determined. From this Thomson deduced a value for the ratio of the electromagnetic unit of electricity to the electrostatic unit, and indicated methods of determining this ratio experimentally. This ratio is of fundamental importance in electromagnetic theory, and is essentially of the nature of a speed. According to Maxwell it is the speed of propagation of electromagnetic waves in an insulating medium for which the units are defined. It was first determined in the Glasgow laboratory by Mr. Dugald McKichan, and has been determined many times since. It is practically identical with the speed of light as ascertained by the best experiments.