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 Association 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 anticipated. 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 space 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 space where the magnetic action is found to exist. The charge of the jar consists of a state of intense and peculiar strain in the glass 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 space—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 quantity 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 accumulated 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 Q2. We suppose it to be ½Q2 ⁄ C, where C is a constant depending on the condenser and called its capacity.
The idea of the charge as a quantity 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 circumstances 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 circumstances 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 quantity 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 ½Lc2. This is analogous to the kinetic energy ½mv2 of a body (say a pendulum bob) of mass m and velocity v, so that L represents a quantity for the conducting arrangement analogous to inertia, and c is the analogue of the velocity of the body. The whole energy at any instant is thus
½Q2 ⁄ C + ½Lc2, or ½Q2 ⁄ C + ½L (dQ ⁄ dt)2.
The loss of energy due to heating of the conducting connection is not completely understood, though its quantitative 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 quantity 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.