Part 8
For the analysis of mechanical phenomena, a metrical notion, derived from experience, and bearing the designation _work_, has proved itself useful. A machine can be set in motion only when the forces acting on it can perform work.
Let us consider, for example, a wheel and axle (Fig. 35) having the radii 1 and 2 metres, loaded respectively with the weights 2 and 1 kilogrammes. On turning the wheel and axle, the 1 kilogramme-weight, let us say, sinks two metres, while the 2 kilogramme-weight rises one metre. On both sides the product
KGR. M. KGR. M.
1 × 2 = 2 × 1.
is equal. So long as this is so, the wheel and axle will not move of itself. But if we take such loads, or so change the radii of the wheels, that this product (kgr. × metre) on displacement is in excess on one side, that side will sink. As we see, this product is characteristic for mechanical events, and for this reason has been invested with a special name, _work_.
In all mechanical processes, and as all physical processes present a mechanical side, in all physical processes, work plays a determinative part. Electrical forces, also, produce only changes in which work is performed. To the extent that forces come into play in electrical phenomena, electrical phenomena, be they what they may, extend into the domain of mechanics and are subject to the laws which hold in this domain. The universally adopted measure of work, now, is the product of the force into the distance through which it acts, and in the C. G. S. system, the unit of work is the action through one centimetre of a force which would impart in one second to a gramme-mass a velocity-increment of one centimetre, that is, in round numbers, the action through a centimetre of a pressure equal to the weight of a milligramme. From a positively charged body, electricity, yielding to the force of repulsion and performing work, flows off to the earth, providing conducting connexions exist. To a negatively charged body, on the other hand, the earth under the same circumstances gives off positive electricity. The electrical work possible in the interaction of a body with the earth, characterises the electrical condition of that body. We will call the work which must be expended on the unit quantity of positive electricity to raise it from the earth to the body _K_ the _potential_ of the body _K_.[31]
We ascribe to the body _K_ in the C. G. S. system the potential +1, if we must expend the unit of work to raise the positive electrostatic unit of electric quantity from the earth to that body; the potential -1, if we gain in this procedure the unit of work; the potential 0, if no work at all is performed in the operation.
The different parts of one and the same electrical conductor in electrical equilibrium have the same potential, for otherwise the electricity would perform work and move about upon the conductor, and equilibrium would not have existed. Different conductors of equal potential, put in connexion with one another, do not exchange electricity any more than bodies of equal temperature in contact exchange heat, or in connected vessels, in which the same pressures exist, liquids flow from one vessel to the other. Exchange of electricity takes place only between conductors of different potentials, but in conductors of given form and position a definite difference of potential is necessary for a spark, which pierces the insulating air, to pass between them.
On being connected, every two conductors assume at once the same potential. With this the means is given of determining the potential of a conductor through the agency of a second conductor expressly adapted to the purpose called an electrometer, just as we determine the temperature of a body with a thermometer. The values of the potentials of bodies obtained in this way simplify vastly our analysis of their electrical behavior, as will be evident from what has been said.
Think of a positively charged conductor. Double all the electrical forces exerted by this conductor on a point charged with unit quantity, that is, double the quantity at each point, or what is the same thing, double the total charge. Plainly, equilibrium still subsists. But carry, now, the positive electrostatic unit towards the conductor. Everywhere we shall have to overcome double the force of repulsion we did before, everywhere we shall have to expend double the work. By doubling the charge of the conductor a double potential has been produced. Charge and potential go hand in hand, are proportional. Consequently, calling the total quantity of electricity of a conductor _Q_ and its potential _V_, we can write: _Q = CV_, where _C_ stands for a constant, the import of which will be understood simply from noting that _C = Q/V_.[32] But the division of a number representing the units of quantity of a conductor by the number representing its units of potential tells us the quantity which falls to the share of the unit of potential. Now the number _C_ here we call the capacity of a conductor, and have substituted, thus, in the place of the old relative determination of capacity, an absolute determination.[33]
In simple cases the connexion between charge, potential, and capacity is easily ascertained. Our conductor, let us say, is a sphere of radius _r_, suspended free in a large body of air. There being no other conductors in the vicinity, the charge _q_ will then distribute itself uniformly upon the surface of the sphere, and simple geometrical considerations yield for its potential the expression _V = q/r_. Hence, _q/V = r_; that is, the capacity of a sphere is measured by its radius, and in the C. G. S. system in centimetres.[34] It is clear also, since a potential is a quantity divided by a length, that a quantity divided by a potential must be a length.
Imagine (Fig. 36) a jar composed of two concentric conductive spherical shells of the radii _r_ and _r₁_, having only air between them. Connecting the outside sphere with the earth, and charging the inside sphere by means of a thin, insulated wire passing through the first, with the quantity _Q_, we shall have _V = (r₁-r)/(r₁r)Q_, and for the capacity in this case _(r₁r)/(r₁-r)_, or, to take a specific example, if _r = 16_ and _r₁ = 19_, a capacity of about 100 centimetres.
We shall now use these simple cases for illustrating the principle by which capacity and potential are determined. First, it is clear that we can use the jar composed of concentric spheres with its known capacity as our unit jar and by means of this ascertain, in the manner above laid down, the capacity of any given jar _F_. We find, for example, that 37 discharges of this unit jar of the capacity 100, just charges the jar investigated at the same striking distance, that is, at the same potential. Hence, the capacity of the jar investigated is 3700 centimetres. The large battery of the Prague physical laboratory, which consists of sixteen such jars, all of nearly equal size, has a capacity, therefore, of something like 50,000 centimetres, or the capacity of a sphere, a kilometre in diameter, freely suspended in atmospheric space. This remark distinctly shows us the great superiority which Leyden jars possess for the storage of electricity as compared with common conductors. In fact, as Faraday pointed out, jars differ from simple conductors mainly by their great capacity.
For determining potential, imagine the inner coating of a jar _F_, the outer coating of which communicates with the ground, connected by a long, thin wire with a conductive sphere _K_ placed free in a large atmospheric space, compared with whose dimensions the radius of the sphere vanishes. (Fig. 37.) The jar and the sphere assume at once the same potential. But on the surface of the sphere, if that be sufficiently far removed from all other conductors, a uniform layer of electricity will be found. If the sphere, having the radius _r_, contains the charge _q_, its potential is _V = q/r_. If the upper half of the sphere be severed from the lower half and equilibrated on a balance with one of whose beams it is connected by silk threads, the upper half will be repelled from the lower half with the force _P = q²/8r² = 1/8V²_. This repulsion _P_ may be counter-balanced by additional weights placed on the beam-end, and so ascertained. The potential is then _V = [sqrt](8P)_.[35]
That the potential is proportional to the square root of the force is not difficult to see. A doubling or trebling of the potential means that the charge of all the parts is doubled or trebled; hence their combined power of repulsion quadrupled or nonupled.
Let us consider a special case. I wish to produce the potential 40 on the sphere. What additional weight must I give to the half sphere in grammes that the force of repulsion shall maintain the balance in exact equilibrium? As a gramme weight is approximately equivalent to 1000 units of force, we have only the following simple example to work out: _40×40 = 8× 1000.x_, where _x_ stands for the number of grammes. In round numbers we get _x_ = 0.2 gramme. I charge the jar. The balance is deflected; I have reached, or rather passed, the potential 40, and you see when I discharge the jar the associated spark.[36]
The striking distance between the knobs of a machine increases with the difference of the potential, although not proportionately to that difference. The striking distance increases faster than the potential difference. For a distance between the knobs of one centimetre on this machine the difference of potential is 110. It can easily be increased tenfold. Of the tremendous differences of potential which occur in nature some idea may be obtained from the fact that the striking distances of lightning in thunder-storms is counted by miles. The differences of potential in galvanic batteries are considerably smaller than those of our machine, for it takes fully one hundred elements to give a spark of microscopic striking distance.
* * * * *
We shall now employ the ideas reached to shed some light upon another important relation between electrical and mechanical phenomena. We shall investigate what is the potential _energy_, or the _store of work_, contained in a charged conductor, for example, in a jar.
If we bring a quantity of electricity up to a conductor, or, to speak less pictorially, if we generate by work electrical force in a conductor, this force is able to produce anew the work by which it was generated. How great, now, is the energy or capacity for work of a conductor of known charge _Q_ and known potential _V_?
Imagine the given charge _Q_ divided into very small parts _q_, _q₁_, _q₂_ ..., and these little parts successively carried up to the conductor. The first very small quantity _q_ is brought up without any appreciable work and produces by its presence a small potential _V__{'}. To bring up the second quantity, accordingly, we must do the work _q__{'}_V__{'}, and similarly for the quantities which follow the work _q__{''}_V__{''}, _q__{'''}_V__{'''}, and so forth. Now, as the potential rises proportionately to the quantities added until the value _V_ is reached, we have, agreeably to the graphical representation of Fig. 38, for the total work performed,
_W = 1/2QV_,
which corresponds to the total energy of the charged conductor. Using the equation _Q_ = _CV_, where _C_ stands for capacity, we also have,
_W = 1/2CV²_, or _W = Q²/2C_.
It will be helpful, perhaps, to elucidate this idea by an analogy from the province of mechanics. If we pump a quantity of liquid, _Q_, gradually into a cylindrical vessel (Fig. 39), the level of the liquid in the vessel will gradually rise. The more we have pumped in, the greater the pressure we must overcome, or the higher the level to which we must lift the liquid. The stored-up work is rendered again available when the heavy liquid _Q_, which reaches up to the level _h_, flows out. This work _W_ corresponds to the fall of the whole liquid weight _Q_, through the distance _h_/2 or through the altitude of its centre of gravity. We have
_W = 1/2Qh_.
Further, since _Q_ = _Kh_, or since the weight of the liquid and the height _h_ are proportional, we get also
_W = 1/2Kh²_ and _W = Q²/2K_.
As a special case let us consider our jar. Its capacity is _C_ = 3700, its potential _V_ = 110; accordingly, its quantity _Q = CV_ = 407,000 electrostatic units and its energy _W = 1/2QV_ = 22,385,000 C. G. S. units of work.
The unit of work of the C. G. S. system is not readily appreciable by the senses, nor does it well admit of representation, as we are accustomed to work with weights. Let us adopt, therefore, as our unit of work the gramme-centimetre, or the gravitational pressure of a gramme-weight through the distance of a centimetre, which in round numbers is 1000 times greater than the unit assumed above; in this case, our numerical result will be approximately 1000 times smaller. Again, if we pass, as more familiar in practice, to the kilogramme-metre as our unit of work, our unit, the distance being increased a hundred fold, and the weight a thousand fold, will be 100,000 times larger. The numerical result expressing the work done is in this case 100,000 times less, being in round numbers 0.22 kilogramme-metre. We can obtain a clear idea of the work done here by letting a kilogramme-weight fall 22 centimetres.
This amount of work, accordingly, is performed on the charging of the jar, and on its discharge appears again, according to the circumstances, partly as sound, partly as a mechanical disruption of insulators, partly as light and heat, and so forth.
The large battery of the Prague physical laboratory, with its sixteen jars charged to equal potentials, furnishes, although the effect of the discharge is imposing, a total amount of work of only three kilogramme-metres.
In the development of the ideas above laid down we are not restricted to the method there pursued; in fact, that method was selected only as one especially fitted to familiarise us with the phenomena. On the contrary, the connexion of the physical processes is so multifarious that we can come at the same event from very different directions. Particularly are electrical phenomena connected with all other physical events; and so intimate is this connexion that we might justly call the study of electricity the theory of the general connexion of physical processes.
With respect to the principle of the conservation of energy which unites electrical with mechanical phenomena, I should like to point out briefly two ways of following up the study of this connexion.
A few years ago Professor Rosetti, taking an influence-machine, which he set in motion by means of weights alternately in the electrical and non-electrical condition with the same velocities, determined the mechanical work expended in the two cases and was thus enabled, after deducting the work of friction, to ascertain the mechanical work consumed in the development of the electricity.
I myself have made this experiment in a modified, and, as I think, more advantageous form. Instead of determining the work of friction by special trial, I arranged my apparatus so that it was eliminated of itself in the measurement and could consequently be neglected. The so-called fixed disk of the machine, the axis of which is placed vertically, is suspended somewhat like a chandelier by three vertical threads of equal lengths _l_ at a distance _r_ from the axis. Only when the machine is excited does this fixed disk, which represents a Prony's brake, receive, through its reciprocal action with the rotating disk, a deflexion _[alpha]_ and a moment of torsion which is expressed by _D = (Pr²/l)[alpha]_, where _P_ is the weight of the disk.[37] The angle _[alpha]_ is determined by a mirror set in the disk. The work expended in _n_ rotations is given by _2n[pi]D_.
If we close the machine, as Rosetti did, we obtain a continuous current which has all the properties of a very weak galvanic current; for example, it produces a deflexion in a multiplier which we interpose, and so forth. We can directly ascertain, now, the mechanical work expended in the maintenance of this current.
If we charge a jar by means of a machine, the energy of the jar employed in the production of sparks, in the disruption of the insulators, etc., corresponds to a part only of the mechanical work expended, a second part of it being consumed in the arc which forms the circuit.[38] This machine, with the interposed jar, affords in miniature a picture of the transference of force, or more properly of work. And in fact nearly the same laws hold here for the economical coefficient as obtain for large dynamo-machines.
Another means of investigating electrical energy is by its transformation into heat. A long time ago (1838), before the mechanical theory of heat had attained its present popularity, Riess performed experiments in this field with the help of his electrical air-thermometer or thermo-electrometer.
If the discharge be conducted through a fine wire passing through the globe of the air-thermometer, a development of heat is observed proportional to the expression above-discussed _W = 1/2QV_. Although the total energy has not yet been transformed into measurable heat by this means, in as much as a portion is left behind in the spark in the air outside the thermometer, still everything tends to show that the total heat developed in all parts of the conductor and along all the paths of discharge is the equivalent of the work 1/2_QV_.
It is not important here whether the electrical energy is transformed all at once or partly, by degrees. For example, if of two equal jars one is charged with the quantity _Q_ at the potential _V_ the energy present is 1/2_QV_. If the first jar be discharged into the second, _V_, since the capacity is now doubled, falls to _V_/2. Accordingly, the energy 1/4_QV_ remains, while 1/4_QV_ is transformed in the spark of discharge into heat. The remainder, however, is equally distributed between the two jars so that each on discharge is still able to transform 1/8_QV_ into heat.
* * * * *
We have here discussed electricity in the limited phenomenal form in which it was known to the inquirers before Volta, and which has been called, perhaps not very felicitously, "statical electricity." It is evident, however, that the nature of electricity is everywhere one and the same; that a substantial difference between statical and galvanic electricity does not exist. Only the quantitative circumstances in the two provinces are so widely different that totally new aspects of phenomena may appear in the second, for example, magnetic effects, which in the first remained unnoticed, whilst, _vice versa_, in the second field statical attractions and repulsions are scarcely appreciable. As a fact, we can easily show the magnetic effect of the current of discharge of an influence-machine on the galvanoscope although we could hardly have made the original discovery of the magnetic effects with this current. The statical distant action of the wire poles of a galvanic element also would hardly have been noticed had not the phenomenon been known from a different quarter in a striking form.
If we wished to characterise the two fields in their chief and most general features, we should say that in the first, high potentials and small quantities come into play, in the second small potentials and large quantities. A jar which is discharging and a galvanic element deport themselves somewhat like an air-gun and the bellows of an organ. The first gives forth suddenly under a very high pressure a small quantity of air; the latter liberates gradually under a very slight pressure a large quantity of air.
In point of principle, too, nothing prevents our retaining the electrostatical units in the domain of galvanic electricity and in measuring, for example, the strength of a current by the number of electrostatic units which flow per second through its cross-section. But this would be in a double aspect impractical. In the first place, we should totally neglect the magnetic facilities for measurement so conveniently offered by the current, and substitute for this easy means a method which can be applied only with difficulty and is not capable of great exactness. In the second place our units would be much too small, and we should find ourselves in the predicament of the astronomer who attempted to measure celestial distances in metres instead of in radii of the earth and the earth's orbit; for the current which by the magnetic C. G. S. standard represents the unit, would require a flow of some 30,000,000,000 electrostatic units per second through its cross-section. Accordingly, different units must be adopted here. The development of this point, however, lies beyond my present task.
FOOTNOTES:
[Footnote 26: A lecture delivered at the International Electrical Exhibition, in Vienna, on September 4, 1883.]
[Footnote 27: If the two bodies were oppositely electrified they would exert attractions upon each other.]
[Footnote 28: The quantity which flows off is in point of fact less than _q_. It would be equal to the quantity _q_ only if the inner coating of the jar were wholly encompassed by the outer coating.]
[Footnote 29: Rigorously, of course, this is not correct. First, it is to be noted that the jar _L_ is discharged simultaneously with the electrode of the machine. The jar _F_, on the other hand, is always discharged simultaneously with the outer coating of the jar _L_. Hence, if we call the capacity of the electrode of the machine _E_, that of the unit jar _L_, that of the outer coating of _L_, _A_, and that of the principal jar _F_, then this equation would exist for the example in the text: _(F + A)/(L + E) = 5_. A cause of further departure from absolute exactness is the residual charge.]
[Footnote 30: Making allowance for the corrections indicated in the preceding footnote, I have obtained for the dielectric constant of sulphur the number 3.2, which agrees practically with the results obtained by more delicate methods. For the highest attainable precision one should by rights immerse the two plates of the condenser first wholly in air and then wholly in sulphur, if the ratio of the capacities is to correspond to the dielectric constant. In point of fact, however, the error which arises from inserting simply a plate of sulphur that exactly fills the space between the two plates, is of no consequence.]
[Footnote 31: As this definition in its simple form is apt to give rise to misunderstandings, elucidations are usually added to it. It is clear that we cannot lift a quantity of electricity to _K_, without changing the distribution on _K_ and the potential on _K_. Hence, the charges on _K_ must be conceived as fixed, and so small a quantity raised that no appreciable change is produced by it. Taking the work thus expended as many times as the small quantity in question is contained in the unit of quantity, we shall obtain the potential. The potential of a body _K_ may be briefly and precisely defined as follows: If we expend the element of work _dW_ to raise the element of positive quantity _dQ_ from the earth to the conductor, the potential of a conductor _K_ will be given by _V = dW/dQ_.]
[Footnote 32: In this article the solidus or slant stroke is used for the usual fractional sign of division. Where plus or minus signs occur in the numerator or denominator, brackets or a vinculum is used.--_Tr._]