Part 14

In view of the fact that the same electrical bodies exert at different distances different forces upon one another, exception might be taken to the measure of quantity here developed. What kind of a quantity is that which now weighs more, and now weighs less, so to speak? But this apparent deviation from the method of determination commonly used in practical life, that by weight, is, closely considered, an agreement. On a high mountain a heavy mass also is less powerfully attracted to the earth than at the level of the sea, and if it is permitted us in our determinations to neglect the consideration of level, it is only because the comparison of a body with fixed conventional weights is invariably effected at the same level. In fact, if we were to make one of the two weights equilibrated on our balance approach sensibly to the centre of the earth, by suspending it from a very long thread, as Prof. von Jolly of Munich suggested, we should make the gravity of that weight, its heaviness, proportionately greater.

Let us picture to ourselves, now, two different electrical fluids, a positive and a negative fluid, of such nature that the particles of the one attract the particles of the other according to the law of the inverse squares, but the particles of the same fluid repel each other by the same law; in non-electrical bodies let us imagine the two fluids uniformly distributed in equal quantities, in electric bodies one of the two in excess; in conductors, further, let us imagine the fluids mobile, in non-conductors immobile; having formed such pictures, we possess the conception which Coulomb developed and to which he gave mathematical precision. We have only to give this conception free play in our minds and we shall see as in a clear picture the fluid particles, say of a positively charged conductor, receding from one another as far as they can, all making for the surface of the conductor and there seeking out the prominent parts and points until the greatest possible amount of work has been performed. On increasing the size of the surface, we see a dispersion, on decreasing its size we see a condensation of the particles. In a second, non-electrified conductor brought into the vicinity of the first, we see the two fluids immediately separate, the positive collecting itself on the remote and the negative on the adjacent side of its surface. In the fact that this conception reproduces, lucidly and spontaneously, all the data which arduous research only slowly and gradually discovered, is contained its advantage and scientific value. With this, too, its value is exhausted. We must not seek in nature for the two hypothetical fluids which we have added as simple mental adjuncts, if we would not go astray. Coulomb's view may be replaced by a totally different one, for example, by that of Faraday, and the most proper course is always, after the general survey is obtained, to go back to the actual facts, to the electrical forces.

We will now make ourselves familiar with the concept of electrical quantity, and with the method of measuring or estimating it. Imagine a common Leyden jar (Fig. 29), the inner and outer coatings of which are connected together by means of two common metallic knobs placed about a centimetre apart. If the inside coating be charged with the quantity of electricity +q, on the outer coating a distribution of the electricities will take place. A positive quantity almost equal to the quantity +q flows off to the earth, while a corresponding quantity-q is still left on the outer coating. The knobs of the jar receive their portion of these quantities and when the quantity q is sufficiently great a rupture of the insulating air between the knobs, accompanied by the self-discharge of the jar, takes place. For any given distance and size of the knobs, a charge of a definite electric quantity q is always necessary for the spontaneous discharge of the jar.

Let us insulate, now, the outer coating of a Lane's unit jar L, the jar just described, and put in connexion with it the inner coating of a jar F exteriorly connected with the earth (Fig. 30). Every time that L is charged with +q, a like quantity +q is collected on the inner coating of F, and the spontaneous discharge of the jar L, which is now again empty, takes place. The number of the discharges of the jar L furnishes us, thus, with a measure of the quantity collected in the jar F, and if after 1, 2, 3, ... spontaneous discharges of L the jar F is discharged, it is evident that the charge of F has been proportionately augmented.

Let us supply now, to effect the spontaneous discharge, the jar F with knobs of the same size and at the same distance apart as those of the jar L (Fig. 31). If we find, then, that five discharges of the unit jar take place before one spontaneous discharge of the jar F occurs, plainly the jar F, for equal distances between the knobs of the two jars, equal striking distances, is able to hold five times the quantity of electricity that L can, that is, has five times the capacity of L.

We will now replace the unit jar L, with which we measure electricity, so to speak, into the jar F, by a Franklin's pane, consisting of two parallel flat metal plates (Fig. 32), separated only by air. If here, for example, thirty spontaneous discharges of the pane are sufficient to fill the jar, ten discharges will be found sufficient if the air-space between the two plates be filled with a cake of sulphur. Hence, the capacity of a Franklin's pane of sulphur is about three times greater than that of one of the same shape and size made of air, or, as it is the custom to say, the specific inductive capacity of sulphur (that of air being taken as the unit) is about 3. We are here arrived at a very simple fact, which clearly shows us the significance of the number called dielectric constant, or specific inductive capacity, the knowledge of which is so important for the theory of submarine cables.

Let us consider a jar A, which is charged with a certain quantity of electricity. We can discharge the jar directly. But we can also discharge the jar A (Fig. 33) partly into a jar B, by connecting the two outer coatings with each other. In this operation a portion of the quantity of electricity passes, accompanied by sparks, into the jar B, and we now find both jars charged.

It may be shown as follows that the conception of a constant quantity of electricity can be regarded as the expression of a pure fact. Picture to yourself any sort of electrical conductor (Fig. 34); cut it up into a large number of small pieces, and place these pieces by means of an insulated rod at a distance of one centimetre from an electrical body which acts with unit of force on an equal and like-constituted body at the same distance. Take the sum of the forces which this last body exerts on the single pieces of the conductor. The sum of these forces will be the quantity of electricity on the whole conductor. It remains the same, whether we change the form and the size of the conductor, or whether we bring it near or move it away from a second electrical conductor, so long as we keep it insulated, that is, do not discharge it.

A basis of reality for the notion of electric quantity seems also to present itself from another quarter. If a current, that is, in the usual view, a definite quantity of electricity per second, is sent through a column of acidulated water; in the direction of the positive stream, hydrogen, but in the opposite direction, oxygen is liberated at the extremities of the column. For a given quantity of electricity a given quantity of oxygen appears. You may picture the column of water as a column of hydrogen and a column of oxygen, fitted into each other, and may say the electric current is a chemical current and vice versa. Although this notion is more difficult to adhere to in the field of statical electricity and with non-decomposable conductors, its further development is by no means hopeless.

The concept quantity of electricity, thus, is not so aerial as might appear, but is able to conduct us with certainty through a multitude of varied phenomena, and is suggested to us by the facts in almost palpable form. We can collect electrical force in a body, measure it out with one body into another, carry it over from one body into another, just as we can collect a liquid in a vessel, measure it out with one vessel into another, or pour it from one into another.

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.

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. 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.

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. 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 = (8P).

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.

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.

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