Page:Encyclopædia Britannica, Ninth Edition, v. 8.djvu/32

Rh ELECTRICITY [ELECTEIC DISTRIBUTION. The law of electric force between two quantities q and q now becomes Force =-. a- The unit of quantity which we have just defined is called the electrostatic unit, in contradistinction to the electromagnetic unit which we shall define hereafter. Since the dimension of unit of force is [LMT~ 2 ], where L,M,T symbolize units of length, mass, and time, we have for the dimension of unit of electrical quantity [Q] [Q] = [LF HL M*!- 1 ]. * Quantitative Results concerning Distribution. Tt has already been indicated that electricity in equili brium resides on the surface of conducting bodies. We must now review shortly the experimental method by which this surface distribution has been more closely investigated. We shall state here some of the general principles arrived at, and one or two of the results, reserving others for quota tion when we come to the mathematical theory of electrical distribution. The most important experiments are due to Coulomb. He used the proof-plane and the torsion balance. Riess, who afterwards made similar experiments, used methods similar to those of Coulomb. Allusion has already been made to the use of the proof- plane, and it has been stated that whsn applied to any part of the surface of an electrified body, it brings away just as much electricity as originally occupied the part of the sur face which it covers. If, therefore, we electrify the mov able ball of the torsion balance in the same sense as the body we are to examine, and note the repulsion caused by the proof-plane when introduced in place of the fixed ball after having touched in succession two parts of the surface of the body, we can, from the indications of the balance, calculate the ratio of the quantities of electricity on the plane in the two cases, and hence the ratio of the electrical densities at the two points of the surface. We suppose, of course, that the proof-plane is small enough to allow us to assume that the electrical density is sensibly uniform over the small area covered by it. In some of his experiments Rie^s used a small sphere (about two lines in diameter) instead of the small disc of the proof-plane as Coulomb used it. The sphere in such cases ought to be very small, and even then, except in the case of plane surfaces, its use is objec tionable, unless the object be merely to determine, by twice touching the same point of the same conductor, the ratio of the whole charges on the conductor at two different limes. The fundamental requisite is that the testing body shall, when applied, alter the form of the testing body as little as possible, 1 and this requisite is best satisfied by a Email disc, and the better the smaller the disc is. The theoretically correct procedure would be to have a small portion of the actual surface of the body movable. If we could remove such a piece so as to break contact with all neighbouring portions simultaneously, then we should, by testing the electrification of this in the balance, get a perfect measure of the mean electric surface density on the removed portion. We shall see that Coulomb did employ a method like this. 1 It is evident from what we have advanced here that the use of the proof-plane to determine the electric density at points of a surface where the curvature is very great, e.g., at edges or conical points is inadmissible. If we attempt to determine the electrical density at the vertex of a cone by applying a proof-sphere there, as Riess did, we shall very evidently get a result much under the mark, owing to the blunting of the point when the sphere is in situ. We should, on the Dther hand, for an opposite reason, get too large a result by apply ing a proof-plane edgewise to a point of a surface where the curvature is continuous. There are various ways of using the torsion balance in researches on distribution. We may either electrify the movable ball independently (as above described), or we may electrify it each time by contact with the proof-plane when it is inserted into the balance. It must be noticed that the repulsion of the movable ball is in the first case proportional to the charge on the proof-plane, but in the second to the square of the charge, so that the indications must be reduced differently. In measuring we may either bring the movable ball to a fixed position, in which case the whole torsion required to keep it in this position is proportional to the charge on the proof-plane (or to its square, if the second of the above modes of operation be adopted), or we may simply observe the angle of equilibrium and calculate the quantity from that. It is supposed, for simplicity of explanation in all that follows, that the former of the two alternatives is adopted, and that the movable ball is always independently charged. The gradual loss of electricity experienced more or less by every insulated conductor has already been alluded to. This loss forms one of the greatest difficulties to be encoun tered in such experiments as we are now describing. If we apply the proof-plane to a part of a conductor and take the balance reading, giving a torsion r l say, and repeat the observation, after time t, we shall get a different torsion T 2, owing to the loss of electricity in the interval. TLis loss, partly if not mainly due to the insulating supports, depends on a great many circumstances, some of which are entirely beyond even the observation of the experimenter. We may admit, however, what experiment confirms within certain small limits, that the rate of loss of electricity is propor tional to the charge, and we shall call tUz (the loss per t unit of time on hypothesis of uniformity) the coefficient of dissipation (8), This coefficient, although, as we have im plied, tolerably constant for one experiment, will vary very much from experiment to experiment, and from day to day ; it depends above all on the weather. Supposing we have determined this coefficient by such an observation as the above, then we can calculate the torsion T, which we should have observed had we touched the body at any interval t after the first experiment ; for we have, provided t be small, T = TJ - & = T 2 + S(t - In particular, if t =  t, we have r -ifo + Tj). Coulomb used this principle in comparing the electric densities at two points A and A of the same conductor. He touched the two points a number of times in succession, first A, then A, then A again, and so on, observing the cor responding torsions TU T/, T. 2 ,T 2 ,&c., the intervals between the operations being very nearly equal. He thus got for the ratio of the densities at A and A the values T - - J&amp;gt; - --, &c. These values ought to be all T! +T 2, ZTg equal: the mean of them was taken as the best result. In certain cases, where the rapidity of the electric dissi pation was too great to allow the above method to be applied, Riess used the method of paired proof-planes. For a description of this, and for some elaborate calculations on the subject of electrical dissipation, the reader is referred to Riess s work. The cage method is well adapted for experiments on distribution. The proof-plane, proof-sphere, or paired proof-planes may all be used in conjunction with it. If the cage be fairly well insualted, and a tolerably deli cate Thomson s electrometer be used, so that the cage may