516 



ELECTRICITY. 



Bbori 19 



. : M. Poi- 



too. 



CM of two 



kodra 



When two 

 unequal 



In coo tact. 



that, t the Mirface of spheroid, differiiij; very little 

 from a sphere, the repulsive force of tin- fluid in each 

 point is proportional to the thicknc-- of the coat .it that 

 point ; and the name i* true for ellip-oid- of revolution, 

 whatever be the ratio of their axes. Hen- 

 approaching to sphere, and in ellipsoids of revolution, 

 the electrical repulsion is greatest in thooe places where 

 the accumulation of the electricity is a maximum. 



In the determination of the electrical state of two 

 or more conducting bodies, placed within the >|)lu-re of 

 each other* activity. M. 1'oisson has laid down the fol- 

 lowing general principle, which holds good whether 

 rach of the conducting bodies is covered over all its 

 surface with only a single fluid, cither vitreous or rc.i/- 

 HOtu, or whether, in consequence of their mutual in- 

 fluence, one or more of the bodies are partly covered 

 by the vitreous and partly by the re.-inoii^ fluid. This 

 principle is thus enunciated: 



It' several electrified conducting Ixxlies are placed 

 in the presence of each other, and arrive at a perma- 

 nent electric state, it is necessary that the resulting 

 ! rising from the actions of all the electric coats, 

 which cover them upon any point in the interior of one 

 f the bodies, be equal to nothing." 



If this force is not equal to nothing, it will act upon 

 the natural fluid which the different bodies contain, and 

 consequently a new quantity of this fluid will be de- 

 composed, and their electric state will be changed. On 

 the contrary, when this force is nothing, it is easy to 

 nee that the coat of electricity which is distributed over 

 each body is in a state of equilibrium at its surface, so 

 that the preceding principles contains the only condi- 

 tion which it is necessary to consider. 



This principle will furnish, in any particular case, as 

 many equations as there are conductors, and these equa- 

 tions will serve to determine the variable thickness of 

 the coat of fluid which envelopes these different bodies. 



In his first memoir, M. Poisson has confined him- 

 self to the case of two spheres formed of matter which 

 is a perfect conductor, and placed at any distance from 

 ach other ; ami after having shewn how to reduce the 

 equations formed from the general principle, to ordina- 

 ry equations with variable differences, and a single in- 

 dependent variable quantity, he resolves the problem 

 in two different cases: 1st, When the two spheres are 

 in contact ; and, 2d, When their distance is very great 

 in relation to one of their radii. 



\\ hen two unequal spheres are in contact, the equa- 

 tions may be integrated in a very simple manner by 

 definite integrals, and they afford the following re- 

 sults. At the point of contact of the two spheres, the 

 thickness of the coat of electricity will be nothing, that 

 is, there will be no electricity at the point of contact ; a 

 rc-ult precisely the same as that which was obtained 

 by Coulomb from direct experiment. See pp. 452, 4SS. 



In the neighlxmrhood of the point of contact, and to 

 a considerable distance from it, the coat is very thin, 

 and the electricity very weak upon the two spheres. 

 When it amounts to a sensible quantity, it is at first 

 most intense on the largest of the two spheres, but af- 

 terwards it increases at the greatest rate upon the small- 

 est sphere, so that at 180 from contact it is always 

 greater in the smaller sphere than on the corresponding 

 point of the greater sphere. This result harmonizes 

 also with Coulomb's experiments to far ns they go. 



When the two spheres are separated, it follows from 

 the theory, that each carries off the whole quantity of 

 electricity with which it was covered ; and when they 

 are at the sphere of their mutual action, the electricity 

 is uniformly distributed upon each sphere. 



In determining the electrical densities of two une- 

 qual glolms in contact, M. I'ois.-im finds, that the ratio 

 in which the electricity is divided bit ween the two 

 -. is id ways less than that of the surfaces; so that, 

 alter separation, tle mean thickness of the coat of' fluid 

 is alwa\- K re.ile,t on the -niallest of the two i; 

 The ratio between these two thick ne-.es tends towards 

 a eon-tant limit, which is equal to the square of the 

 ratio of the circumference to the diameter divided by 

 six, which is very nearly aa 5 to 3 ; that is. if a very 

 -mall spherical conducting body is placet! u|>on an 

 electrified spherical conductor of considerable -i/c, the 

 electricity will he divided between the two bodi, 

 the ratio of about 5 times the Mirtiicc of the small 

 sphere to three times the surface of the great sphere. 

 The experiments of Coulomb, of which we have givcu 

 a full account in pa^c l.V_', enable us again to examine 

 the theory by experiment. Coulomb found, that the ra- 

 tio to which the electrical density constantly approach- 

 ed, was 0, which is the same as 6 to 3, while the theo- 

 ry gives the ratio of 1.67, or of 5 to 3. This difference 

 is, at lir-t sight, a little more than might have been ex- 

 pected; but \vlven we consider the difficulty of deter- 

 mining such a limit experimentally, we shall rather be 

 surprised at the coincidence between the theory and 

 experiment. It ought to be considered, too, that Cou- 

 lomb always found the ratio below 2, or below (j to 3 ; 

 anil the highest ratio which he appears, from his Ta- 

 ble, to have found, is 1.65 : which is almost exactly the 

 same as that obtained by M. 1'oisson, and is, perliaps, 

 not far from the limiting ratio. 



M. Poison proceeds to apply the analysis to a new 

 case, where these two fluids occur at the same time on 

 the surface of the same boly. This takes place in the 

 case of two spheres placed at a distance, which is great 

 when compared to one of the twc radii. If the smaller 

 of the two spheres is not electrified directly, but mere- 

 ly from being within the atmo-phere of the greater 

 sphere, it will, of course, be possessed of an electricity 

 opposite to that of the great sphere. This electricity 

 accumulates towards the point that is least distant from 

 the great sphere, while the electricity similar to that of 

 the great sphere accumulates on the opposite point. 

 The two opposite electricities on those two opposite 

 points are almost equal ; and the neutral line which sepa- 

 rate - them, almost coincides with the great circle which 

 is perpendicular to the line joining the sectors of the 

 globes, and divides the smaller globe into two equal 

 parts. By means of very simple formulas, the electri- 

 cal density, and the character of the electricity, may 

 be determined for any part of the two surfaces. This 

 result is confirmed by Coulomb's experiments, so far as 

 they go. See Exp. 3. and 4, col. 2. p. 4-53. 



In ins second Memoir, M. I'oisson has given the ge- 

 neral integrals of the two equations of the problem, first 

 under the form of a series, and then under a finite form, 

 by means of definite integrals. By the nature of these 

 equations, their integrals contain an arbitrary periodical 

 function, which seems to indicate, that the problem is 

 indeterminate, or that the distribution of the electric 

 fluid, the law of which depends on these integrals, may 

 take place in an infinite variety of ways. M. Foisson, 

 however, has demonstrated, in a rigorous manner, that 

 tins function is foreign to the question, and that the 

 term which contains it ought to be suppressed. After 

 doing this, he obtains series which contain only de- 

 terminate quantities, and whicli express the thickness 

 of the coat of fluid, or what is the same thing, the elec- 

 tric density at any point on the two surfaces of the 

 spheres. Except in the case where the two spheres are 



Theoretiul 



icily. 



Case of tw 

 sphere! at a 

 distance. 



