V con stant in hollow oondue- ior. Indirect evidence for the law of inverse square. Heneril problem jf elec- irical listribu-
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ELECTROSTAT1CAL THEORY.] is always a level surface, it follows, from what we have already proved about a space bounded by a surface of con stant potential, that, inside a holloiv conductor the potential is constant, provided there be no electrified bodies within. This is true, no matter how we electrify the conductor or what electrified bodies there may be outside. Hence, if we inclose any conductor A completely within another B, then electrify B and put A in metallic communication with it, A will not become charged either + or - ; for, A being at the same potential as B, electricity will not tend to flow from the one to the other. This is in reality Biot s 1 ex periment with the hemispheres, to which we have already alluded ; only the point of view is slightly changed. The most striking experiment ever made in illustration of the present principle is that described by Faraday in his Experimental Researches. He constructed a hollow cube (12 feet in the edge) of conducting matter, and insulated it iu the lecture-room of the Royal Institution. We quote in his own words the part of his description which bears on the present question : "1172. I put a delicate gold leal electrometer within the cube, and then charged the whole by an outside communication, very strongly for some time 2 together ; but neither during the charge or after the discharge did the electrometer or air within show the least sign of electricity ] went into the cube and lived in it, and using all other tests of electrical states, I could not find the least influ ence upon them, though all the time the outside of the cube was powerfully charged, and large sparks and brushes were darting oft from every point of its outer surface." The proposition that the potential is constant inside a hollow conductor containing no electrified bodies may be regarded as one of the most firmly established in the whole of experimental science. The experiments on which it rests are of extreme delicacy. It is of the greatest theoretical importance; for we can deduce from it the law of the inverse square. Taking the particular case of a spherical shell, uninfluenced by other bodies, on which of course the electrical distribution must from symmetry be uniform, it can be demonstrated mathematically that, if we assume the action between two elements of electricity to be a function of the distance between them, then that function must be the inverse square, in order that the potential may be constant throughout the interior. A demonstration of this proposition was given by Caven dish, who saw its importance ; a more elaborate proof was afterwards given by Laplace ; for a very elegant and simple demonstration we refer the mathematical reader to Clerk Maxwell s Electricity, vol. i. 74. This must be regarded as by far the most satisfactory evidence for the law of the inverse square ; for the delicacy of the tests involved infinitely surpasses that of the measurements made with the torsion balance; and now that we have instruments of greatly increased sensitiveness, like Thom son s quadrant electrometer, the experimental evidence might be still further strengthened. In the problem to determine the distribution of elec tricity in a given system of conductors, the data are in most cases either the charge or the potential for each con ductor. If the conductor is insulated it can neither give nor lose electricity, its charge is therefore given. If, on the other hand, it be connected with some inexhaustible source of electricity at a constant potential, its potential is given. Such a source the earth is assumed to be ; and we shall henceforth take the potential of the earth as zero, and reckon the potential of all other bodies with reference to it. If all our electrical experiments were con- J The experiment was first made by Cavendish. There is an account of H in his hitherto unpublished papers. 4 Faraday was looking for what he called the absolute charge of matter ; incidentally the experiment illustrates the point we are dis cussing. 27 ducted in a space inclosed by a perfectly conducting enve lope, the potential of this envelope would be the natural zero of our reckoning. It will be useful to analyse more closely the distribution on a system of conductors, in order to see how far the above data really determine the solution of the electrical problem. For this purpose the following proposition is useful. If e v e. 2 , . . . . e n be the charges at the points Principle 1, 2, .... n of any system, and V the potential at P, ofelec- and if V be the potential at P due to <?/, e z , . . . . < at lrical 1, 2, .... 72, then the potential at P due to e l + e 1 , S itj n{ e 2 + e 2> at 1, 2, . . . . is Y + V. This principle fol lows at once from the definition of the potential as a sum formed by the mere addition of parts due to all the single elements of the system. Applied to a system cf conductors in equilibrium, it may evidently be stated thus: If E^ E 2 , . . . . E n and Vj, V 2 , .... Y n be the respective charges and potentials for the conductors 1, 2, 3 .... n in a state of equilibrium, E/, E/, .... E,, and Y 1 , Y^ .... V B corresponding charges and potentials for another state of equilibrium, then Ej + E/ E B + , V 1 + V/ V n + V n will be corresponding charges and potentials for a third state of equilibrium. Suppose that in the system of conductors 1, 2, 3, n the con- Particu- ductor 1 is kept at potential 1 and all the others at potential zero, lar case then it can be shown that there is one and only one distribution of of general electricity fulfilling these conditions. Mathematically stated, the problem, problem is to determine a function V, which shall satisfy the equa tion v 2 V = throughout the space unoccupied by conductors, and have the values 1, 0, 0, was respectively at each point of the surfaces of 1, 2, n respectively. Consider the integral T -i dV dxdydz . (16), where the integration is extended all over the space unoccupied by conductors. If we consider all the values which this integral may have, consistent with the boundary conditions V = l, V = 0, &c. at the surfaces of 1, 2, &c., it is obvious that there must be a minimum value ; for the integral is essentially positive, and cannot become less than zero. dSV 2 /~/YsYv 2 Vdxdydz (17) by partial integration. The surface terms vanish, since 8V = at every surface. Hence v 2 V = is the condition for a maximum or minimum value of I, and since we know that a minimum value exists, there must be a solution of this equation. It can, moreover, be shown, by a method which we shall apply below to the more general problem, that there is only one solution of v 2 V = con sistent with the given conditions, and this will of course be that which makes I a minimum. If our mathematical methods were powerful enough to determine V, we might proceed to find the surface density for each conductor by means of the formula 1 dV <T*~ --T- -T- : then the charges on the conductors could be found 47T dv by means of the integral - - -j-rfS . In very few cases indeed could we actually find these cnargos ; we have, however, de- monstrated their existence and shown that our problem is definite. Let these charges on 1, 2, . . . n be called q l -,, q l 2 ( l n - Coeffi- Corresponding to the data 0, 1,0, .... for the potentials dents cf of 1, 2, ... n, we should get a series of charges q 2 V q 22) .... capacity <7 9 n , and so on ; <?, ,, q n <,,q,., . . are called the coefficients of * lul ,!"~ ic- i ,- *11*22X88 , i o Q ductiuo. self-induction or capacity for the conductors 1,2,6,... , <7j o, <?! 3 , &c., are called the coefficients of induction of 1 on 2, 1 on 3, etc. It is obvious that these coefficients depend solely on the form and relative position of the conductors. It follows, from the principle of the superposition, that, if 1, 2, ... n be at the potentials Y 1? 0, ... 0, then the
charges on them will bo ^ jYj, q l ^V l .... q- n Y We