Page:Encyclopædia Britannica, Ninth Edition, v. 8.djvu/34

ELECTRICITY [ELECTROSTATICAL THEORY. 180 respectively, the electric density had the relative values 20, 77, 96, I OO. When the spheres are unequal the distribution is no longer alike on each. On the small sphere it is less uniform, and the density at the point of the small sphere diametrically opposite the point of con tact is greater than anywhere else on the body. The distribution on the larger sphere is more uniform than on the smaller, and the more unequal the spheres are the more uniform is the distribution on the larger, and the .smaller the unelectrified part in the neighbourhood of the point of contact. The following results of Coulomb are useful illustrations of distribution on elongated and pointed bodies : Three equal spheres (2 in. diameter) in contact, with their centres in the same straight line : the mean densities were 1 34, I OO, 1 34 on the spheres 1, 2, and 3 respectively. Six equal spheres as before: mean densities on 1, 2, and ? = 1-56, 1-05, I OO. Twelve equal spheres: mean densities on 1, 2, and 6 = 1 70, [14, 1-00. Twenty-four equal spheres : mean densities on 1, 2, and 12 = 175, 1-07, I OO. Large (8 in. diameter) sphere with four small (2 in.) spheres applied to it, all the centres in line: the mean density on large sphere being 1, that on the small one next it was 60 that on the extreme small one 2 08. Large sphere 1 , and twenty-four (2 to 25) small ones : mean densities on 1, 2, 13, 24, 25 = 1-00, "60, 1 28, 1 46, 2 17. MATHEMATICAL THEORY OF ELECTRICAL EQUILIBRIUM. We take as the basis of our theory the assumptions already laid down under the head Provisional Theory, and in addition the precise elementary law of electrical action established by Coulomb. We shall also suppose that we have only perfect conductors and perfect non-conductors to deal with, the medium being in all cases the same. viz., air. When we have to deal with electrified non-con ductors we shall suppose the electrification to be rigid, i.e. incapable of disturbance by any electric force we have to consider. In our mathematical outline we have in view the requirements of the physical more than the mathematical student, and shall pass over many points of great interest and importance to the latter, for full treatment of which we must refer him to original sources, such as the classical papers of Green, the papers of Sir William Thomson, and the works of Ganss. Of peculiar interest mathematically is the elegant and powerful memoir of the last Allgemcine Lehr- sdtze in Ueziehung auf die im vcrkehrten Verhaltnisse des Quadrats der Entfemung wirkenden Anziehungs- und Abstossungskrafte, in which will be found detailed discussions of the continuity of the integrals used in the potential theory, &c. The works of Green and Thomson are too well known in this country to require farther remark. Oefini- When, in what follows, we speak of the electric field, we tions. mean simply a portion of space which we are considering with reference to its electrical properties ; it will be found conducive to clearness to regard that space as bounded. In general the natural boundary would be the walls of the experimenting room ; but, for mathematical purposes, we shall, unless the contrary is stated, suppose our field to be bounded by a sphere of radius so great that the action at a point on its circumference due to an electrified body in the field is infinitely small. The resultant force at a point in the electric field is the force which would be exerted on a unit of + electricity placed there without disturbing the electrical distribution else where. It is plain that the resultant force has a definite magnitude and direction at every point in the field, and consequently is in modern mathematical language a vector. A curve drawn in the field such that its tangent at every point is in the direction of the resultant force at that point is called a line of force. We can draw such a line through every point of space, and if we suspend at any point a small conducting needle, it is obvious, from what we have already laid down about induction, that it will take up a position very nearly parallel to the line of force ; so that if we start from any point and carry the centre of the needle always in the direction in which the needle points we should trace out a line of force. The potential at any point is the work done by a unit of + electricity in passing from that point to the infinitely distant boundary of the electric field, the electric distri bution being supposed undisturbed. It is usual to call the infinitely distant boundary a place of zero potential. Zero is to be understood in the sense of " point or posi tion from which we reckon." 1 Consider two points P, Q, infinitely near each other Force in the field, and draw a curve from P passing through terms Q to oo . Then, if F be the component parallel to 1 Q of the resultant force at P, we have by our definition or in differential notation hence F= - ~ds V =J fds =" ( Kdx + Ydy + Zdz) . (1), (2), and where V denotes the potential at P, and X,Y,Z the components parallel to the co-ordinate axes of the resultant electric force. We clearly have as particular cases of (1) v A. = i dx v- ~ dy (3). We may remark that, in all cases which we shall consider at pre sent, the work done in passing from any point to any other point is the same whatever the intermediate path of our exploring unit. Hence V as above denned is a single valued function, and th for mula; (3) gives the components of resultant force when V is known. The work done by a unit of + electricity in passing by any path from P to Q is called the electromotive force from P to Q; it is obviously equal to the difference of the poten tials at the two points. Thus Y i ,-V Q =/ p Q (X ( ^+ Ydy+Zd:) . . . (4). is the electromotive force from P to Q. Suppose we concentrate m units of electricity at any point P, and Expre require the potential due to this at a point Q, distant D from P. sion Applying (2), and, since any path to co may be chosen, taking the of V i integral along the production of PQ to oo , we get terms -ao definit V = / dr =- (5, ./i> r" D If we have any number of discrete points with charges HI,, w_, K? 3 ,.. .. at distances D 1; D 2 , D 3 , ...... from Q, since the work done by the exploring unit under the action of the whole is got by adding tip the work done under the action of each part separately, we clearly have From this we may pass to the case of a continuous volume distribu tion. If p be the volume density at the point r; and V the potential at xyz, we have where D denotes + /s <l~ a ) :! + ( T 7~2/) !! + (C- 2 ) 2 > and the integral is to be extended over every part of the field where there is any charge, or, which is the same thing, over the whole field, on the understanding that p = where there is no charge. If, as will generally be the case, the electricity is distributed on a surface in such a way that on an element dS of surface there is a quantity <rdS of electricity, where a- is a finite surface density, then where I) has the same meaning as before, and the integral is rx- tended all over the electrified surface or surfaces. 1 It may be well here to warn the reader that measurement of potential is relative, just as much as measurement of distance is, and

to caution him against the fallacious idea of absolute zero of potential.