Hence the electromotive force required to produce a spark between convex surfaces, as in Lane's electrometer, is less than if the surfaces had been plane and at the same distance. When the air-space is large, the path of the sparks, and therefore the electromotive force required to produce them, is exceedingly irregular. The accompanying figure is from a photograph of a succession of sparks taken between the same electrodes from four Leyden jars charged by Holtz's machine. A portion of the path near the positive electrode is nearly straight, there is then a sharp turn, which, in all the sparks represented, is in the same direction. Beyond this the course of the spark is very irregular, although its general direction is deflected towards the same side as the first sharp turn. Theory of two circular disks on the same axis, their radii being small compared with the distance between them. A circular disk may be considered as an ellipsoid, two of whose axes are equal, while the third is zero, and we may apply the method of ellipsoidal co-ordinates to the calculation of the potential*. In the * See Ferrers' Spherical Harmonics, p. 136. case before us everything is symmetrical about the axis, so that we have to consider only the zonal harmonics, and of these only those of even order, unless we wish to distinguish between the surface density on opposite sides of the same element of the disk, for this depends on the harmonics of odd orders. Let a be the radius of the first disk, b that of the second, and c the distance between them. We shall use ellipsoidal co-ordinates confocal with the first disk. Let and r, be the greatest and least distances respectively of a given point from the edge of the disk, and let r, then if is the distance of the point from the plane of the disk, and r its distance from the axis, If the surface-density of the electricity on the disk is a function of the distance from the axis, it may be expressed in the form 2n and P is the zonal harmonic of order 2n. Only even orders are admissible, for since every element of the disk corresponds to two values of μ, numerically equal but of opposite signs, a term involving an harmonic of odd order would give the surface-density everywhere zero. The potential arising from this distribution at any point whose ellipsoidal co-ordinates are w = αμ and η= av In this expression Q', (v) denotes a series, the terms of which are numerically equally to those of Q., (v), the zonal harmonic of the second kind, but with the second and all even terms negative. If we put i for 1, we may write This expression is an infinite series, the terms of which increase without limit when v is diminished without limit. TWO DISKS. It may, however, be expressed in the finite form* 389 (11) (12) that is to say P',(v) is a zonal harmonic of the first kind with all its terms positive, and Z ̧ (v) is a rational and integral function of v of 2n − 1 degrees, which is such as to cancel all the terms of P', (v) tan-1 which do not vanish when v becomes infinite. The expression (11) is applicable to small as well as great values of v. Thus we find when v is 0, as it is at the surface of the disk, The potential at any point of the disk is therefore the sum of a series of terms, the general form of which is On the axis, μ = 1 and av=z, and the potential is the sum of a series of terms, the general form of which is Since we have to determine the value of the potential arising from the first disk at a point in the second disk for which z=c at a distance r from the axis, and if we write where b is the radius of the second disk, and p is a quantity corresponding to μ in the first disk, then the most convenient expression for the potential due to the first disk at a point (p) in the second, is where U denotes the value of the potential at the axis, and where, after the differentiations, va is to be made equal to c. To investigate the mutual action of the two disks, let us assume that the surface-density on the second disk is the sum of a number of terms of which the general form is The potential at the surface of the second disk arising from this distribution will be the sum of a series of terms of the form π 1 (2n !)2 2 b 2 (!)* Ban Pan (P). (19) The potential arising from the presence of the first disk is given in equation (17). Having thus expressed the most general symmetrical distribution of electricity on the two disks and the potentials thence arising, we are able to calculate the potential energy of the system in terms of the squares and products of the two sets of coefficients A and B. If W denotes the potential energy, W = 1 ffovds, Vd (20) when the integration is to be extended over every element of surface ds. Confining our attention to the second disk, the part of W thence arising is and the part arising from the term in the density whose coefficient is B is 2n The part of the value of V which arises from the electricity on the second disk itself is the sum of a series of terms of the form (19). The surface-integral of the product of any two of these of different orders is zero, so that in finding the potential energy of the disk on itself we have to deal only with terms of the form B2 1 (2n!) 1 * 4 b 2TM (n!)* 4n + 1 (23) The energy arising from the mutual action of the disks consists of terms whose coefficients are products of A's and B's, and in calculating these we meet with the integral* (1 − p3)"P ̧„„(p) dp− (− 1)" 22m. m + n ! m! m! 2n! 2m+2n+1!m (24) n!n!n!" We have, therefore, for the harmonic of order zero. Surface-density on the first disk, σ = of the first disk. A 1 where A is the charge * I am indebted for the general value of this integral to Mr W. D. Niven, of Trinity College. |