Laplace [Mec. Cel. 1. 2] gave the first direct demonstration that no function of the distance except the inverse square can satisfy the condition that a uniform spherical shell exerts no force on a particle within it. If we suppose that B, the charge of the inner sphere, is always accurately zero, or, what comes to the same thing, if we suppose B1 or B to be zero, then 9 Differentiating twice with respect to b, a being constant, and dividing by a, we find We may notice, however, that though the assumption of Cavendish, that the force varies as some inverse power of the distance, appears less general than that of Laplace, who supposes it to be any function of the distance, it is the most general assumption which makes the ratio of the force at two different distances a function of the ratio of those distances. If the law of force is not a power of the distance, the ratio of the forces at two different distances is not a function of the ratio of the distances alone, but also of one or more linear parameters, the values of which if determined by experiment would be absolute physical constants, such as might be employed to give us an invariable standard of length. Now although absolute physical constants occur in relation to all the properties of matter, it does not seem likely that we should be able to deduce a linear constant from the properties of anything so little like ordinary matter as electricity appears to be. NOTE 20, ARt. 272. On the Electric Capacity of a Disk of sensible Thickness. Consider two equal disks having the same axis, let the radius of either disk be a, and the distance between them b, and let b be small compared with a. Let us begin by supposing that the distribution on each disk is the same as if the other were away, and let us calculate the potential energy of the system. We shall use elliptical co-ordinates, such that the focal circle is the edge of the lower disk. In other words we define the position of a given point by its greatest and least distances from the edge of the lower disk, these distances being a (a + B) and a (a- ẞ). The distance of the given point from the axis is and its distance from the plane of the lower disk is (1) (2) If A, is the charge of the lower disk, the potential at the given 1 point is If A, is the charge of the upper disk, the density at any point is (3) (4) (5) We have now to multiply the charge of an element of the upper disk into the potential due to the lower disk, and integrate for the whole surface of the upper disk, Between the limits of integration we may write with a sufficient degree of approximation, so that when b is very small compared with a, the value of γ cannot differ greatly from that given by equation (11). Hence we may write the expression (10) The corresponding quantity for the action of the upper disk on itself is got by putting 4, 4, and 6 = 0, and is = In the actual case A,A,=E, where E is the whole charge, and the capacity is showing that the capacity of two disks very near together is equal to that of an infinitely thin disk of somewhat larger radius. If the space between the two disks is filled up, so as to form a disk of sensible thickness, there will be a certain charge on the curved surface, but at the same time the charge on the inner sides of the disks will disappear, and that on the outer sides near the edges will be diminished, so that the capacity of a disk of sensible thickness is very little greater than that given by (15). We may apply this result to estimate the correction for the thickness of the square plates used by Cavendish. The factor by which we must multiply the thickness in order to obtain the correction for the diameter of an infinitely thin plate of equal capacity is 1 log 2π a The correction is in every case much smaller than Cavendish supposed. NOTE 21, ARTS. 277, 452, 473, 681. Calculation of the Capacity of the Two Circles in Experiment VI. 9.3 The diameter of one of the circles was 9.3 inches, so that its capacity when no other conductor is in the field is = 2.960. The distance between their centres was 36, 24, and 18 inches, which we may call C1, C2) and C3. π The height of the centres of the circles above the floor was about 45 inches, so that the distance of the image of the circle would be about 90 inches and that of the image of the other circle would be about r = (902 + c2)3. Hence, if P is the potential of the circles when the charge of each is 1, where the first term is due to the circle itself, the second and third to the other circle, as in Note 11, and the two last to the images of the two circles. We thus find for the three distances P1 = 0.3438, P1 = 0·3567, P1 = 0·3689. 3 The capacity is 2P-1, and the number of "inches of electricity," according to the definition of Cavendish, is 4P-', or for the three cases. 11.636, 11.212, 10.844, The large circle was 18.5 inches in diameter and its centre was 41 inches from the floor, so that its charge would be 12.69 inches of electricity. Hence the relative charges are as follows: I am not aware of any method by which the capacity of a square can be found exactly. I have therefore endeavoured to find an approximate value by dividing the square into 36 equal squares and calculating the charge of each so as to make the potential at the middle of each square equal to unity. The potential at the middle of a square whose side is 1 and whose charge is 1, distributed with uniform density, is 4 log (1+√√2) = 3.52549. In calculating the potential at the middle of any of the small squares which do not touch the sides of the great square I have used this formula, but for those which touch a side I have supposed the value to be 3.1583, and for a corner square 2.9247. If the 36 squares are arranged as in the margin, and if the charges of the corner squares be taken for unity, the charges will be as follows: А В ССВА B DEED B A B C D E F 1.000 ⚫599 •562 •265 •210 •201 and the capacity of a square whose side is 1 will be 0-3607. The ratio of the capacity of a square to that of a globe whose diameter is equal to a side of the square is therefore 0·7214. |