Calling the potential of the outer shell A, and that of the inner B, we find by what precedes, In the first part of the experiment the shells communicate by the short wire and are both raised to the same potential, say V. Putting AB = V and solving equations (16), (17), we find for the charge of the inner shell B = 21b bf (2a)-a{f(a + b) -ƒ (a - b)} (18) In the original experiment of Cavendish the hemispheres forming the outer shell were removed altogether from the globe and discharged. The potential of the inner shell or globe would then be In the form of the experiment as repeated at the Cavendish Laboratory, the outer shell was left in its place, but was connected to earth, so that A=0. In this case we find for the potential of the inner shell when tested by the electrometer Let us now assume with Cavendish, that the law of force is some inverse power of the distance, not differing much from the inverse square, that is to say, let If we suppose q to be a small numerical quantity, we may expand f(r) by the exponential theorem in the form and if we neglect terms involving q2, equations (19) and (20) become (23) (24) (25) Laplace [Meo. 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 B or B, to be zero, then bf (2a)-af (a+b) — af (a - b) = 0. 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 7' = ααβ, and its distance from the plane of the lower disk is z = a (ao — 1) 3 (1 − ß3)3. (1) (2) If A, is the charge of the lower disk, the potential at the given point is = Aa1 cosec a, or, if we write a2 = y2+1, 4 = Aa1 2 (-tany). (3) (4) (5) If A, is the charge of the upper disk, the density at any point is 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 y 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 A1 = A, and b = 0, and is = 1 In the actual case A, A, E, where E is the whole charge, and the capacity is or, since in our approximation we have neglected (-) be expressed with sufficient accuracy in the form (14) 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 2π log 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 = 2.960. The distance = The diameter of one of the circles was 9.3 inches, so that its capacity when no other conductor is in the field is 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 is 1, r = (902 + c2)3. Hence, if P is the potential of the circles when the charge of each 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, 2 P1 = 0·3567, P= 0·3689. |