Chapter V · THE EXACT EVALUATION OF

I. DISCOVERY OF THE FAILURE OF STOKES’S LAW

Although complete evidence for the atomic nature of electricity is found in the fact that all of the charges which can be placed upon a body as measured by the sum of speeds , and all the changes of charge which this body can undergo as measured by the differences of speed () are invariably found to be exact multiples of a particular speed, yet there is something still to be desired if we must express this greatest common divisor of all the observed series of speeds merely as a velocity which is a characteristic constant of each particular drop but which varies from drop to drop. We ought rather to be able to reduce this greatest common divisor to electrical terms by finding the proportionality factor between speed and charge, and, that done, we should, of course, expect to find that the charge came out a universal constant independent of the size or kind of drop experimented upon. The attempt to do this by the method which I had used in the case of the water drops (p. 55), namely, by the assumption of Stokes’s Law, heretofore taken for granted by all observers, led to the interesting discovery that this law is not valid.[45] According to this law the rate of fall of a spherical drop under gravity, namely, is given by in which is the viscosity of the medium, the radius and the density of the drop, and the density of the medium. This last quantity was neglected in (6), p. 55, because, with the rough measurements there possible, it was useless to take it into account, but with our oil drops in dry air all the other factors could be found with great precision.

When we assume the foregoing equation of Stokes and combine it with equation (5) on p. 55, an equation whose exact validity was proved experimentally in the last chapter, we obtain, after substitution of the purely geometrical relation , the following expression for the charge carried by a drop loaded with electrons which we will assume to have been counted by the method described: According to this equation the elementary charge should be obtained by substituting in this the greatest common divisor of all the observed series of values of () or (). Thus, if we call this ( we have

But when this equation was tested out upon different drops, although it yielded perfectly concordant results so long as the different drops all fell with about the same speed, when drops of different speeds, and, therefore, of different sizes, were used, the values of obtained were consistently larger the smaller the velocity under gravity. For example, ex for one drop for which per second came out , while for another of almost the same speed, namely, , it came out 5.482; but for two drops whose speeds were five times as large, namely, .0536 and .0553, came out 5.143 and 5.145, respectively. This could mean nothing save that Stokes’s Law did not hold for drops of the order of magnitude here used, something like cm. (see Section IV below), and it was surmised that the reason for its failure lay in the fact that the drops were so small that they could no longer be thought of as moving through the air as they would through a continuous homogeneous medium, which was the situation contemplated in the deduction of Stokes’s Law. This law ought to begin to fail as soon as the inhomogeneities in the medium—i.e., the distances between the molecules—began to be at all comparable with the dimensions of the drop. Furthermore, it is easy to see that as soon as the holes in the medium begin to be comparable with the size of the drop, the latter must begin to increase its speed, for it may then be thought of as beginning to reach the stage in which it can fall freely through the holes in the medium. This would mean that the observed speed of fall would be more and more in excess of that given by Stokes’s Law the smaller the drop became. But the apparent value of the electronic charge, namely, is seen from equation (13) to vary directly with the speed ( imparted by a given force. Hence should come out larger and larger the smaller the radius of the drop, that is, the smaller its velocity under gravity. Now, this was exactly the behavior shown consistently by all the oil drops studied. Hence it looked as though we had discovered, not merely the failure of Stokes’s Law, but also the line of approach by means of which it might be corrected.

In order to be certain of our ground, however, we were obliged to initiate a whole series of new and somewhat elaborate experiments.

These consisted, first, in finding very exactly what is the coefficient of viscosity of air under conditions in which it may be treated as a homogeneous medium, and second, in finding the limits within which Stokes’s Law may be considered valid.

II. THE COEFFICIENT OF VISCOSITY OF AIR

The experiments on the coefficient of viscosity of air were carried out in the Ryerson Laboratory by Dr. Lachen Gilchrist,[46] and Dr. I. M. Rapp.[47] Dr. Gilchrist used a method which was in many respects new and which may fairly be said to be freer from theoretical uncertainties than any method which has ever been used. He estimated that his results should not be in error by more than .1 or .2 of 1 per cent. Dr. Rapp used a form of the familiar capillary-tube method, but under conditions which seemed to adapt it better to an absolute evaluation of for air than capillary-tube arrangements have ordinarily been.

These two men, as the result of measurements which were in progress for more than two years, obtained final means which were in very close agreement with one another as well as with the most careful of preceding determinations.