the velocity of the sphere. This radius is called the critical radius. But it was not known how near it was possible to approach to the critical radius. Arnold’s experiments showed that the inertia of the medium has no appreciable effect upon the rate of motion of a sphere so long as the radius of that sphere is less than .6 of the critical radius.
Application of this result to the motion of our oil drops established the fact that even the very fastest drops which we ever observed fell so slowly that not even a minute error could arise because of the inertia of the medium. This meant that the fifth condition necessary to the application of Stokes’s Law was fulfilled. Furthermore, our drops were so small that the second condition was also fulfilled, as was shown by the work of both Ladenburg and Arnold. The third condition was proved in the last chapter to be satisfied in our experiments. Since, therefore, Arnold’s work had shown very accurately that Stokes’s Law does hold when all of the five conditions are fulfilled, the problem of finding a formula for replacing Stokes’s Law in the case of our oil-drop experiments resolved itself into finding in just what way the failure of assumptions 1 and 4 affected the motion of these drops.
[IV]. CORRECTION OF STOKES’S LAW FOR INHOMOGENEITIES IN THE MEDIUM
The first procedure was to find how badly Stokes’s Law failed in the case of our drops. This was done by plotting the apparent value of the electron
against the observed speed under gravity. This gave the curve shown in Fig. 4, which shows that though for very small speeds
varies rapidly with the change in speed, for speeds larger than that corresponding to the abscissa marked 1,000 there is but a slight dependence of
on speed. This abscissa corresponds to a speed of .1 cm. per second. We may then conclude that for drops which are large enough to fall at a rate of 1 cm. in ten seconds or faster, Stokes’s Law needs but a small correction, because of the inhomogeneity of the air.