CHAPTER VTHE 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 dropunder gravity, namely,is given byin whichis the viscosity of the medium,the radius andthe density of the drop, andthe 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) onp. 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 chargecarried by a drop loaded withelectrons which we will assume to have been counted by the method described:According to this equation the elementary chargeshould 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 dropsall fell with about the same speed, when drops of different speeds, and, therefore, of different sizes, were used, the values ofobtained were consistently larger the smaller the velocity under gravity. For example, ex for one drop for whichper 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 likecm. (seeSection IVbelow), 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.Henceshould 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 offor 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.
It will be seen from Table IX that every one of the five different methods which have been used for the absolute determination offor air leads to a value that differs by less than one part in one thousand from the following mean value,.It was concluded, therefore, that we could depend upon the value offor the viscosity of air under the conditions of our experiment to at least one part in one thousand. Very recently Dr. E. Harrington[48]has improved still further the apparatus designed by Dr. Gilchrist and the author and has made with it in the Ryerson Laboratory a determination ofwhich is, I think, altogether unique in its reliability and precision. I give to it alone greater weight thanto all the other work of the past fifty years in this field taken together. The final value isand the error can scarcely be more than one part in two thousand.
III. LIMITS OF VALIDITY OF STOKES’S LAW
In the theoretical derivation of Stokes’s Law the following five assumptions are made: (1) that the inhomogeneities in the medium are small in comparison with the size of the sphere; (2) that the sphere falls as it would in a medium of unlimited extent; (3) that the sphere is smooth and rigid; (4) that there is no slipping of the medium over the surface of the sphere; (5) that the velocity with which the sphere is moving is so small that the resistance to the motion is all due to the viscosity of the medium and not at all due to the inertia of such portion of the media as is being pushed forward by the motion of the sphere through it.
If these conditions were all realized then Stokes’s Law ought to hold. Nevertheless, there existed up to the year 1910 no experimental work which showed that actual experimental results may be accurately predicted by means of the unmodified law, and Dr. H. D. Arnold accordingly undertook in the Ryerson Laboratory to test how accurately the rates of fall of minute spheres through water and alcohol might be predicted by means of it.
His success in these experiments was largely due to the ingenuity which he displayed in producing accurately spherical droplets of rose-metal. This metal melts at about 82° C. and is quite fluid at the temperature of boiling water. Dr. Arnold placed some of this metal in a glasstube drawn to form a capillary at one end and suspended the whole of the capillary tube in a glass tube some 70 cm. long and 3 cm. in diameter. He then filled the large tube with water and applied heat in such a way that the upper end was kept at about 100° C., while the lower end was at about 60°. He then forced the molten metal, by means of compressed air, out through the capillary into the hot water. It settled in the form of spray, the drops being sufficiently cooled by the time they reached the bottom to retain their spherical shape. This method depends for its success on the relatively slow motion of the spheres and on the small temperature gradient of the water through which they fall. The slow and uniform cooling tends to produce homogeneity of structure, while the low velocities allow the retention of very accurately spherical shape. In this way Dr. Arnold obtained spheres of radii from .002 cm. to .1 cm., which, when examined under the microscope, were found perfectly spherical and practically free from surface irregularities. He found that the slowest of these drops fell in liquids with a speed which could be computed from Stokes’s Law with an accuracy of a few tenths of 1 per cent, and he determined experimentally the limits of speed through which Stokes’s Law was valid.
Of the five assumptions underlying Stokes’s Law, the first, third, and fourth were altogether satisfied in Dr. Arnold’s experiment. The second assumption he found sufficiently realized in the case of the very smallest drops which he used, but not in the larger ones. The question, however, of the effect of the walls of the vessel upon the motion of drops through the liquid contained in the vessel had been previouslystudied with great ability by Ladenburg,[49]who, in working with an exceedingly viscous oil, namely Venice turpentine, obtained a formula by which the effects of the wall on the motion might be eliminated. If the medium is contained in a cylinder of circular cross-section of radiusand of length,then, according to Ladenburg, the simple Stokes formula should be modified to readArnold found that this formula held accurately in all of his experiments in which the walls had any influence on the motion. Thus he worked under conditions under which all of the first four assumptions underlying Stokes’s Law were taken care of. This made it possible for him to show that the law held rigorously when the fifth assumption was realized, and also to find by experiment the limits within which this last assumption might be considered as valid. Stokes had already found from theoretical considerations[50]that the law would not hold unless the radius of the sphere were small in comparison with,in whichis the density of the medium,its viscosity, andthe 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 isless 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 electronagainst the observed speed under gravity. This gave the curve shown in Fig. 4, which shows that though for very small speedsvaries rapidly with the change in speed, for speeds larger than that corresponding to the abscissa marked 1,000 there is but a slight dependence ofon 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 tenseconds or faster, Stokes’s Law needs but a small correction, because of the inhomogeneity of the air.
i004Fig. 4
Fig. 4
Fig. 4
To find an exact expression for this correction we may proceed as follows: The average distance which a gas molecule goes between two collisions with its neighbors, a quantity well known and measured with some approach to precision in physics and called “the mean free path”of a gas molecule, is obviously a measure of the size of the holes in a gaseous medium. When Stokes’s Law begins to fail as the size of the drops diminish, it must be because the medium ceases to be homogeneous, as looked at from the standpoint of the drop, and this means simply that the radius of the drop has begun to be comparable with the mean size of the holes—a quantity which we have decided to take as measured by the mean free path.The increase in the speed of fall over that given by Stokes’s Law, when this point is reached, must then be some function of.In other words, the correct expression for the speedof a drop falling through a gas, instead of beingas Arnold showed that it was when the holes were negligibly small—as the latter are when the drop falls through a liquid—should be of the formIf we were in complete ignorance of the form of the functionwe could still express it in terms of a series of undetermined constants,etc., thusand so long as the departures from Stokes’s Law were small asFig. 4showed them to be for most of our drops, we could neglect thesecond-order terms inand have thereforeUsing this corrected form of Stokes’s Law to combine with (9) (p. 20), we should obviously get the chargein just the form in which it is given in (13), save that wherever a velocity appears in (13) we should now have to insert in place of this velocity.And since the velocity of the drop appears in the ³⁄₂ power in (13), if we denote now by e the absolute value of the electron and byas heretofore, the apparent value obtained from the assumption of Stokes’s Law, that is, from the use of (13), we obtain at onceIn this equationcan always be obtained from (13), whileis a known constant, but,,andare all unknown. Ifcan be found our observations permit at once of the determination of bothand,as will be shown in detail under Section VI (seep. 105).
However, the possibility of determiningif we knowcan be seen in a general way without detailed analysis. For the determination of the radius of the drop is equivalent to finding its weight, since its density is known. That we can find the charge on the drop as soon as we can determine its weight is clear from the simple consideration that the velocity under gravity is proportional to its weight, while the velocity in a given electrical field is proportional to the chargewhich it carries. Since we measure these two velocities directly, we can obtain either the weight, if we know the charge, or the charge, if we know the weight. (See equation 9,p. 70.)
V. WEIGHING THE DROPLET
The way which was first used for finding the weight of the drop was simply to solve Stokes’s uncorrected equation (11) (p. 91) for a in the case of each drop. Since the curve ofFig. 4shows that the departures from Stokes’s Law are small except for the extremely slow drops, and sinceappears in the second power in (11), it is clear that, if we leave out of consideration the very slowest drops, (11) must give us very nearly the correct values of.We can then find the approximate value ofby the method of the next section, and after it is found we can solve (15) for the correct value of.This is a method of successive approximations which theoretically yieldsandwith any desired degree of precision. As a matter of fact the whole correction term,is a small one, so that it is never necessary to make more than two approximations to obtainwith much more precision than is needed for the exact evaluation of.
As soon aswas fairly accurately known it became possible, as indicated above, to make a direct weighing of extraordinarily minute bodies with great certainty and with a very high degree of precision. For we have already shown experimentally that the equationis a correct one and it involves no assumption whatever as to the shape, or size, or material of the particle. If we solve this equation for the weightof the particle we getIn this equationis known with the same precision as,for we have learned how to count.It will presently be shown thatis probably now known with an accuracy of one part in a thousand, hencecan now be determined with the same accuracy for any body which can be charged up with a counted numberof electrons and then pulled up against gravity by a known electrical field, or, if preferred, simply balanced against gravity after the manner used in the water-drop experiment and also in part of the oil-drop work.[51]This device is simply an electrical balance in place of a mechanical one, and it will weigh accurately and easily to one ten-billionth of a milligram.
Fifty years ago it was considered the triumph of the instrument-maker’s art that a balance had been made so sensitive that one could weigh a piece of paper, then write his name with a hard pencil on the paper and determine the difference between the new weight and the old—that is, the weight of the name. This meant determining a weight as small as one-tenth or possibly one-hundredth of a milligram (a milligram is about ¹⁄₃₀₀₀₀ of an ounce). Some five years ago Ramsay and Spencer, in London, by constructing a balance entirely out of very fine quartz fibers and placing it in a vacuum, succeeded in weighing objects as small as one-millionth of a milligram, that is, they pushed the limitof the weighable down about ten thousand times. The work which we are now considering pushed it down at least ten thousand times farther and made it possible to weigh accurately bodies so small as not to be visible at all to the naked eye. For it is only necessary to float such a body in the air, render it visible by reflected light in an ultra-microscope arrangement of the sort we were using, charge it electrically by the capture of ions, count the number of electrons in its charge by the method described, and then vary the potential applied to the plates or the charge on the body until its weight is just balanced by the upward pull of the field. The weight of the body is then exactly equal to the product of the known charge by the strength of the electric field. We made all of our weightings of our drops and the determination of their radii in this way as soon as we had locatedwith a sufficient degree of precision to warrant it.[52]Indeed, even beforeis very accurately known it is possible to use such a balance for a fairly accurate evaluation of the radius of a spherical drop. For when we replacein (18) byand solve for a we obtainThe substitution in this equation of an approximately correct value ofyieldswith an error but one-third as great as that contained in the assumed value of,foris seen from this equation to vary as the cube root of.This is the method which, in view of the accurate evaluation of,it is now desirable touse for the determination of the weight or dimensions of any minute body, for the method is quite independent of the nature of the body or of the medium in which it is immersed. Indeed, it constitutes as direct and certain a weighing of the body as though it were weighed on a mechanical balance.
VI. THE EVALUATION OFAND
Withandknown, we can easily determineandfrom the equationfor if we write this equation in the formand then plot the observed values ofas ordinates and the corresponding values ofas abscissae we should get a straight line, provided our corrected form of Stokes’s Law (15) (p. 101) is adequate for the correct representation of the phenomena of fall of the droplets within the range of values ofin which the experiments lie. If no such linear relation is found, then an equation of the form of (15) is not adequate for the description of the phenomena within this range. As a matter of fact, a linear relation was found to exist for a much wider range of values ofthan was anticipated would be the case.
i005Fig. 5
Fig. 5
Fig. 5
i006Fig. 6
Fig. 6
Fig. 6
The intercept of this line on the axis of ordinates, that is, the value ofwhenis seen from (20) to beand we have but to raise this to the ³⁄₂ power to obtain the absolute value of.Again,is seen from (20) to be merely the slope of this line divided by the intercept on theaxis.
In order to carry this work out experimentally it is necessary to varyand find the corresponding values of.This can be done in two ways. First, we may hold the pressure constant and choose smaller and smaller drops with which to work, or we may work with drops of much the same size but vary the pressure of the gas in which our drops are suspended, for the mean free pathis evidently inversely proportional to the pressure.
Both procedures were adopted, and it was found that a given value ofalways corresponded to a given value of,no matter whetherwas kept constant andreduced to, say, one-tenth of its first value, orkept about the same andmultiplied tenfold. The result of one somewhat elaborate series of observations which was first presented before the Deutsche physikalische Gesellschaft in June, 1912, and again before the British Association at Dundee in September, 1912,[53]is shown in Figs.5and6. The numerical data from which these curves are plotted are given fairly fully in Table X. It will be seen that this series of observations embraces a study of 58 drops. These drops represent all of those studied for 60 consecutive days, no single one being omitted.
Mean of first 17 numbers in the last column = 61.120Mean of last 6 numbers in the last column = 61.138
Mean of all numbers in last column = 61.138Mean of first 23 numbers = 61.120
They represent a thirty-fold variation in(from .016, drop No. 1, to .444, drop No. 58), a seventeen-fold variation in the pressure(from 4.46 cm., drop No. 56, to 76.27 cm., drop No. 10), a twelvefold variation in(fromcm., drop No. 28, tocm., drop No. 1), and a variation in the number of free electrons carried by the drop from 1 on drop No. 28 to 136 on drop No. 56.