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ON THE DROPLET METHOD FOR DETERMINING THE ELECTRON CHARGE
The droplet method for determining the electron charge and, consequently, Avogadro’s number, proposed in 1909 by Millikan, remains to this day the most accurate, surpassed only by the X-ray method. However, the values obtained by different authors in measurements with droplets differ substantially from one another and from the value \(e = 4.8024\) electrostatic units, to which, according to the latest data, the X-ray method leads. These discrepancies, amounting to several units in the third decimal place, considerably exceed the probable errors, which indicates that some factors have not been taken into account. It has long been clear that the main cause of the discrepancies should be sought in an inaccurate determination of the viscosity of air. If the matter were only an inaccurate knowledge of the viscosity, then the ratio \(e^{2/3}/\eta_{23}\) (where \(\eta_{23}\) is the coefficient of viscosity of air at \(23^\circ\) C) would be the same in all experiments. In reality it proves to be different, ranging approximately from 334 to 337 units \(CGSE\).
The paper under review*) is devoted to a systematic investigation of the droplet method with the aim of clarifying the nature of these discrepancies. The electric field in Hopper’s experiments was almost vertical. Drops were introduced through an orifice \(0.368\) mm in diameter in the upper capacitor plate, located at a distance of \(2\) mm from the center. The velocity was measured by photographing the droplets at the center of the capacitor at intervals of \(1/5\) second. The apparatus made it possible, by means of a small displacement of the air, to return the drops to their initial position and repeat the measurement process. The experiments were carried out with drops of various oils.
As a result, the influence of certain previously unaccounted-for factors was found.
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The time for which the air remains in the apparatus affects the value of \(e^{2/3}/\eta\); namely, when the air remained in the apparatus for several weeks, \(e^{2/3}/\eta\) increased by \(0.3\%\) in comparison with fresh air. The author believes that contamination of the air by the walls of the apparatus occurs.
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The presence of walls entails noticeable deviations from Stokes’ law in the direction of decreasing \(e^{2/3}/\eta\).
) V. D. Hopper, Nature 163*, 733 (1949).
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Oil droplets, after remaining in air for a long time, oxidize or adsorb air, as a result of which the velocity of their fall under the action of gravity increases linearly with time, while the velocity of their rise under the action of the electric field decreases linearly with time. For different oils these effects are expressed to different degrees.
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The cleanliness of the wire mesh on which the spraying is carried out has a very significant influence. In some cases even contamination of the droplets by metallic inclusions was observed.
Taking all this into account, as well as a number of other corrections, the author processed measurements with forty-seven droplets, while varying the potential difference between the plates (2000 and 4000 V) and the value \(1/pr\) (where \(p\) is the pressure and \(r\) is the droplet radius) from 21.2 to 174.4. As a result, extrapolating to \(\frac{1}{pr}=0\), he obtained the value
\[ \frac{e^{2/3}}{\eta}=(335.75\pm 0.11)\,10^{-5}\ \mathrm{CGSE}. \]
The results of the measurements are shown in part in the figure. Assuming \(e=4.8024\times 10^{-10}\), the value \(\eta_{23}=1826.5\cdot 10^{-7}\ \mathrm{CGSE}\) was found. Recalculations of earlier data, both Hopper’s own and those of other authors, with the necessary, rather significant corrections taken into account, led to values of \(e^{2/3}/\eta\) differing only in the fourth decimal place. Approximately the same discrepancies occur among values of \(\eta\) obtained by other methods (rotating cylinder, capillary method).
Thus Hopper succeeded in establishing certain sources of error in the droplet method which had previously escaped attention, and in somewhat improving the convergence of the results. Hopper himself believes that the accuracy of the electrical and mechanical measurements carried out in the process of determining \(e^{2/3}/\eta\) is sufficiently high, and that the accuracy of determining the electron charge is entirely determined by the accuracy of determining the viscosity. However, this opinion is hardly fully justified. The discrepancies among the values of \(e^{2/3}/\eta\) have not yet been eliminated and exceed the probable errors by about an order of magnitude. If the task is set of attaining an accuracy comparable with the accuracy of the X-ray method, then, along with improving the technique of measuring \(\eta\), further development of the droplet method itself will undoubtedly be required.
R. G.