The Millikan Oil-Drop Experiment. The charge of an electron was first measured
by the American physicist Robert Millikan during \(1909-1913 .\) In his
experiment, oil was sprayed in very fine drops (about \(10^{-4} \mathrm{~mm}\)
in diameter) into the space between two parallel horizontal plates separated
by a distance \(d\). A potential difference \(V_{A B}\) was maintained between the
plates, causing a downward electric field between them. Some of the oil drops
acquired a negative charge because of frictional effects or because of
ionization of the surrounding air by x rays or radioactivity. The drops were
observed through a microscope. (a) Show that an oil drop of radius \(r\) at rest
between the plates remained at rest if the magnitude of its charge was
\(q=\frac{4 \pi}{3} \frac{\rho r^{3} g d}{V_{A B}}\) where \(\rho\) is oil's
density. (Ignore the buoyant force of the air.) By adjusting \(V_{A B}\) to keep
a given drop at rest, Millikan determined the charge on that drop. provided
its radius \(r\) was known. (b) Millikan's oil drops were much too small to
measure their radii directly. Instead, Millikan determincd \(r\) by cutting off
the electric field and measuring the terminal speed \(u_{\mathrm{i}}\) of the
drop as it fell. (We discussed terminal speed in Section \(\left.5.3 .\right)\)
The viscous force \(F\) on a sphere of radius \(r\) moving at speed \(v\) through a
fluid with viscosity \(\eta\) is given by Stokes's law: \(F=6 \pi \eta r v .\)
When a drop fell at \(v_{1}\), the viscous force just balanced the drop's weight
\(w=m g\). Show that the magnitude of the charge on the drop was \(q=18 \pi
\frac{d}{V_{A B}} \sqrt{\frac{\eta^{3} v_{1}^{3}}{2 \rho g}}\) (c) You repeat
the Millikan oil-drop experiment. Four of your measured values of \(V_{A B}\)
and \(v_{1}\) are listed in the table: $$
\begin{array}{lcccc}
\text { Drop } & 1 & 2 & 3 & 4 \\
\hline V_{A B}(\mathrm{~V}) & 9.16 & 4.57 & 12.32 & 6.28 \\
v_{1}\left(10^{-5} \mathrm{~m} / \mathrm{s}\right) & 2.54 & 0.767 & 4.39 &
1.52
\end{array} $$ In your apparatus, the separation \(d\) between the horizontal
plates is \(1.00 \mathrm{~mm}\). The density of the oil you use is \(824
\mathrm{~kg} / \mathrm{m}^{3}\). For the viscosity \(\eta\) of air, use the value
\(1.81 \times 10^{-5} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\). Assume
that \(g=9.80 \mathrm{~m} / \mathrm{s}^{2}\).
Calculate the charge \(q\) of each drop. (d) If electric charge is quantized
(that is, exists in multiples of the magnitude of the charge of an electron),
then the charge on each drop is -ne, where \(n\) is the number of excess
electrons on each drop. (All four drops in your table have negative charge.)
Drop 2 has the smallest magnitude of charge observed in the experiment, for
all 300 drops on which measurements were made, so assume that its charge is
due to an excess charge of one electron. Determine the number of excess
electrons \(n\) for each of the other three drops. (e) Use \(q=-n e\) to calculate
\(e\) from the data for each of the four drops, and average these four values to
get your best experimental value of \(e .\)
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