The fundamental charge discovered by Millikan has a specific value of
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[Numbers change] In the early 1900s, Robert Millikan discovered the peculiar property that charge came in little packets, no smaller than e = 1.602 x 10^-19 C - he had measured the charge of the electron. Here's (roughly) how he did it: He removed an electron from an initially neutral droplet of oil with a diameter of 0.8 um. In a vacuum, he positioned the droplet between two metallic plates separated by 5 mm and fiddled with the potential (voltage) across the plates until the droplet would hover against the force of gravity: Droplets of this size with +e charge would hover, but only for a particular voltage (otherwise they would sink or rise). Given the parameters stated here, and the fact that the density of the oil was 933 kg/m^3, what was the voltage that made the droplets hover? (Give your answer with 0.1 V precision)
Madhur L.
Actual data from one of Millikan's early experiments are as follows: $$\begin{array}{l} a=0.000276 \mathrm{cm} \\ \rho=0.9561 \mathrm{g} / \mathrm{cm}^{3} \end{array}$$ Average time of fall $=11.894 \mathrm{s}$ Rise or fall distance $=10.21 \mathrm{nm}$ Plate separation $=16.00 \mathrm{mm}$ Average potential difference between plates $=5085 \mathrm{V}$ Sequential rise times in seconds: 80.708,22.336,22.390 22.368,140.566,79.600,34.748,34.762,29.286,29.236 Find the average value of $e$ by requiring that the difference in charge for drops with different rise times be equal to an integral number of elementary charges.
$\textbf{The Millikan Oil-Drop Experiment.}$ 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$^{-4}$ mm in diameter) into the space between two parallel horizontal plates separated by a distance $d$. A potential difference $V_{AB}$ 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 $$q = \frac{4\pi}{3} \frac{\rho{r^3gd}}{V_{AB}}$$ where $\rho$ is oil's density. (Ignore the buoyant force of the air.) By adjusting $V_{AB}$ 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 determined $r$ by cutting off the electric field and measuring the $terminal \space speed \space v_t$ of the drop as it fell. (We discussed terminal speed in Section 5.3.) 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: $F = 6\pi \eta rv$. When a drop fell at $v_t$, the viscous force just balanced the drop’s weight $w =$ mg. Show that the magnitude of the charge on the drop was In your apparatus, the separation $d$ between the horizontal plates is 1.00 mm. The density of the oil you use is 824 kg/m3. For the viscosity $\eta$ of air, use the value 1.81 $\times$ 10$^{-5}$ N $\cdot$ s/m$^2$. Assume that $g =$ 9.80 m/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 $q = -ne$ 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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