Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• In order to find the electron’s charge e, Millikan needed to know the mass M of his oil drops, and this was actually the source of his greatest uncertainty in determining e. (His value of e was about 0.5% low as a result.) However, to show that all charges are multiples of some basic unit charge, it was not necessary to know M, which cancels out of the charge ratios. The following problem illustrates this point and gives some more details of Millikan’s experiment.

By switching the E field off and on, Millikan could time an oil drop as it fell and rose through a measured distance l (falling under the influence of gravity, rising under that of E and gravity). The downward and upward speeds are given by (3.42) as

Both speeds were measured in the form v = l/t, where t is the time for the droplet to traverse l.

(a) By adding the two equations (3.67) and rearranging, show that

 

where the quantity K = E/(6πrηl) was constant as long as Millikan watched a single droplet and did not vary E.

From (3.68) we see that the charge q is proportional to the quantity (1/td) + (1/tu). Thus if it is true that q is always an integer multiple of e, it must also be true that (1/td) + (1/tu) is always an integer multiple of some fixed quantity. Table 1 shows a series of timings for a single droplet (made with a commercial version of the Millikan experiment used in a teaching laboratory). That tu changes abruptly from time to time indicates that the charge on the droplet has changed as described in Section 3.11. (The times td should theoretically all be the same, of course. The small variations give you a good indication of the uncertainties in all timings. For td you should use the average of all measurements of td.)

Table 1

A series of measurements of td and tu, the times for a single droplet to travel down and up a fixed distance l. All times are in seconds.

td:	15.2	15.0	15.1	15.0	14.9	15.1	15.1	15.0	15.2	15.2
tu:	6.4	6.3	6.1	24.4	24.2	3.7	3.6	1.8	2.0	1.9

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(b) Calculate (1/td) + (l/tu) and show that (within the uncertainties) this quantity is always an integer multiple of one fixed number, and hence that the charge is always a multiple of one fixed charge.

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(c) Use Equation (3.44) for vd and the additional data in Table 3.8 to find the radius r of the droplet.

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(d) Finally, use (3.68) to find the four different charges q on the droplet. What is the best estimate of e based on this experiment?

Additional data for the oil-drop experiment.

* For very small droplets, there is a small correction to the formula (3.42) for the terminal velocity. For the experiment reported here, this correction amounts to a 12% reduction in the viscosity. We have included this correction in the value given.