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Millikan’s measurement of the charge on the electron is one of the few truly crucial experiments in physics and, at the same time, one whose simple directness serves as a standard against which to compare others. Figure 3-4 shows a sketch of Millikan’s apparatus. With no electric field, the downward force on an oil drop is mg and the upward force is bv. The equation of motion is
where b is given by Stokes’ law:
3-
and where is the coefficient of viscosity of the fluid (air) and a is the radius of the drop. The terminal velocity of the falling drop vf is
b 6 a
mg bv m
dv dt
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Millikan’s Oil-Drop
Experiment
Light source
(+) (–)
(+)
(–)
Atomizer
Telescope Fig. 3-4 Schematic diagram of the Millikan oil-drop apparatus. The drops are sprayed from the atomizer and pick up a static charge, a few falling through the hole in the top plate. Their fall due to gravity and their rise due to the electric field between the capacitor plates can be observed with the telescope. From measurements of the rise and fall times, the electric charge on a drop can be calculated. The charge on a drop could be changed by exposure to x rays from a source (not shown) mounted opposite the light source.
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(see Figure 3-5). When an electric field is applied, the upward motion of a charge qn is given by
Thus the terminal velocity vr of the drop rising in the presence of the electric field is
In this experiment, the terminal speeds were reached almost immediately, and the drops drifted a distance L upward or downward at a constant speed. Solving Equa- tions 3-12 and 3-13 for qn , we have
where Tf L / vf is the fall time and Tr L / vr is the rise time. If any additional charge is picked up, the terminal velocity becomes , which is related to the new charge by Equation 3-13:
The amount of charge gained is thus
The velocities vf , vt , and are determined by measuring the time taken to fall or rise the distance L between the capacitor plates. If we write qn ne and where n is the change in n, Equations 3-14 and 3-15 can be written
and
n
T r
Tr ^
e mgTf
n
Tf
Tr ^
e mgTf
q n qn n e
v r
mgTf
T r
Tr
q n qn
mg vf
( v r vr )
v r
q n mg b
q n
v r
qn
mg vf
( vf vr )
mgTf
Tf
Tr
vr
qn mg b
q (^) n mg bv m
dv dt
vf
mg b
Droplet
Buoyant forcebv
Weightmg
e
Fig. 3-5 An oil droplet carrying an ion of charge e falling at terminal speed, i.e., mg bv.
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Rise and fall times of a single oil drop with calculated number
of elementary charges on drop
T (^) f Tr n 1/ Tf 1/ T (^) r n (1/ n )(1/ T (^) r 1/ T (^) f ) 11.848 80.708 0.09655 18 0. 11.890 22.366 0.03234 6 0.005390 0.12887 24 0. 11.908 22. 11.904 22. 11.882 140.566 0.03751 7 0.005358 0.09138 17 0. 11.906 79.600 0.005348 1 0.005348 0.09673 18 0. 11.838 34.748 0.01616 3 0.005387 0.11289 21 0. 11.816 34.
1/ T r 1/ Tr (1/ n )(1/ T r 1/ Tr )
