Millikan Oil Drop Apparatus
-Introduction
®
2
012-13093B
Figure 2 shows the forces acting on the drop when it is rising under the influence of an elec-
tric field. In Figure 2,
E
is the electric intensity,
q
is the charge carried by the drop, and
v
r
is
the velocity of rise. Adding the forces vectorially yields:
In both cases there is also a small buoyant force exerted by the air on the droplet. Since the
density of air is only about one-thousandth that of oil, this force may be neglected.
Eliminating
k
from equations ( 1 ) and ( 2 ) and solving for
q
yields:
To eliminate
m
from equation ( 3 ), one uses the expression for the volume of a sphere:
where
a
is the radius of the droplet, and
is the density of the oil.
To calculate
a
, one employs Stokes’ Law, relating the radius of a spherical body to its velocity of fall in a viscous
medium (with t
he coefficient of viscosity,
).
Stokes’ Law, however, becomes incorrect when the velocity of fall of the droplets is less than 0.1 cm/s. (Droplets
having this and smaller velocities have radii, on the order of 2 microns, comparable to the mean free path of air
molecules, a condition which violates one of the assumptions made in deriving Stokes’ Law.) Since the velocities
of the droplets used in this experiment will be in the range of 0.01 to 0.001 cm/s, the viscosity must be multiplied
by a correction factor. The resulting effective viscosity is:
where
b
is a constant,
p
is the atmospheric pressure, and
a
is the radius of the drop as calculated by the uncorrected
form of Stokes’ Law, equation ( 5 ).
Substituting
eff
in equation (6) into equation (5), and then solving for the radius a gives:
Substituting equations (4), (5), and (6) into equation (3) yields:
The electric intensity is given by:
Where
V
is the potential difference across the parallel plates separated by the distance
d
.
Figure 2
kv
r
mg
Eq
Eq
mg kv
r
( 2 )
+
=
q
mg v
f
v
r
+
Ev
f
---------------------------- ( 3 )
=
m
4
3
---
a
3
( 4 )
=
a
9
v
f
2
g
----------- ( 5 )
*
=
eff
1
1
b
pa
------
+
---------------
( 6 )
**
=
a
b
2
p
------
2
9
v
f
2
g
--------------
+
b
2
p
------ ( 7 )
–
=
q
6
E
------
9
3
2
g
1
b
pa
------
+
3
----------------------------------
v
f
v
r
+
v
f
( 8 )
=
E
V
d
--- ( 9 )
=