We distinguish two different cases, depending on the sign of the electric
field. In both cases we pick the positive direction of the z
axis to be up, and project the forces on it. Gravity is always down,
the sign of the electrostatic force is defined by the voltage on the upper
plate. Initially, if the electrostatic force on the droplet is positive
(i.e., oriented up) the droplet will be moving up, and the viscous drag
force is then negative (oriented down). The velocity is measured by measuring
time, and by convention time is taken to positive, t > 0,
when the droplet is moving up.
Alternatively, if the electric field is oriented up (with V <
0), the electric field is aiding gravity, and the droplet is moving
down, ending up with a ballance of forces that is shown in the sketch below.
The case when there is no electric field (V = 0) is a
special case of "Case 2," since the droplet is moving down due to gravity.
After replacing 
and 
, the ballance
of forces can be summed in the following three equations:
Where ne is the charge of the droplet, V
is voltage, s is the distance between the plates, a
is the radius of the droplet, 
and 
are the densities of the
oil and the air respectively, and 
is the viscosity of air at room temperature. The velocity is measured
by measuring time 
(or 
) during
which the droplet travels a standard distance d. NOTE:
defined this way,
can be both positive and negative, depending the sign of the field.
The measurement begins by ionizing the chamber and letting the falling
oil droplets `sweep' electrons or ions. By slowly changing V
we let only one droplet remain, and with it we proceed to measure
and multiple
times. The data is analyzed with a collection of measurements for
many drops and many charges. It is convenient to rewrite these equations
as:
where we defined 
and 
.
Defined this way, both A and B are positive.
B
is determined from the measurement of .
In the expression for A, we see that the only quantity that
depends on a particular droplet is its radius, a. So
if one performs a serious of measurements with different charges on the
same drop, one can prove that the charge is quantized, merely by
plotting the distribution of An = 
.
However, it turns out that in our setup it is
hard to keep the same droplet for a long time. Therefore, the best
strategy is to take triplets of measurements (field up, field down, no
field), and then from
and calculate
the charge of each droplet q = ne.
Task #1: plot the distribution of the charge of each droplet, q. Show that the charge is quantized. NOTE: it is vital to obtain as many values of q as possible, and also to try to get decent statistics in the first couple of peaks, i.e., q = e, 2e, 3e...
Task #2: determine e. There are several ways to do this, and the choice is left to you. Be creative! The key is to assign n to each value of q. Here is what the students did in the past:
Links: the PASCO manual with the technical description of the equipment can be found here.