Millikin’s oil drop experiment 1

Demonstrating the Quantization of Electrical Charge –

Millikan’s Experiment

Objectives of the experiment To demonstrate that electrical charge is quantized, and to determine the elementary

electron charge by observing the motion of charged oil drops in an electric field.

Introduction

The charge of the electron is one of the most important fundamental constants in nature.

The ratio of the mass to charge of the electron may be readily determined via the

observation of the path of a pre-accelerated electron through a magnetic field. However,

the determination of the charge alone is a little more difficult. The first attempt was

performed by Thomson in 1896, using a cloud chamber, and resulted in a value close to

1.1×10-19 Coulombs, but with a large error. The more precise measurements of Millikan

were performed in 1911 in his now famous oil drop experiment.

The apparatus he used is

shown schematically in Fig. 1.

For each oil droplet with a

charge q there is a

gravitational force downwards

(mg), and a buoyancy force

upwards (bv, b is the

buoyancy and v is the terminal

velocity of the oil drop) which are equal when the terminal velocity is reached (i.e. mg =

bv). The equation of motion is thus,

dt

dvmbvmg =− (1)

The buoyancy may be obtained from Stoke’s law

ab πη6= (2)

where a is the radius of the drop and η is the viscosity of air.

The terminal velocity of the oil drop is thus

a

mg

b

mgv f

πη6== (3)

Now if an electric field, E, is applied to the two plates shown in Fig. 1, then for a positive

voltage applied to the top plate, there will be a corresponding upward motion of the oil

drop which possesses charge q.

dt

dvmbvmgqE =−− (4)

[Note the change in sign of the buoyancy force].

In this instance the terminal velocity is

+/- V

-/+ V

Fig. 1. Schematic diagram of the Millikan apparatus

Millikin’s oil drop experiment 2

a

mgEq

b

mgEqvr

πη6

−=

−= (5)

In Millikan’s experiment the terminal velocities are achieved rapidly, and only the

motion of the oil drops with terminal velocity is observed. If the oil drops are observed

to move a distance L in times Tf (fall) and Tr (rise), then it is possible to solve the

equations (3) and (5) for q.

( )rf

f

vvEv

mgq += (6)

This is however expressed in terms of the effective mass of the oil drop moving through

the air, where

)(3

4 3

airoilam ρρπ −= (7)

This can be used in conjunction with equation 3 to calculate an expression for a

g

va

airoil

f

)(2

9

ρρ

η

−= (8)

and thus m can be calculated and thus the corresponding charge

g

vvv

Eq

airoil

f

rf)(

2)(

19

3

ρρ

ηπ

−+= (9)

Noting that the electric field can be expressed in terms of the voltage between the two

plates and their distance, then

g

vvv

V

dq

airoil

f

rf)(

2)(9

3

ρρ

ηπ

−+= (10)

The two equations (8) and (10) are required for the rise and fall method for determining

q.

There is a second technique for determining q, which is the float technique. For these

measurements the appropriate equations are;

g

va

airoil

f

)(2

9

ρρ

η

−= and

g

v

V

dq

airoil

f

)(

29

33

ρρ

ηπ

−= (11)

You might attempt deriving the latter two equations (11) yourself.

Millikin’s oil drop experiment 3

Apparatus

The equipment to be used in the measurement is shown in Fig. 2. The Millikan

equipment should be connected to the voltage control system and timer as shown Fig. 3.

Fig. 2. Diagram of the Millikan

experimental apparatus

Fig. 3. The connections and switch functions for the voltage control system and

timer.

Millikin’s oil drop experiment 4

Make sure the equipment is correctly connected before switching on, if in doubt consult a

demonstrator. The oil atomizer should be filled such that the bent capillary tube stands

about 2 mm in the oil, and the spray nozzle should be positioned such that it points

towards the bore holes in the plastic cover. The oil used has a density of 877 kgm-3 at 15 oC, and 871 kgm-3 at 25 oC. The capacitor plates have a diameter of 8 cm and a separation

of 6(±0.05) mm. The telescope itself is equipped with an eyepiece with a magnification

of 10, and the objective magnification of the telescope is 2(±0.05). The telescope is used

to locate the position of the oil drops by measurements against the scale provided. The

length of the micrometer scale is 10 mm, with 0.1 mm graduations.

Procedure Setting up the equipment

- Turn the lens holder of the micrometer eyepiece until you can clearly see the

micrometer scale.

- If necessary, turn the eyepiece to orient the micrometer scale vertically. For this purpose

you should slightly loosen the fastening screw. Since falling droplets are observed on the

micrometer scale as rising droplets due to the reversion of the image in the microscope,

the scale start (0) should point upward and the scale end should point downward (10).

- Use the knurled knob to push the measuring microscope close to the plastic cover. The

illuminated capacitor plates can be seen at the top and bottom in the circular-viewing

field. The beginning and end of the micrometer scale are at a small distance to the

capacitor plates.

Do not attempt the following without consulting the demonstrator first.

To eliminate disturbing light reflections or to correct the observation region, if you are

not satisfied with the illumination:

- Loosen the fastening screw of the capacitor and move the capacitor.

- You can also adjust the lamp with the help of the adjusting screw (recessed head screw).

Observing oil droplets - Use the rubber ball to spray oil between the capacitor plates so that oil droplets can be

seen in the entire observation field.

- By moving the measuring microscope, create a plane, in which a selected oil droplet is

clearly seen as light point.

Objective magnification

Due to the objective magnification M, a fall or rise distance s of the oil droplet between

the capacitor plates is represented on the scale section

Msx =

If the image of an oil droplet moves in the time ∆t on the scale over a distance ∆x, the

velocity of the oil droplet is

tM

xv

∆

∆=

The objective magnification is M = 2 quite accurately. For more exact measurements, you

should determine the magnification:

Millikin’s oil drop experiment 5

- Remove the plate capacitor and put a suitable scale vertically on the base plate.

- Adjust the microscope so that external scale and micrometer scale can be clearly seen

next to one another. By comparing the two scales, determine the exact magnification.

- Then, do not move the eyepiece any more.

Timer/Counter Operation

o Set mode to “tE,F”.

o Press start until corresponding LED is lit

o Cable from clock1 on Millikan control box should be connected to E.

o Cable from clock2 on Millikan control box should be connected to F.

o Zero timer: press “→0←”.

o Times can be read out using button tE,F – when E LED is lit, first time is given i.e.

time between start of clock and event E. When F LED is list this gives the time

between events E and F.

Fall/rise method

The fall velocity vf and the rise velocity vr are determined from the fall time tf and rise

time tr for a pre-selected distance s. The following equations can then be used for the

radius a and the charge q of the droplet (see the introduction).

g

va

airoil

f

)(2

9

ρρ

η

−=

g

vvv

V

dq

airoil

f

rf)(

2)(9

3

ρρ

ηπ

−+=

- Zero counter press “→0←”.

- First turn switch U and switch t downward.

- Use switch U to turn on the capacitor voltage and adjust it using a rotary potentiometer

so that a selected oil droplet rises.

As soon as the oil droplet is in the upper area of the capacitor:

- Use switch U to turn off the capacitor voltage.

As soon as the oil droplet is next to a pre-selected graduation scale mark:

- Use switch t to start measuring the fall time.

As soon as the oil droplet has fallen over a pre-selected distance:

- Use switch U to turn on the capacitor voltage, to end measurement of the fall time and

start measurement of the rise time.

As soon as the oil droplet has risen over the same pre-selected distance s :

- Use switch t to end time measurement.

- Read fall time, rise time (press button tE,F) and capacitor voltage U and record with fall

or rise distance s.

Float Method

The float potential U and the fall speed v are determined from the fall time t for a pre-

selected distance s. The following applies for the radius a and the charge q of the droplet:

g

va

airoil

f

)(2

9

ρρ

η

−= and

g

v

V

dq

airoil

f

)(

29

33

ρρ

ηπ

−=

Millikin’s oil drop experiment 6

- Zero counter press “→0←”.

- First turn switch U and switch t downward.

- Use switch U to turn on capacitor voltage, then adjust it using a rotary potentiometer so

that a selected oil droplet floats.

- Use switch U to turn off the capacitor voltage.

As soon as the oil droplet is next to a selected scale graduation mark:

- Use switch t to start time measurement.

As soon as the oil droplet has fallen over a pre-selected distance:

- Use switch U to turn the capacitor voltage back on and thus stop time measurement.

- Read fall time t and capacitor voltage U and record with fall or rise distance s.

Analysis Use both methods to deduce the charge and radius of the oil drops. The viscosity of air,

η, is 1.824×10-5

Nsm-2

, and you will need to calculate the density of the air. A plot of the

charge against radius should resemble that shown in Fig. 4.

Cunningham found that there was a small deviation from Stoke’s equation for the friction

for small oil drops with a radius a (this is the deviation shown for small diameters in Fig.

4) . This results is a modification of the equation for the charge such that

3

1

'

+

=

a

A

qq

where the constant A takes the value 0.07776×10-6

m at standard pressure and at 25oC.

Reanalyse your results using this correction. tReferences

Adapted from Leybold instruction sheets 559 421, 575 451 and 559 411.

Fig 4. Measurements of

the charge and radius of

oil drops.

Millikin’s oil drop experiment 7

P5.6e