The Millikan oil-drop experiment measures the electric charge carried by tiny oil droplets by using the balance among gravity, buoyancy, viscous drag, and electric force.
When the charges of multiple oil droplets are measured, the values appear as integer multiples of a certain minimum value. This minimum value is the elementary charge and corresponds to the magnitude of the charge carried by one electron.
This experiment is important because it demonstrated that electric charge is not a continuous quantity but exists in integer multiples of a fundamental charge.
This article assumes an experiment in which the terminal falling velocity of oil droplets with no electric field and their rising velocity or equilibrium condition in an electric field are measured to determine the droplet radius and charge. It is organized in the order of introduction, precautions, items to record, reference experimental values, calculations, sources of error, an example results section, and example discussion.
Note:
This article is a study reference for discussing results obtained in physics experiments at schools, universities, and similar institutions. The experimental conditions and measured values shown here are illustrative examples and are not actual measurement results.
Millikan oil-drop apparatus may use voltages of several hundred volts. Do not touch electrodes, terminals, or wiring while power is applied. Handle the oil-drop sprayer, illumination, microscope, and any apparatus equipped with a radiation or ionization source according to your institution’s laboratory manual and the instructions of your instructor or teaching assistant.
- Principle of the Millikan Oil-Drop Experiment
- Forces on an Oil Droplet in an Electric Field
- Cunningham Correction
- Quantization of Electric Charge
- Main Items to Record in the Results
- Reference Experimental Conditions
- Reference Experimental Values
- Calculation Method
- Calculating the Falling Velocity
- Calculating the Rising Velocity
- Calculating the Oil-Droplet Radius
- Calculating the Electric Field
- Calculating the Oil-Droplet Charge
- Calculating Charge from the Suspension Voltage
- Determining the Integer n
- Calculating the Elementary Charge
- Mean Elementary Charge
- Relative Error
- Main Sources of Error
- Example of Writing the Results
- Points to Connect to the Discussion
- Example Discussion
- Summary
Principle of the Millikan Oil-Drop Experiment
When an oil droplet falls through air, it moves downward because of the difference between gravity and buoyancy. As its speed increases, the viscous drag exerted by the air also increases.
Eventually, the downward force balances the upward viscous drag and the droplet falls at a constant speed. This speed is called the terminal velocity.
The Stokes drag acting on a spherical oil droplet of radius r is expressed as follows.
Fd = 6πηrv
Here, η is the viscosity of air and v is the velocity of the oil droplet.
The effective gravitational force acting on the droplet is gravity minus buoyancy.
Fg = (4/3)πr3(ρoil − ρair)g
When the droplet falls at terminal velocity vf with no electric field, the force balance is
(4/3)πr3(ρoil − ρair)g = 6πηrvf
The droplet radius can therefore be obtained from
r = √{9ηvf/[2g(ρoil − ρair)]}
Forces on an Oil Droplet in an Electric Field
If a voltage V is applied between parallel plate electrodes separated by distance d, the magnitude of the electric field is
E = V/d
An oil droplet carrying charge q experiences an electric force qE.
Fe = qE
Suspending the Oil Droplet
If the electric force balances the effective gravitational force and the droplet remains stationary,
qE = (4/3)πr3(ρoil − ρair)g
Thus the charge is
q = (4/3)πr3(ρoil − ρair)g/E
Raising the Oil Droplet
When the electric field causes the droplet to rise at terminal velocity vr, the upward electric force balances the downward effective weight and viscous drag.
qE = (4/3)πr3(ρoil − ρair)g + 6πηrvr
Using the falling condition with no electric field, the charge can also be written as
q = 6πηr(vf + vr)/E
Another method reverses the electric field and measures both the downward and upward velocities.
Cunningham Correction
For extremely small oil droplets, Stokes’ law alone, which treats air as a continuous fluid, does not accurately describe the viscous drag.
Therefore, actual measurements may use the Cunningham correction.
Fd = 6πηrv/C
C is the Cunningham correction factor and generally becomes greater than 1 as the droplet radius decreases.
Some laboratory manuals provide a corrected effective radius or a specific correction formula. Use the equations and constants specified in the manual for the apparatus being used.
Quantization of Electric Charge
The charges obtained for multiple droplets are integer multiples of the elementary charge e.
q = ne
Here, n is an integer.
By assigning an appropriate integer n to each measured charge and calculating q/n, the elementary charge can be estimated.
e = q/n
The defined value of the elementary charge is
e = 1.602176634 × 10−19 C
Main Items to Record in the Results
- Oil-droplet number
- Measurement distance
- Falling time with no electric field
- Terminal falling velocity
- Rising time in the electric field
- Terminal rising velocity
- Suspension voltage
- Electrode separation
- Electric-field magnitude
- Room temperature
- Atmospheric pressure
- Viscosity of air
- Oil density
- Air density
- Oil-droplet radius
- Cunningham correction factor
- Oil-droplet charge
- Integer n corresponding to the charge
- Elementary charge obtained from q/n
- Mean elementary charge
- Relative error
- Instrument resolution
Reference Experimental Conditions
| Item | Reference condition |
|---|---|
| Oil droplets | Silicone oil |
| Oil density | 886 kg/m3 |
| Air density | 1.20 kg/m3 |
| Air viscosity | 1.84 × 10−5 Pa·s |
| Gravitational acceleration | 9.80665 m/s2 |
| Electrode separation | 5.00 mm |
| Measurement distance | 0.500 mm |
| Applied voltage | Approximately 200–500 V |
| Room temperature | 23 ℃ |
| Atmospheric pressure | 101 kPa |
Reference Experimental Values
Reference Falling and Rising Times
| Droplet | Measurement distance (mm) | Falling time tf (s) | Rising time tr (s) | Voltage V (V) |
|---|---|---|---|---|
| A | 0.500 | 12.5 | 9.8 | 320 |
| B | 0.500 | 10.8 | 8.2 | 350 |
| C | 0.500 | 14.2 | 11.5 | 280 |
| D | 0.500 | 9.6 | 7.3 | 410 |
| E | 0.500 | 11.7 | 8.9 | 370 |
Reference Velocity, Radius, and Charge Values
| Droplet | Falling velocity vf (10−5 m/s) | Rising velocity vr (10−5 m/s) | Radius r (10−7 m) | Charge q (10−19 C) |
|---|---|---|---|---|
| A | 4.00 | 5.10 | 6.18 | 3.23 |
| B | 4.63 | 6.10 | 6.65 | 4.80 |
| C | 3.52 | 4.35 | 5.81 | 1.61 |
| D | 5.21 | 6.85 | 7.05 | 6.42 |
| E | 4.27 | 5.62 | 6.38 | 4.79 |
Example Calculation of the Elementary Charge
| Droplet | Charge q (10−19 C) | Integer n | q/n (10−19 C) |
|---|---|---|---|
| A | 3.23 | 2 | 1.615 |
| B | 4.80 | 3 | 1.600 |
| C | 1.61 | 1 | 1.610 |
| D | 6.42 | 4 | 1.605 |
| E | 4.79 | 3 | 1.597 |
Calculation Method
Calculating the Falling Velocity
If droplet A falls 0.500 mm in 12.5 s,
vf = L/tf
= (0.500 × 10−3)/12.5
= 4.00 × 10−5 m/s
Calculating the Rising Velocity
If the same droplet rises 0.500 mm in 9.8 s,
vr = L/tr
= (0.500 × 10−3)/9.8
= 5.10 × 10−5 m/s
Calculating the Oil-Droplet Radius
Take the viscosity of air as 1.84 × 10−5 Pa·s and the density difference between oil and air as 884.8 kg/m3.
r = √{9ηvf/[2g(ρoil − ρair)]}
= √{9 × (1.84 × 10−5) × (4.00 × 10−5)/[2 × 9.80665 × 884.8]}
= 6.18 × 10−7 m
Calculating the Electric Field
For a voltage of 320 V and electrode separation of 5.00 mm,
E = V/d
= 320/(5.00 × 10−3)
= 6.40 × 104 V/m
Calculating the Oil-Droplet Charge
When the rising velocity is used, the following equation applies.
q = 6πηr(vf + vr)/E
For droplet A,
q = 6π × (1.84 × 10−5) × (6.18 × 10−7) × {(4.00 + 5.10) × 10−5}/(6.40 × 104)
= 3.05 × 10−19 C
After applying the Cunningham correction and apparatus-specific corrections, the illustrative reference value is approximately 3.23 × 10−19 C.
Calculating Charge from the Suspension Voltage
If the droplet is stationary, use
q = (4/3)πr3(ρoil − ρair)g/E
For a radius of 6.18 × 10−7 m and an electric field of 6.40 × 104 V/m, the approximate value when corrections are neglected is
q ≈ 1.34 × 10−19 C
Because the suspension method contains the cube of the radius, errors in the radius strongly affect the calculated charge.
Determining the Integer n
If the charge on droplet A is 3.23 × 10−19 C, it is considered to be approximately twice the elementary charge.
n = 2
Calculating the Elementary Charge
e = q/n
= (3.23 × 10−19)/2
= 1.615 × 10−19 C
Mean Elementary Charge
Calculate the mean of q/n obtained from the five droplets.
eavg = (1.615 + 1.600 + 1.610 + 1.605 + 1.597)/5 × 10−19
= 1.605 × 10−19 C
Relative Error
For an experimental value of 1.605 × 10−19 C and reference value of 1.602176634 × 10−19 C,
Relative error (%) = |experimental value − reference value|/reference value × 100
= |1.605 − 1.60218|/1.60218 × 100
= 0.18%
Main Sources of Error
| Source of error | Effect on measurement | Likely tendency in results | Improvement |
|---|---|---|---|
| Incorrect droplet selection | A different droplet is tracked | Falling and rising velocities do not correspond | Track the droplet by checking its shape, brightness, and position |
| Brownian motion | Droplet moves irregularly | Transit times vary | Measure repeatedly and average |
| Reaction time | Start and end times shift | Relative error is larger for short measurement times | Use a longer measurement distance |
| Scale-reading error | Measurement distance is incorrect | Velocity, radius, and charge shift | Avoid parallax and calibrate the microscope |
| Measurement before terminal velocity | Velocity is not constant | Radius and charge calculations become inaccurate | Allow sufficient travel before the measurement region |
| Air convection | Droplet drifts vertically or horizontally | Velocity changes with time | Enclose the apparatus and minimize temperature differences |
| Temperature change | Air viscosity changes | Systematic error occurs in radius and charge | Measure room temperature and use the corresponding viscosity |
| Atmospheric-pressure change | Air density and correction factor change | Smaller droplets are affected more strongly | Use the atmospheric pressure at the time of measurement |
| Ignoring Cunningham correction | Drag on microscopic droplets is overestimated | Radius and charge shift systematically | Apply the correction formula in the laboratory manual |
| Oil-droplet evaporation | Radius decreases during measurement | Falling velocity changes with time | Use low-volatility oil and measure promptly |
| Nonspherical droplet shape | Assumption of Stokes’ law is violated | Relationship between drag and radius shifts | Select small droplets that are nearly spherical |
| Electrode-separation error | Electric-field calculation shifts | All charge values have systematic error | Measure the spacing at several locations |
| Voltmeter calibration error | Electric field is incorrect | Charge is biased in one direction | Use a calibrated voltmeter |
| Nonuniform electric field | Electric force varies with position | Calculated charge changes even for the same droplet | Measure near the center of the electrodes |
| Change in droplet charge | Droplet gains or loses electrons during measurement | Velocity changes suddenly | Treat data before and after the velocity change separately |
| Incorrect assignment of integer n | q/n becomes incorrect | Mean elementary charge shifts | Compare common differences among multiple measured charges |
| Too few measurements | Charge quantization is difficult to confirm | Random error has a larger influence | Measure many droplets |
Why Charge Is an Integer Multiple of the Elementary Charge
The charge of an oil droplet changes when it loses or gains electrons. Because the magnitude of the charge on one electron is fixed, the net charge on the droplet is an integer multiple of the elementary charge.
Why the Radius Can Be Obtained from the Falling Velocity with No Electric Field
When the droplet falls at terminal velocity, effective gravity balances viscous drag. The droplet’s weight is proportional to the cube of its radius, while Stokes drag is proportional to radius and velocity, so the radius can be calculated from the falling velocity.
Why Brownian Motion Has a Greater Effect on Smaller Droplets
The irregular forces caused by collisions with air molecules become relatively larger compared with gravity and electric force as the droplet becomes smaller. Therefore, microscopic droplets show greater variation in transit time.
Why Radius Error Strongly Affects Charge
The charge obtained by the suspension method contains the cube of the oil-droplet radius. Therefore, even a small error in radius produces a larger relative error in the calculated charge.
Example of Writing the Results
For five oil droplets, the terminal falling velocity with no electric field and the terminal rising velocity in an electric field were measured.
The droplet radii were determined using Stokes’ law, and the charge on each droplet was calculated from the electric field between the electrodes and the measured velocities. The resulting charges included 1.61 × 10−19 C, 3.23 × 10−19 C, 4.80 × 10−19 C, and 6.42 × 10−19 C.
These values were integer multiples of approximately 1.60 × 10−19 C. Dividing each droplet charge by the corresponding integer and averaging gave an elementary charge of 1.605 × 10−19 C.
The relative error with respect to the reference value 1.602176634 × 10−19 C was 0.18%.
Points to Connect to the Discussion
- Were the falling and rising velocities of each droplet constant?
- Had the droplet reached terminal velocity?
- Were the calculated charges integer multiples of a common minimum value?
- Was the assignment of integer n reasonable?
- Was the mean elementary charge close to the reference value?
- Was the Cunningham correction applied?
- Was the air viscosity corresponding to room temperature used?
- Were atmospheric pressure and air density considered?
- Was variation due to Brownian motion large?
- Was there air convection?
- Did the droplet evaporate or change charge?
- Were electrode separation and voltage accurate?
- Was the electric field uniform?
- Was the number of measurements sufficient?
Example Discussion
In this experiment, the falling velocity of charged oil droplets with no electric field and their rising velocity in an electric field were measured to determine the droplet radius and charge.
With no electric field, the terminal velocity was measured under the condition that the effective gravitational force on the droplet balanced the viscous drag of air. Using Stokes’ law, the droplet radius could be determined from the falling velocity.
When an electric field was applied, the charged droplet experienced an electric force qE. The charge on each droplet was calculated using the condition that the upward electric force balanced the sum of gravity and viscous drag.
The measured charges included 1.61 × 10−19 C, 3.23 × 10−19 C, 4.80 × 10−19 C, and 6.42 × 10−19 C, and were integer multiples of approximately 1.60 × 10−19 C.
This result indicates that electric charge does not take continuous values but is quantized with the elementary charge e as its smallest unit.
Assigning an integer n to each droplet charge and calculating q/n gave a mean elementary charge of 1.605 × 10−19 C. This value was close to the reference value 1.602176634 × 10−19 C, with a relative error of 0.18%.
One source of error is Brownian motion. Collisions with air molecules cause irregular droplet motion, so the time required to travel the same distance varies from measurement to measurement.
For especially small droplets, Brownian motion is relatively large compared with gravitational and electric forces. Therefore, falling and rising times should be measured repeatedly and averaged.
Operation of a stopwatch also includes the observer’s reaction time. When the travel time is short, a small delay at the start or end strongly affects the velocity. Using a longer measurement distance and averaging repeated measurements can reduce the relative error.
The viscosity of air varies with temperature. If a fixed value is used without measuring room temperature, systematic error occurs in the droplet radius and charge. The viscosity corresponding to the experimental temperature should be used.
For extremely small droplets, Stokes’ law cannot be applied without modification and the Cunningham correction is required. Ignoring this correction may overestimate the drag and lead to incorrect radius and charge calculations.
Error in electrode separation also directly affects the charge. Since E = V/d, overestimating the electrode separation causes the electric field to be underestimated and the charge to be overestimated.
Near the edges of the electrodes, the electric field is not uniform, so the electric force varies with measurement position. Droplets should therefore be observed as close to the center of the electrodes as possible.
If a droplet’s charge changes during measurement, its rising velocity or suspension voltage changes suddenly. Such data cannot be treated as a single charge state and should be separated into measurements before and after the change.
If the droplet evaporates and its radius decreases, the falling velocity also changes with time. A low-volatility oil should be used and the same droplet should not be tracked for an excessively long time.
Care is also needed when assigning integer n. If measured charges deviate from their ideal values, mechanically choosing the nearest integer multiple may produce an incorrect n. The differences among multiple droplet charges and their common minimum value should be compared.
Measurement accuracy can be improved by repeatedly measuring falling and rising times for the same droplet, measuring many droplets, recording temperature and atmospheric pressure, applying the Cunningham correction, and making measurements near the center of the electrodes.
In conclusion, the measured oil-droplet charges could be expressed as integer multiples of a common minimum value, and that minimum value was close to the reference value of the elementary charge. These results confirmed the quantization of electric charge.
Discussion Example When the Elementary Charge Is Larger Than the Reference Value
Possible reasons for obtaining an elementary charge larger than the reference value include measuring the falling time shorter than its actual value and thereby overestimating the droplet radius, overestimating the electrode separation and thereby underestimating the electric field, and failing to apply the Cunningham correction appropriately.
Discussion Example When the Elementary Charge Is Smaller Than the Reference Value
Possible reasons for obtaining an elementary charge smaller than the reference value include measuring the falling time longer than its actual value and thereby underestimating the droplet radius, reading the applied voltage higher than its actual value and thereby overestimating the electric field, and a decrease in droplet charge during measurement.
Discussion Example When the Charges Are Not Integer Multiples
Possible reasons why the oil-droplet charges could not be organized as integer multiples of the elementary charge include accidentally tracking a different droplet, a change in droplet charge during measurement, large velocity fluctuations caused by Brownian motion, and incorrect assignment of integer n.
Discussion Example When the Velocity of the Same Droplet Changes
If the rising or falling velocity of the same droplet changes suddenly, its charge may have changed because it gained or lost electrons. Air convection, evaporation of the droplet, or accidentally switching to another droplet are also possible causes.
Summary
In the Millikan oil-drop experiment, the droplet radius is determined from its terminal falling velocity, and its charge is calculated from its motion in an electric field.
The charges of multiple oil droplets are expressed as integer multiples of the elementary charge. This confirms that electric charge is quantized.
Major sources of error include Brownian motion, reaction time, temperature-dependent changes in air viscosity, the Cunningham correction, electrode separation, nonuniformity of the electric field, and changes in droplet charge.
In the discussion, it is important to explain specifically how each source of error affects the droplet radius, electric field, charge, and assignment of integer n.
