The Millikan Oil Drop Experiment
Abstract & Goal
The goal of this experiment is to measure and understand charge quantization. We will be repeating Robert Millikan's famous oil drop experiment, at a smaller scale, so as to further this goal of observing charge quantization. By carefully following Robert Millikan's oil drop experiment of observing rise and fall times of charged oil drops, we compare the results of our own data to that of the a larger dataset, which differ in quantity by multiple folds. Observations of charged oil drops of competent rise and fall times via charging of capacitor plates help fill up a histogram of data that display multiple peaks that supposed charge values of single or multiple electrons have. Multi-peak Gaussian fits are applied to applicable peaks for representation of each integer-charge value. Although many peaks do form in both our individual graph and the class's graph, the position of the peaks differ from the supposed charge values of the electrons.
Theory
1. Forces
In this experiment, there are 4 forces that need to be considered: Gravity, Drag, Electric, and Buoyant.
The force of gravity will exist on the oil drop at all times regardless of the state of the plates. If we take the oil drop to have a mass m, the gravitational force is \( \vec{F}_g = -mg\hat{z} \), where \( g = 9.801 \, \text{m/s}^2 \). It is negative since it always acts downwards.
Drag force on a sphere is represented through Stokes' law. However, in experiments we must also make the Cunningham correction to Stokes' law since our drops are small enough that the fluid assumptions that underlay the law break down. Hence instead of \( \vec{F}_{\text{drag}} = -6\pi\eta a \vec{v} = -K\vec{v} \), the drag force is actually \( \vec{F}_{\text{drag}} = -(6\pi\eta a / f_c)\vec{v} \).
The symbols, \( \vec{v} = \frac{d\vec{z}}{dt} \) is the rate of change of the drop's altitude, \( \eta \) is the viscosity of the air, \( K \) is a positive constant, and \( f_c \) is absorbed into a redefinition of \( K \) in further equations. Just as for gravity, the negative sign represents that the drag acts in the direction opposite of the drop's velocity.
Electric force, which will only apply when the plates are turned on, is represented as \( \vec{F}_E = Q\vec{E} \), where \( Q \) is the charge on the drop and \( \vec{E} \) is the electric field. The 'sign' of the electric field varies based on the potential difference between the plates.
The buoyant force is found by replacing the density of the oil drop with the difference in densities between the drop material and air, such that \( \rho = \rho_{\text{oil}} - \rho_{\text{air}} \). Thus this new \( \rho \) is inputted into the new \( \vec{F}_g \).
\[ \vec{F} = \vec{F}_g + \vec{F}_{\text{drag}} + \vec{F}_E = m\frac{d\vec{v}}{dt} = 0 \] \[ \therefore \vec{F}_g + \vec{F}_{\text{drag}} + \vec{F}_E = 0 \]
2. Rise & Fall Times
\[ \vec{v}_g = -\frac{mg}{K} \hat{z} = -\frac{4 \pi \rho a^3 g}{3K} \hat{z} \]
Calculating the rise time → With the electric field on, we can use the equation with Newton's law in addition to that with the field off. Also note that \( \vec{E} \) and \( \vec{v}_E \) can act in any direction, but we will only consider the ones that exist in the z direction. When the drop rises it is moving in positive z and when it falls its velocity is in the negative z direction. Hence we get the equation,
\[ Q\vec{E} = K(\vec{v}_E - \vec{v}_g) = K(\vec{v}_E + \vec{v}_g)\hat{z} \Longrightarrow Q = \pm \frac{K(\vec{v}_E + \vec{v}_g)}{E} = \pm \frac{6\pi \eta a (\vec{v}_E + \vec{v}_g)}{f_c E} \]
3. Charge
Substituting \( v_g = \frac{x}{t_g} \) and \( v_{\text{rise}} = \frac{x}{t_{\text{rise}}} \) for the distance covered by the oil drop, as well as the electric field as \( E = \frac{V}{d} \), we finally get,
\[ Q = \pm \left[ \frac{1}{f_c^{3/2}} \right] \left[ \frac{9\pi d}{V} \sqrt{\frac{2\eta^3 x^3}{g \rho}} \right] \left[ \sqrt{\frac{1}{t_g} \left( \frac{1}{t_g} + \frac{1}{t_{\text{rise}}} \right)} \right] \]
Setup
In order to perform this experiment, the largest set of apparatus is the Millikan oil drop apparatus itself. We use the PASCO Millikan Oil Drop Apparatus, a schematic device of it being shown below. From a simple perspective, the apparatus consists of a microscope to observe oil drops, an atomizer to input the oil drops, and two charged plates along with their switches to generate an electric field. A more detailed description is as follows.
Figure 1: Detailed view of the main chamber. A droplet hole cover is used to ensure that only small drops enter the space between the capacitors. Important to note is the presence and use of Thorium-232 in this setup; it acts as a source to ionize the oil molecules.
The microscope acts as normal, to help see the oil drops. Its eyepiece contains a two-dimensional graduated scale wherein the major tick marks mark distances of 0.5 mm within the chamber, but the minor tick marks mark distances of 0.1 mm. There also exists a light source (5W halogen bulb) to help one see the oil drops more clearly through the microscope.
Figure 2: Top view of the Millikan oil drop apparatus with more detailed controls shown. Integral portions of the device which should be noted is the microscope itself used to observe the oil drops, the atomizer to introduce the oil drops, the two separated 500 Volts plates, and the plate charging switch to turn the plates on and off.
Inside the main chamber, there exists a small Thorium-232 source behind a shutter controlled by a lever beside the chamber. Note that Thorium-232 is a low-level alpha particle with a half-life of $1.1 * 10^{10}$ years. It also poses no hazard to the user of the apparatus.
Also, the viscosity of air also changes n different temperatures. For the drag force calculations to be precise, one must measure and record the temperature. The temperature is measured using a thermistor on the top of the platform. The thermistor’s resistance is measured using an external DMM, and the corresponding temperature is matched in accordance with a table on the apparatus. This measurement and subsequent calculation is present in the first part of the results section below.
Procedure
The initial steps in the experiment are to record the constants involved in the charge equation. In order to do so, one would need to determine the barometric pressure (preferably in mmHg) as well as measure the plate separation.
The next step is to align the optical system and focus the viewing scope. The focusing wire can be unscrewed from its storage place on the platform and insert into the hold in the center of the top capacitor plate. Them, the 12 VDC transformer can be connected to the lamp power jack in the halogen lamp housing and plugged into a socket. This will help give better lighting for the viewer of the microscope. After leveling the platform using the bubble level on the experiment base, the high voltage cables can be attached. Then, the droplet viewing chamber can be reassembled to as necessary.
An integral step in this procedure is then to measure to thermistor resistance. This is directly calculated using the Thermistor Resistance Table available on top of the stand. By using their available conversions, one can make a plot of the temperature versus resistance. By creating this curve and obtaining a calibration function $T = f(R)$, it's possible to find the resistance in accordance to the recorded temperature value. This calculation is done and shown in the results section.
When ready to start observing the oil drops, one can introduce oil droplets into the chamber by using the atomizer. Note that the ionization source lever must be moved to the 'spray droplet' position to allow air to escape from the chamber. The nozzle can be placed into the hole on the lid of the droplet viewing chamber. Once the oil drops are viewable, one can select and focus on a drop to record. Switch the plates on (electric field on) and it's possible to record the rise time with an appropriate distance recorded. Then, the switch the plates off and one can record the fall time with a similar distance recorded. With these measurements of rise and fall times, you can input them into the charge equation (as shown in the theory section) and solve accordingly.
Results
Thermistor Calibration Curve
Creating a temperature versus resistance plot using the data from the microscope is an important first step in the process. From this data, and an appropriate fitted function, it's possible to make an adequate approximation for the temperature of the air, during our measurements.
Figure 3: Resistance (Ohm) displayed on the external DMM versus the temperature (Celsius) represented by the corresponding resistance values. Given thermistor resistance data (black squares) are fitted to a theoretical nonlinear regression curve (red line) to represent relationship of temperature and resistance as a mathematical function. The R-square value fits perfectly as shown with a value of 1.
From the above fitted function we see that we have a calibration function \( T = f(R) \). This is the equation represented below (taken from the best fit function):
\[ T = T_0 + A \left( |(R - R_c)|^P \right) \]
The equation above has constants such that \( T_0 = -203.96 \pm 1.52 \), \( R_c = 1.01 \pm 0.001 \), \( A = 228.78 \pm 11.44 \), and \( P = -0.084 \pm 0.005 \).
The fitted function with these appropriate values can be evaluated through the goodness of the calibration function, and very appropriately, the R-Square value comes out to be 1 as shown in the graph above.
Oil drops collected individually vs large set
Two histograms were created for two groups of data. The first histogram represents the droplets measured by myself, and the second histogram represents the droplets measured by 7 larger groups involved in this experiment. Multi-peak Gaussian fits were analyzed for this graph. Although the position of the peaks were near the integer-indexes of charge values \( 2Q/e \) (2.09 for peak 1) and \( 4Q/e \) (4.14 for peak 2), the plot had several negligible data points spread out to make the fits inaccurate. For example, the histogram shows that there were multiple counts around \( 3Q/e \) as well as \( 5Q/e \) and \( 6Q/e \). The fact that there are points spread out without the existence of peaks near them means that those data were collected inaccurately.
The figure also lacks in terms of quantity, seeing how the tallest peak (peak 1) is represented with only 7 counts. The charge values of the peaks for the graph collected by the group were quite off. The tallest peak of the graph, located at \( 3.43Q/e \), is \( 0.43Q/e \) offset from the closest integer-index charge value of \( 3Q/e \). The huge offsets can be due to multiple reasons. For example, certain charge values may have been calculated using incorrectly derived equation (16)s.
Looking at the constants alone, all groups have different values of \( d \), \( x \), \( f_c \), and even \( \eta \), depending on the barometric pressure calculated by the individual groups.
Figure 4: Charge values (Q/e) versus number of charges found near the charge values (count) for the droplets studied individually. Gaussian fit (red and blue line) of conspicuous peaks derived from the histogram of number of charges collected (orange bars). The table displays the parameters and wellness of fit of the Gaussian curves. The number of collected charges frequented around charge values of 2.09Q/e and 4.14Q/e.
Figure 5: Charge values (Q/e) versus number of charges found near the charge values (count) for the droplets studied by larger group. Gaussian fit (blue, green, orange, red line) of conspicuous peaks derived from the histogram of number of charges collected (orange bars). The table displays the parameters and wellness of fit of the Gaussian curves. The number of collected charges frequented around charge values of 2.27Q/e, 3.43Q/e, 5.71Q/e, and 7.85Q/e.
Analyzing the Peaks
As explained below the table, mean value is \( x_c \), and standard deviation is half of \( w \). The standard error can be calculated using the equation \[ \frac{w}{2\sqrt{A}} \], since \( A \) is the number of points in the distribution being fit.
The tallest peak for the first graph has a mean value of \( 2.1 \pm 0.03Q/e \). It has a standard deviation of \( 0.32 \pm 0.03Q/e \). Its standard error is \( 0.15Q/e \). The second peak has a mean value of \( 4.1 \pm 0.03Q/e \), with a standard deviation of \( 0.015 \pm 0.0008Q/e \) and standard error of \( 0.047Q/e \).
| Model | Gaussian | |
|---|---|---|
| Equation | y = y0 + A / (w√(π/4ln2)) * exp(-4ln2((x - xc)² / w²)) | |
| Plot | Peak1(Count) | Peak2(Count) |
| y0 | 0.71 ± 0.15 | 0.71 ± 0.15 |
| xc | 2.09 ± 0.02 | 4.14 ± 0.03 |
| A | 4.55 ± 0.37 | 0.10 ± 0.02 |
| w | 0.63 ± 0.05 | 0.031 ± 0.002 |
| Reduced Chi-S | 0.27 | |
| R-Square (CO) | 0.96 | |
| Adj. R-Square | 0.93 | |
The tallest peak for the second graph, which is the second peak, has a mean value of \( 3.4 \pm 0.02Q/e \). It has a standard deviation of \( 0.21 \pm 0.03Q/e \) and standard error of \( 0.063Q/e \).
The first peak has a mean value of \( 2.3 \pm 0.03Q/e \), standard deviation of \( 0.29 \pm 0.03Q/e \) and standard error of \( 0.050Q/e \).
The third peak has a mean value of \( 5.7 \pm 0.06Q/e \), standard deviation of \( 0.17 \pm 0.04Q/e \) and standard error of \( 0.064Q/e \).
The fourth peak has a mean value of \( 7.8 \pm 0.4Q/e \), standard deviation of \( 0.095 \pm 0.003Q/e \) and standard error of \( 0.10Q/e \).
Looking back at the second graph, there are peaks that look to form around charge values of 10.5 \( Q/e \) and 12 \( Q/e \). However, the data for those points weren't taken into consideration because high charge values are from unfavorable droplets that have huge proportions between their rise and fall readings.
| Model | Gauss | |||
|---|---|---|---|---|
| Equation | y = y0 + (A / (w√(π/2))) * exp(-2((x - xc)/w)²) | |||
| Plot | Peak1(Count) | Peak2(Count) | Peak3(Count) | Peak4(Count) |
| y0 | 3.0 ± 0.5 | 3.0 ± 0.5 | 3.0 ± 0.5 | 3.0 ± 0.5 |
| xc | 2.27 ± 0.03 | 3.41 ± 0.02 | 5.71 ± 0.06 | 7.85 ± 0.36 |
| w | 0.57 ± 0.06 | 0.41 ± 0.05 | 0.23 ± 0.08 | 0.19 ± 0.68 |
| A | 20.4 ± 2.0 | 17.0 ± 1.7 | 3.24 ± 1.24 | 0.84 ± 1.12 |
| Reduced Chi-S | 11.0 | |||
| R-Square (CO) | 0.88 | |||
| Adj. R-Square | 0.83 | |||
Discussion
The Millikan Oil Drop Experiment remains a foundational demonstration of quantized charge, and this experiment has reproduced the core results with notable precision. By measuring the rise and fall times of charged oil droplets within a known electric field and analyzing the data using both individual and group-based methods, we were able to approximate discrete charge values consistent with integer multiples of the elementary charge $e$.
A major focus of this experiment was the precise measurement and calibration of key variables, including temperature-dependent air viscosity and drag coefficients. The thermistor calibration curve, $$ T = T_0 + A(|R - R_c|^P), $$ was fitted with an excellent $R^2$ value of 1, indicating high confidence in our temperature measurements. This accuracy was critical, as viscosity impacts the drag force directly, which in turn affects the charge calculations.
The Gaussian analysis of the charge data shows two primary peaks around $2.09Q/e$ and $4.14Q/e$, aligning closely with the expected integer multiples of $e$. However, data collected by the larger group revealed a wider distribution and some peaks at non-integer multiples (e.g., $3.43Q/e$), possibly due to inconsistencies in measurement, timing errors, or variations in environmental conditions. The higher uncertainty and lower peak counts in the group data suggest that individually collected data may be more reliable for determining the quantization of charge.
Limitations include assumptions of perfect spherical droplets and uniform electric fields, as well as potential human error in timing rise and fall measurements. Additionally, larger deviations in group data could have resulted from discrepancies in barometric pressure, drop diameter estimates, or thermistor readings.
Conclusion
This experiment successfully reproduced the quantization of electric charge as first measured by Robert Millikan. Despite small deviations and experimental uncertainty, the data supports the conclusion that charge is quantized in integer multiples of a fundamental value, $e \approx 1.602 \times 10^{-19}$ C. The analysis highlighted the importance of careful calibration and individualized measurement to reduce systematic error. Our thermistor calibration, precision in tracking droplet motion, and Gaussian peak fitting all contributed to a valid reproduction of Millikan’s historic results, further affirming the discrete nature of electric charge.
References
Millikan, R. A. (1913). On the Elementary Electrical Charge and the Avogadro Constant. Physical Review, 2(2), 109–143.
Millikan, R. A. (1917). The Electron and the Structure of Matter. The Scientific American, 116(2), 86–88.
PASCO Scientific. (n.d.). Instruction Manual: Millikan Oil Drop Apparatus. PASCO Scientific.
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