In the early 1900s, there was still much debate about the makeup of the matter around us. Scientists were conflicted as to whether quantities like mass and electric charge were discrete (implying the existence of fundamental, smallest particles) or whether they could take any value (however arbitrarily large or small). The physicist J. J. Thompson had recently discovered the electron in 1897, and showed that it was a negatively charged particle of with fixed charge to mass ratio, $q/m$. Thompson's discovery implied the existence of a fundamental unit of charge – what we today term $e$ – but his experiment was unable to measure it. (He could measure the ratio $e/m$ only… not $e$ or $m$ separately.)

It was in this environment that Robert A. Millikan began work. Starting in 1909 (in the Ryerson Laboratories building, still standing here at the University of Chicago), Millikan and his PhD graduate student Harvey Fletcher designed the experiment that you will carry out today. The two scientists created small drops of oil that each held a tiny net charge on them and placed them between two plates that could be charged to a given voltage difference. By looking at how the drops fall with the voltage turned on and off, Millikan and Fletcher were able to measure the total charge on hundreds of drops and use the collected data to argue that charge comes only in discrete units.

This lab is intended to serve as a completely remote make-up assignment for students who missed a normal in-lab day of lab this quarter. You will work on this assignment alone and submit your report directly via email to your TA for grading.

Click on the link below to start your individual lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.)

For this project, you will be using a very realistic simulation of the Millikan oil drop experiment. The apparatus modeled by the simulation is shown in Fig. 1.

Figure 1: Schematic of the Millikan oil drop apparatus (Source)

The main part of the apparatus is a open chamber with a metal plate at the top and a metal plate on the bottom. Small drops of oil can be sprayed above the top plate, some of which fall through a hole and float downward through the chamber. Because of the way the drops are formed – by spraying liquid through a small aperture so that it breaks up into tiny clumps – each drop is a slightly different size and has a slightly different net electrical charge. (Some drops have charge stripped off during formation, while others gain excess charge in the process.) A microscope is positioned to look into the chamber so that a person can observe the falling drops.

The two plates are connected to a power supply and can be adjusted so that there is a voltage difference between them.

• When the voltage difference is zero, the drops feel a force downward due to gravity as well as an upward drag force due to air resistance. The drops quickly come to terminal velocity, but also jiggle about due to random collisions with air molecules.
• When the voltage is non-zero, some of the drops slow down or even reverse direction and rise upward because they feel an additional electric force. If a particular drop has net charge, $q_i$, it will feel an electrical force $F_i = q_iE = q_i(V/D)$ where $E = V/d$ is the electric field between the plates, $V$ is the voltage difference, and $d$ is the separation distance between the plates. Since drops have different amounts of charge, they experience different forces and will respond differently when the voltage is turned on.

The apparatus has one additional feature: an x-ray source that can be turned on and off. When the x-ray source is turned on, it may randomly interact with different drops and knock electrons loose. The result is that when the x-ray source is on, some drops will spontaneously change their total charge – either by directly losing one or more electrons due to the x-rays, or gaining one or more of the extra electrons which are now zooming around.

In the next section, we are going to use a realistic simulation of the Millikan oil drop to collect data and look for evidence of discrete charge, but first let's look at a very simplified demonstration.

Open the demonstration below. You should see a screen like the one shown in Fig. 2.

Figure 2: The starting screen of the simplified Millikan oil drop demonstration. Note that the time shown at the top of the screen is not accurate and should be ignored.

In the above demonstration, you are given a simplified experiment with a single drop. You have the ability to “Run”, “Pause”, and “Reset” the simulation (which returns the drop to the starting position). Play around with the demo to understand the basic principles.

• What happens when the drop falls with voltage set to V = 0? What force(s) apply?
• What happens when you increase the voltage? What force(s) apply now? Is the charge on the drop positive or negative? (Why?)
• Is it possible to find a value of the electric potential which balances the charge?

Create a new drop.

• Is the mass the same? Is the charge the same? Does it balance at the same voltage?

If we knew all the parameters of the simulation, we could calculate the net charge on each of the drops. However, for now let's just focus on what is qualitatively happening to a single drop when it is placed between two plates. In the next section, we'll look at a more complete (and therefore more complicated) simulation and take data with that.

Make sure you understand all the questions above before moving on to the next part.

The simulation we will use for the rest of the project is located here:

Open up the simulation. You should see a screen like that shown in Fig. 3 below. This simulation has a lot more going on in it, so we will walk through the controls carefully to make sure you understand what's going on.

Figure 3: Screenshot of simulation in its starting state

The first thing to note is that the image you see is inverted. Drops falling downward due to graving will appear as drops falling upward in the simulation.

This feature is meant to mimic the inverted image that is produced when the single optical lens microscope is used to magnify the drops in the chamber.

To produce drops, push the spray button. This will produce a number of white dots on the screen which – if you do nothing else – will jiggle about randomly, but slowly fall. (Again, they fall upward in the simulation, which means they are falling downward in real life.) Note that each individual drop falls at roughly a constant velocity, but that different drops fall at different velocities. The drops are different sizes (produced by random chance when the oil is sprayed), but all of them are small enough (and are subject to strong drag due to air resistance) that they quickly reach terminal velocity.

The drag force which the drops experience is given by something called Stokes' law. We won't derive or motivate it here, but it states that for round objects moving through a fluid (air, in this case), the drag force is equal to $F_{drag} = 6\pi\eta rv$ where $\eta$ is the viscosity of air, $r$ is the drop radius, and $v$ is the drop velocity. (If you don't know what viscosity is, don't worry! It has to do with how easy it is to move through a fluid… honey is more viscous than water which is more viscous than air, for example.)

As the drops continue falling, they enter the region of the image where there are yellow indicator lines. These lines are separated by a fixed amount, and are included so that you can measure rise or fall velocities (by using the stopwatch to time how long it takes to travel a certain vertical distance.).

The yellow lines on the microscope image are separated by 0.1 mm. The stopwatch measures the time in the simulation which is different from real time. Do not use your own stopwatch or phone to time the drops!

You can turn on the voltage and see the drops quickly scatter – some moving upward, some downward. Adjusting the voltage strength changes the electric force. Reversing the voltage changes the sign of the force.

The electric field between the plates is assumed to be perfectly vertical. The strength is given by $E=V/d$, where the plate separation is $d$ = 6 mm.

Turning the rays on activates the x-ray source next to the chamber which can cause drops to randomly change their total net charge. It may be difficult to see, but with the rays on, you should occasionally see a drop suddenly start to fall or rise faster or slower as the electric force changes.

The x-rays are a random process. You may need to wait 30 seconds or a minute before any specific particle is affected.

For this method, we will attempt to balance the downward force due to gravity with an upward electric force. When the voltage has been adjusted so that the drop is neither falling nor rising, then the two forces are equal and opposite:

where $V_0$ is the balancing voltage. (Note that if the particle is stationary, the velocity is zero, so the drag force is zero.) We want to determine $q$ and we know $V_0$, $d$, and $g$; however, the particle mass $m$ is still an unknown. We can determine it by measuring the terminal velocity when the drop falls with the voltage turned off.

With no voltage, now the downward gravitational force and upward drag force are balanced:

where $v_t$ is the terminal velocity. While we don't know the mass or the radius of an individual drop, we do know that they are related by the density as $m = (4/3)\pi r^3\rho$. Substituting this into Eq. (2) and rearranging, we can solve for the radius,

Substituting this back into Eq. (2) finally yields

Returning to Eq. (1), we can now substitute this on the left-hand side and solve for the charge:

where we have rolled all the constant values into a prefactor, $k = 18\pi d\sqrt{\frac{\eta^3}{2\rho g}}$. For our simulation we have:

• an oil mass density of $\rho = 875~\textrm{kg}/\textrm{m}^3$,
• an air viscosity of $\eta = 7.25\times 10^{−6}~\textrm{Ns}/\textrm{m}^2$, and
• a plate separation of $d = 6~\textrm{mm}$.

That means that the prefactor is equal to $k = 5.06\times 10^{−11}~\textrm{CV(s/m)}^{3/2}$.

Thus, this measurement technique requires measuring two quantities: the balancing voltage $V_0$ and the terminal velocity as the drop falls downward, $v_t$.

We want you to carefully follow three drops, and determine the charge on each one several times.

As a sample procedure, consider the following:

• Create a spray of drops. Allow them to fall with the voltage off until many have drifted into the area with yellow lines.
• Turn the voltage on to a value of a few hundred volts. Many drops will scatter away quickly, but some may be moving slowly or be nearly stationary. Choose one of these and try to carefully adjust the voltage until it is balanced – neither falling, nor rising. (Note that the drop will never truly stop. It will only appear to jiggle around a fixed point. There will be some uncertainty in the value required to balance your drop.) This yields $V_0$.
• Turn the voltage off and time how long it takes the drop to fall a certain distance. This yields $v_t = x/t$, where $x$ is the fall distance and $t$ is the fall time.
• If you turn the voltage back on, the drop will again be balanced. Turn the voltage up temporarily to move the drop back to a more convenient location on the screen and balance it again.
• At this point, you have enough information to determine $q$ for this drop from Eq. (5). However, this is more you can do. You can…
  • …repeat the terminal velocity measurement several times to get a better estimate of that value and it's uncertainty; or
  • …turn on the x-ray source and wait for the charge on the drop to change.

Make sure that you have collected several measurements of the terminal velocity (which should not change since the drop remains a fixed size) and several measurements of the balancing voltage (which should change every time you use the x-ray source to change the drop charge) for three drops. Collect your data in a well-formatted table and calculate the net charge $q$ for each balancing voltage.

A video demo of how to balance a drop and measure velocity is shown in Fig. 4.

Now that you have taken data for a few drops, you understand what poor Millikan and Fletcher had to go through! In order to say something about the discreteness of charge, you have to collect a lot of drops.

Since it is unlikely that you can say much with your three drops alone, you will add your handful of charge values to a much larger data collected by 132 students in Summer 2021 (using the same simulation you are using today). This will give you many data points to use to look for any evidence for the discretization of electric charge.

The following Google Colaboratory script includes several interactive histograms that visualize this data.

Open up the notebook and run the cells. At the end of the file you will find two histograms:

• The first shows a histogram of absolute charge, $q$.
  • Do you see evidence that some values of charge are more likely than others?
  • Do your charge values fall into one or more of the charge clusters created by the class set or are your data far away?
• The second shows a histogram of the ratio of absolute charge to fundamental charge, $q/e$.
  • As you adjust the guess for $e$, do you find a value that makes the position of the peaks line up with integer ratios of $q/e$ = 1, 2, 3, etc?
  • How does this value compare to the literature value of the fundamental electric charge?

In both histograms you can change the histogram bin size and the maximum charge displayed. Adjust both (if needed) to make the histogram as clear and easy to read as you can.

Answer these questions and include screenshots of both histograms (after you've made your adjustments) in your lab notebook.

For this make-up assignment, you do not need to write a summary or conclusion report. Instead, just make sure that you fill out all the sections of the notebook in enough detail and answer all the questions asked.

When you are finished, save the notebook as a PDF and email it to your TA. (Do not submit the notebook on Canvas.)

Your TA will grade the notebook out of 8 points and use this score in place of the scores you normally would have received for your missed lab.