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 section is to be completed individually BEFORE your first meeting with the TA and your lab group.

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.)

We will try to call out places where you need to write in your lab notebook by using the NOTEBOOK tag, like in the following:

NOTEBOOK: Fill out the top portion of the first page.

You should write down additional things in your notebook whenever you think it is useful – whether or not we specifically ask for it. These can include notes to yourself (e.g. to record/remember what you've done) or to your reader (e.g. to communicate an outcome or discuss a result). Remember to fill this out as you go along. Do not wait until you have completed the experiment. The notebook is meant to be a record of everything you've done in lab – good and bad – and is only useful if it is complete and honest.

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, $v_t$.

For the first week, we want you to carefully follow one drop, and determine the charge on it 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.

For your first meeting, 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 one drop. 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.

It is possible to compute uncertainties on each charge. Both the terminal velocity and balancing voltage will have uncertainties that can be propagated through to the final value for $q$. You do not need to calculate this final uncertainty for the first part of the project yourself, but you will discuss uncertainties during the TA/group meeting and decide if (or how) to apply them in the second part.

Some guidance and reminders follow. It will be helpful if you read through this before your meeting.

The first part of the experiment is expected to take you about 2 hours on your own (including the demo) before the meeting. This simulation is very realistic, and therefore at times challenging to use and collect data with. Reach out to your TA if you are having difficulty. If you finish in less than an hour, you likely have not collected enough data to yet be familiar with the subtleties of the simulation; find an additional drop or continue using your current drop if it is still on screen.

At the discussion, your group and the TA will talk about measurement technique and will decide on a plan for collecting and analyzing more drop data.

NOTEBOOK: Take notes during your meeting. What did your group talk about? What results (from you or your groupmates) are important to keep in mind? What information did the TA provide to guide you? What did your group decide to do next in Part 2?

Do not begin on this part until AFTER your first meeting with your group and TA.

Using the methods which your group discussed at the first meeting, collect more charge data. You should aim to collect data for about 1.5-2 hours more.

Prior to the meeting, you will be need to upload your charge values using the following form. Your TA will then be able to pool your data together in order to help the group understand and interpret the results.

Remember… Millikan's original experiment was used to show that charge is a discrete quantity and that there is a minimal unit of charge, $e$. Do you see any evidence for this in your own data?

At the meeting, the TA will bring together your group's data, plot it, and lead a discussion on the results. Be prepared to discuss your methods and share data, and be ready to participate in a discussion about how to interpret the results.

Your TA will use the following Google Colab notebook to help the group visualize the data.

After your second meeting, you will again need to write up your summary and your conclusions. Include any data tables, plots, etc. from the experiment or discussions as necessary in order to show how your data support your conclusions.

This part doesn't need to be long; one or two pages should be sufficient. What is important, however, is that your writing should be complete and meaningful. Address both the qualitative and quantitative aspects of the experiment, and make sure you cover all the “take-away” topics in enough depth. Don't include throw-away statements like “Looks good” or “Agrees pretty well.” Instead, try to be precise.

Remember… your goal is not to discover some “correct” answer. In fact, approaching any experiment with that mind set is the exact wrong thing to do. You must always strive to reach conclusions which are supported by your data, regardless of what you think the “right” answer should be. Never, under any circumstances should you state a conclusion which is contradicted by the data. Stating that the results of your experiment are inconclusive, or do not agree with theoretical predictions is completely acceptable if that is what your data indicate. Trying to shoehorn your data into agree with some preconceived expectation when you cannot support that claim is fraudulent.