In the decade following 1897, many different methods were evolved for determining ionic charges. Some methods depended on measuring the total charge of a number of ions used as nuclei for cloud droplet formation. Other methods were more indirect: experiments, for example, which combined with the kinetic theory of gases, could give crude values for Avogadro's number, $N$. By dividing the Faraday constant (the charge carried in electrolysis by ions formed from one gram-atom of a univalent element) by $N$, one could determine the average charge per ion. None of these methods measured individual charges.

The precursor to the oil drop experiment comes from John S. Townsend's research in 1897. Townsend created clouds of charged water droplets via electrolysis, calculated their mass from their free-fall speed, and used an electrometer to determine the total charge. While imprecise, he did calculate $e$ to be $\approx 3\times10^{-10} \mathrm{esu}$, roughly $1 \times 10^{-19}$ coulombs.

As an assistant professor at The University of Chicago, Robert A. Millikan and his graduate student Harvey Fletcher* began working together experiments with single oil droplets in Ryerson Laboratory in 1906. The experiment had been tried previously with water droplets, but they evaporated too quickly for precise measurements to be made. With the oil-drop method, they were able to show that the charge on his droplets was always $ne$ with $e = 1.594(17) \times 10^{-19}$ coulomb and $n$ a positive or negative integer. This is within 1% of the best modern value, but due to using the incorrect value for the viscosity of air it differs by around $6\sigma$.

Many texts have a brief discussion of the experiment, but an expanded account may be found in Harnwell and Livingood's Experimental Atomic Physics. The basic idea is that a tiny droplet of oil, having some excess electric charge, is trapped inside a parallel plate capacitor. The geometry allows the application of a known electric field, while the illumination and optical microscope allow continuous observation of the droplet. The microscope contains a calibrated reticle which provides a length scale. Together with a computer/timer measurements of the droplet's speed may be made.

The experiment can be carried out at two levels: first the limited objective of showing that the charge of an oil droplet varies by discontinuous amounts, equal to some multiple of a basic unit, and second, the ultimate objective of measuring the elementary unit of charge. Historically, the experiment has two goals:

• Determine whether charge is a smooth, continuous medium and
• If discrete, determine the value of the elementary charge.

*While Harvey Fletcher is not as famous, he was a master of the science of acoustics and speech perception. He recorded the first vinyl record, made the first hearing aid, and was the first to broadcast live stereo sound.

When a droplet of oil is falling in air at terminal velocity, then the gravitational and drag forces acting on it must be balanced:

$mg = kv_f$ where $k$ is the frictional drag coefficient and $v_f$ is the velocity of freefall.

If we apply an electric field to the droplet, we have an additional force $qE$ where $q$ is the charge on the drop and $E$ is the electric field.

$mg + kv_r = qE$ where $v_r$ is the rising velocity

In both instances, we are assuming that the velocity is constant (usually valid, as the acceleration time is in ms) and that we can neglect the buoyant force of air (air is ~$10^{-3}$ times the density of oil)

If we have data from both the falling and rising drop, we can combine terms to eliminate $k$ to find:

$ q = \dfrac{mg(v_f+v_r)}{Ev_f}$

Looks easy right? We just need to know $g$, $E$, and $m$ in addition to the velocity measurements

We can't exactly take an individual droplet and put it on a scale, so we need another way to determine mass here. If our drop is spherical, then we could find the mass from its radius and density:

$m = \frac{4}{3}\pi a^3\rho$ where $\rho$ is the density of the oil (which we can find easily, it should be .886 g/cm^3 for the oil we're using) and $a$ is the radius of the sphere.

This leads to another problem: we can't accurately measure the radii of our spheres with our setup! This is a common problem in microscopy, as we'd have to be quite certain that our sphere was in focus when we made our measurement to get an accurate value. Plus, with the length scales we're working on, the uncertainties would be rather large.

To get around this, we make the assumption that we can apply Stoke's Law to our situation. For a sphere moving in a viscous fluid,

$F_{drag} = 6\pi \eta a v$

where $\eta$ is the coefficient of viscosity. (This is what threw of Millikan's results for the specific value of $e$.

If the drop is at terminal velocity and we model the drag using Stoke's law, then we find

$|F_{drag}| = mg$

combining this with our past equations, we can then state the drop's radius as

$a = \sqrt{\dfrac{9}{2}\dfrac{\eta v_f}{g\rho}}$

First, let's define the mass of a drop in measurable terms:

$m = \dfrac{4}{3}\pi \left(\sqrt{\dfrac{9}{2}\dfrac{\eta v_f}{g\rho}}\right)^3 \rho$

We can move some terms about to express this as

$m = \dfrac{4}{3}\pi v_f^{3/2} \left(\sqrt{\dfrac{9}{2}\dfrac{\eta\rho^{2/3}}{g\rho}}\right)^3$ or, $\dfrac{mg}{v_f^{3/2}} = \dfrac{4}{3}\pi \sqrt{\dfrac{1}{g\rho} \left(\dfrac{9\eta}{2}\right)^3}$

Why on earth did we do that? Well, you'll notice that the right-hand side is independent of the properties of a specific drop. Thus, if we define

$C_1 = \dfrac{4}{3}\pi \sqrt{\dfrac{1}{g\rho} \left(\dfrac{9\eta}{2}\right)^3}$,

$mg = C_1 v_f^{3/2}$

This lets us find the mass of any drop from its terminal velocity, which is quite handy!

Returning to the equation for q, we can finally create an expression for the electric charge on a drop in terms of measurable quantities:

$q = \dfrac{mg(v_f + v_r)}{Ev_f}$

$q = \dfrac{4}{3}\pi v_f^{3/2} \sqrt{\dfrac{1}{g\rho} \left(\dfrac{9\eta}{2}\right)^3}\dfrac{(v_f + v_r)}{Ev_f}$

Finally, if we sub in the equation for the electric field within a plate capacitor ($E = V/d$)

$q = C_1\left(\dfrac{d\cdot(v_f+v_r)}{V}\right)\sqrt{v_f}$

where $d$ is the distance between capacitor plates and $V$ is our accelerating voltage.

It turns out that Stokes' Law is not a good description if the velocity of the droplet's fall is less than $0.1 \mathrm{cm/sec}$. (Droplets having this and smaller velocities have radii of about 2 microns, which is comparable to the mean free path of air molecules; a condition which violates one of the assumptions made in deriving Stokes' Law.) Since the velocities of droplets in this experiment will be in the range of $0.01$ to $0.001 cm/sec$, a correction factor must be applied to eq. . This factor is

$\left(\dfrac{1}{1+b/pa}\right)^{3/2}$

Where $p$ is the atmospheric pressure (easy to measure), $a$ is the droplet radius (already calculated), and $b$ is a constant term, which for our setup should be $b=6.17\times 10^{-4}\mathrm{cm}\cdot\mathrm{Hg \;cm}$

It should be noted that Millikan was doing experiments in 1911 (two years before the oil-drop experiment) involving corrections to the Stokes equation, so the need for this subtle correction would have been more apparent to him. (see The Isolation of an Ion, a Precision Measurement of its Charge, and the Correction of Stokes's Law)

$q = C_1 G(v_f) \frac{d}{V} (v_f+v_r)\sqrt{v_f}$

where $C_1 = \dfrac{4}{3}\pi \sqrt{\dfrac{1}{g\rho} \left(\dfrac{9\eta}{2}\right)^3}$ is a constant for the experiment, $G(v_f) = \left(\dfrac{1}{1+b/pa}\right)^{3/2} = \left({1+\dfrac{b}{p\sqrt{\dfrac{9}{2}\dfrac{\eta v_f}{g\rho}}}}\right)^{2/3} = \left(1+\epsilon\right)^{2/3}$ is a velocity-dependent correction to Stokes' Law $\epsilon = \dfrac{b}{p\sqrt{\dfrac{9}{2}\dfrac{\eta v_f}{g\rho}}}$

Finally $\left(\dfrac{d\cdot(v_f+v_r)}{V}\right)\sqrt{v_f}$ contains our measured velocities (i.e. the bulk of our data)

Before you begin, you'll need to make sure to remove any excess oil from the capacitor and its housing.

• Disassemble the capacitor and wipe down all surfaces with dry Kimwipes.

One variable you'll need to know is how far apart the plates of the capacitor are from one another.

• Measure the plastic spacer thickness (take care not to scratch it).

Place a bubble level on the capacitor's bottom plate and ensure that the chamber isn't skewed in any direction.

• Reinstall the plastic spacer, oriented so that the laser beam passes through one hole and such that there is a clear path for light to the microscope
• Turn on the laser and make sure that it is cleanly passing through the spacer.
• Replace the top capacitor plate (make sure the side with a recess is pointed upwards)
• Place the cylindrical housing around the capacitor, aligning its windows to allow light from the laser to pass to the microscope

Before you completely finish assembling the setup, you should use this opportunity to focus the microscope.

• Insert a thin wire into the hole in the top plate, and adjust the camera focus to get a sharp image of the wire on the screen.
• Use the focusing lever to get the pair of reticule lines into sharp focus as well
• show diagram

Now you can remove the wire, place the droplet hole cover on top of the capacitor plate, and then place the housing cover on top of the chamber and secure everything with the hold down clamp.

In order to create our charged oil droplets, we have a sprayer filled with some non-toxic oil.

• Prime the sprayer by squeezing the bulb several times, until you can see a mist of droplets emerging from the end.
• Place the tip of the atomizer into the central hole in the chamber cover and give it a squeeze. You should see something looking like a cloud of points or bubbles; spray again if needs be.

We should note that while Millikan employed a much higher voltage in his setup (up to 10,000 V), he also had his capacitor plates spaced much further apart. As such, by moving our plates closer together we can get similar field strengths while employing much lower voltages.

• To clear out most of the highly charged drops, turn on the field for several seconds.

As an alternative method, a drop can be balanced so as to be stationary by carefully setting the electric field. The required voltage (along with the free-fall time) should be enough to determine the charge on a given droplet.