## Millikan Oil Drop Experiment Lab Conclusion Essay

This is my absolute favorite physics experiment; I actually wrote a miniature essay on my lab report containing, in essence, the following:

Millikan's experiment is amazingly indirect. You don't directly measure the charge of the electron, nor do you compute it from an equation (like, say, the one relating electrostatic force to charge, though you do use this equation to deduce the non-unit charge on each oil drop). You don't even deduce it by fitting a curve to a set of data points. You *do* deduce it from a frequency graph, but it is, again, not any kind of fitting of the actual frequency data; rather, you identify spikes in the frequency counts and mark off the corresponding charges. Then you find their *greatest common divisor* and argue that your data was random enough that this must be the charge of one electron (since it is vanishingly unlikely after enough droplets that each one contained only, say, even numbers of electrons). It is actually a simple form of image recognition.

You don't need to luck into one-electron droplets. You just need to keep measuring until you have an unambiguous set of frequency spikes at sufficiently many different charges and apply (what?!) a number-theoretic computation.

## Addendum

The business of taking the greatest common divisor is a bit tricky in the presence of measurement errors: after all, the charges you find are not only subject to error from your lab equipment but also from your identification of the center of each spike. You *can* assume the charges are all integers by converting your limited-precision floating point numbers to fixed point, but those integers are almost certain to have a GCD of 1. For example, if you measure charges of 201 and 302, you'll find that the fundamental charge is not 100 (which is the obviously correct answer) but rather 1.

You can, of course, eyeball it: say, you can take various ratios and fit them to nearby rational numbers with a small common denominator (in the above example, the ratio is approximately 1.5025, so you easily find 1.5 = 3/2 as a likely "correct" ratio). A better way is to use an "error-tolerant" version of Euclid's algorthm. In short, proceed as usual by dividing (with remainder) the smallest number into all the others and repeating, except that rather than waiting for all the remainders to be 0 (indicating that the last remainder was the GCD), you wait for them all to be "small" in some sense. Say, an order of magnitude smaller than the previous one.

Take the above example: Euclid's algorithm gives you the following sequence of remainders: 302, 201, 101, 100, 1 (each one is the remainder of the division of the previous two). This suggests that 100 is the correct GCD, as indeed it is. Amazingly, the algorithm actually wiped out the measurement errors and got the *exact* correct GCD; I don't know if this kind of "focusing" effect is typical or if I just happened to use the right numbers.

This only increases my love for this experiment.

**Millikan oil-drop experiment**, first direct and compelling measurement of the electric charge of a single electron. It was performed originally in 1909 by the American physicist Robert A. Millikan, who devised a straightforward method of measuring the minute electric charge that is present on many of the droplets in an oil mist. The force on any electric charge in an electric field is equal to the product of the charge and the electric field. Millikan was able to measure both the amount of electric force and magnitude of electric field on the tiny charge of an isolated oil droplet and from the data determine the magnitude of the charge itself.

Millikan’s original experiment or any modified version, such as the following, is called the oil-drop experiment. A closed chamber with transparent sides is fitted with two parallel metal plates, which acquire a positive or negative charge when an electric current is applied. At the start of the experiment, an atomizer sprays a fine mist of oil droplets into the upper portion of the chamber. Under the influence of gravity and air resistance, some of the oil droplets fall through a small hole cut in the top metal plate. When the space between the metal plates is ionized by radiation (e.g., X-rays), electrons from the air attach themselves to the falling oil droplets, causing them to acquire a negative charge. A light source, set at right angles to a viewing microscope, illuminates the oil droplets and makes them appear as bright stars while they fall. The mass of a single charged droplet can be calculated by observing how fast it falls. By adjusting the potential difference, or voltage, between the metal plates, the speed of the droplet’s motion can be increased or decreased; when the amount of upward electric force equals the known downward gravitational force, the charged droplet remains stationary. The amount of voltage needed to suspend a droplet is used along with its mass to determine the overall electric charge on the droplet. Through repeated application of this method, the values of the electric charge on individual oil drops are always whole-number multiples of a lowest value—that value being the elementary electric charge itself (about 1.602 10^{−19} coulomb). From the time of Millikan’s original experiment, this method offered convincing proof that electric charge exists in basic natural units. All subsequent distinct methods of measuring the basic unit of electric charge point to its having the same fundamental value.

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