Millikan Experiment

The Millikan experiment is one of the most fundamental experiments in classical physics and had a huge impact on how we understand our universe. It is named after the American physicist Robert Andrews Millikan (1868 - 1953) and is commonly presented in school classes or university lectures. With the help of this experiment, it can be shown that an elementary charge $e$ exists, and its quantity can be determined. The main idea is based on the fact that small drops of oil, that are created with the help of a sprayer, lost some of their electrons during this process and are hence slightly positively charged. These drops are then attracted by a positively charged capacitor plate and lifted upwards against the gravitational force. Measuring the relation between the charge and the radius of each drop results in discrete charge distribution and therefore proves the existence of the smallest charge quantity called the elementary charge. The setup is relatively simple. It mainly consists of a voltage source providing a variable voltage $U$ and a horizontal plate capacitor with a plate distance $d$. In addition, a microscope is used to magnify the area between these two plates and a lamp might be used to enlighten this part for improving the vision. With the help of a sprayer, small oil drops can be created and sprayed into the volume between the capacitor plates. The exact position of the drops is determined on a small scale. It is, however, very important to keep in mind that using a microscope might switch top and bottom, i.e. if an oil drop moves upwards, then in reality it sinks downwards. A drop of oil with the radius $r$ and the density $\varrho_o$ experiences the following gravitational force: $$F_G = mg = \varrho_o \frac{4}{3}\pi r^3$$ In the second step, the mass $m$ was replaced with the product of density and volume. The volume was then calculated using the formula for a perfect sphere. Since the oil drops are formed in the air, an additional buoyancy $F_B$ has to be taken into account which is given as $$F_B = \varrho_a \frac{4}{3}\pi r^3$$ where $\varrho_a$ is the density of the air. In order to calculate the charge of each oil drop, an electric field $E$ by adjusting the voltage of the power source to a certain value where any arbitrarily chosen drop does not sink any more. In this case, the electric force $$F_E = qE = q\frac{U}{d}$$ does fully compensate the gravitational force and buoyancy leading to the following equation: $$F_E = F_G + F_B$$ Inserting the equations mentioned above and solving the resulting equation for the charge $q$ leads to $$\boxed{q = \frac{(\varrho_o - \varrho_a)\frac{4}{3}\pi r^3 gd}{U}}$$ In principle, all values are now known for determining the charge of each oil drop. However, a problem arises when measuring the radius $r$, because the drops are very small. Therefore, another relation is taken into account. If the voltage source is switched off, all sinking drops reach a fixed speed after a short amount of time. If this condition is fulfilled, Stokes friction $$F_S = 6\pi\eta v r$$ compensates for the difference between the gravitational force and buoyancy: $$F_S = F_G - F_A$$ Inserting all formulas from above and solving the equation for $r$ results in the following formula for the radius $r$: $$\boxed{r = \sqrt{\frac{9v\eta}{2(\varrho_o - \varrho_a)g}}}$$ Thus, in order to measure the charge of each oil drop, first its radius $r$ has to be determined by measuring the sinking speed, and afterward, the result has to be inserted into the formula for calculating the charge $q$. This procedure has to be repeated for all drops and the values must be inserted into a graph with the radius on the $x$-axis and the charge on the $y$-axis. The possible results obtained from measuring the radius and charge of around 100 oil drops are clearly distributed around certain lines. All these lines are multiples of the value $1.6\cdot 10^{-19}\,\mathrm{C}$ which is very close to the known value of the elementary charge.