Millikan's Oil Drop Experiment: How to Determine the Charge of an Electron
The Discovery of the Electron's Charge
In 1897 J. J. Thomson demonstrated that cathode rays, a new phenomenon, were made up of small negatively charged particles, which were soon named electrons. The electron was the first subatomic particle ever discovered. Through his cathode ray experiments, Thomson also determined the electrical charge-to-mass ratio for the electron.
Millikan's oil-drop experiment was performed by Robert Millikan and Harvey Fletcher in 1909. It determined a precise value for the electric charge of the electron, e. The electron's charge is the fundamental unit of electric charge, because all electric charges are made up of groups (or the absence of groups) of electrons. This discretisation of charge is also elegantly demonstrated by Millikan's experiment.
The unit of electric charge is a fundamental physical constant and crucial to calculations within electromagnetism. Hence, an accurate determination of its value was a big achievement, recognised by the 1923 Nobel prize for physics.
Millikan's experiment is based around observing charged oil droplets in free fall and in the presence of an electric field. A fine mist of oil is sprayed across the top of a perspex cylinder with a small 'chimney' that leads down to the cell (if the cell valve is open). The act of spraying will charge some of the released oil droplets through friction with the nozzle of the sprayer. The cell is the area enclosed between two metal plates that are connected to a power supply. Hence an electric field can be generated within the cell and its strength varied by adjusting the power supply. A light is used to illuminate the cell and the experimenter can observe within the cell by looking through a microscope.
As an object falls through a fluid, such as air or water, the force of gravity will accelerate the object and speed it up. As a consequence of this increasing speed, the drag force acting on the object, that resists the falling, also increases. Eventually these forces will balance (along with a buoyancy force) and therefore the object no longer accelerates. At this point the object is falling at a constant speed, which is called the terminal velocity. The terminal velocity is the maximum speed the object will obtain while free falling through the fluid.
Millikan's experiment revolves around the motion of individual charged oil droplets within the cell. To understand this motion the forces acting on an individual oil droplet need to be considered. As the droplets are very small, the droplets are reasonably assumed to be spherical in shape. The diagram below shows the forces and their directions that act on a droplet in two scenarios: when the droplet free falls and when an electric field causes the droplet to rise.
The most obvious force is the gravitational pull of the Earth on the droplet, also known as the weight of the droplet. Weight is given by the droplet volume multiplied by the density of the oil (ρoil) multiplied by the gravitational acceleration (g). Earth's gravitational acceleration is known to be 9.81 m/s2 and the density of the oil is usually also known (or could be determined in another experiment). However, the radius of the droplet (r) is unknown and extremely hard to measure.
As the droplet is immersed in air (a fluid) it will experience an upward buoyancy force. Archimedes' principle states that this buoyancy force is equal to the weight of fluid displaced by the submerged object. Therefore, the buoyancy force acting on the droplet is an identical expression to the weight except the density of air is used (ρair). The density of air is a known value.
The droplet also experiences a drag force that opposes its motion. This is also called air resistance and occurs as a consequence of friction between the droplet and the surrounding air molecules. Drag is described by Stoke's law, which says that the force depends on the droplet radius, viscosity of air (η) and the velocity of the droplet (v). The viscosity of air is known and the droplet velocity is unknown but can be measured.
When the droplet reaches its terminal velocity for falling (v1), the weight is equal to the buoyancy force plus the drag force. Substituting the previous equations for the forces and then rearranging gives an expression for the droplet radius. This allows the radius to be calculated if v1 is measured.
When a voltage is applied to the brass plates an electric field is generated within the cell. The strength of this electric field (E) is simply the voltage (V) divided by the distance separating the two plates (d).
If a droplet is charged it will now experience an electrical force in addition to the three previously discussed forces. Negatively charged droplets will experience an upwards force. This electrical force is proportional to both the electric field strength and the droplet's electrical charge (q).
If the electric field is strong enough, from a high enough voltage, the negatively charged droplets will start to rise. When the droplet reaches its terminal velocity for rising (v2), the sum of the weight and drag is equal to the sum of the electrical force and the buoyancy force. Equating the formulae for these forces, substituting in the previously obtained radius (from the fall of the same droplet) and rearranging gives an equation for the droplet's electrical charge. This means that the charge of a droplet can be determined through measurement of the falling and rising terminal velocities, as the rest of the equation's terms are known constants.
Firstly, calibration is performed such as focusing the microscope and ensuring the cell is level. The cell valve is opened, oil sprayed across the top of the cell and the valve is then closed. Multiple droplets of oil will now be falling through the cell. The power supply is then turned on (to a sufficiently high voltage). This causes negatively charged droplets to rise but also makes positively charged droplets fall quicker, clearing them from the cell. After a very short time this only leaves negatively charged droplets remaining in the cell.
The power supply is then turned off and the drops begin to fall. A droplet is selected by the observer, who is watching through the microscope. Within the cell, a set distance has been marked and the time for the selected droplet to fall through this distance is measured. These two values are used to calculate the falling terminal velocity. The power supply is then turned back on and the droplet begins to rise. The time to rise through the selected distance is measured and allows the rising terminal velocity to be calculated. This process could be repeated multiple times and allow average fall and rise times, and hence velocities, to be calculated. With the two terminal velocities obtained, the droplet's charge is calculated from the previous formula.
This method for calculating a droplet's charge was repeated for a large number of observed droplets. The charges were found to all be integer multiples (n) of a single number, a fundamental electric charge (e). Therefore, the experiment confirmed that charge is quantised.
A value for e was calculated for each droplet by dividing the calculated droplet charge by an assigned value for n. These values were then averaged to give a final measurement of e.
Millikan obtained a value of -1.5924 x 10-19 C, which is an excellent first measurement considering that the currently accepted measurement is -1.6022 x 10-19 C.
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