The electrical charge carried by an electron is a fundamental constant of physics. During the years 1909Ç1933, the American physicist Robert A. Millikan originated the oil drop experiment to demonstrate and measure this quantum of electrical charge. Small drops of oil were placed between two charged horizontal parallel plates, subjecting them to a combination of electrical, gravitational, and viscous retarding forces. The drops were small enough to have acquired random charges equal to small integral multiples of the quantum of charge. Millikan analyzed the motion of the drops under the influence of an electric field to determine the charge on an electron.
The size and mass of oil drops vary, and values must be determined indirectly, making Millikanºs original experiment complex. This apparatus substitutes small plastic spheres of known uniform size and density (unavailable to Millikan) to simplify analysis of results. The occasional sphere fragments or clusters can be quickly distinguished by their size and velocity compared to most spheres, and thus can be disregarded.
Stationary particles in an electric field: If m is the mass of the sphere under observation, the gravitational force on it is:
Fg = mg (1)
If the electric field E is varied until the electric force FE on the sphere equals the gravitational force Fg, the sphere will remain stationary. The viscous force will be zero because there is no net movement, so:
Fg = FE or mg = FE (2)
The electric field E is defined as the force exerted on a unit charge at a given point; so the force FE on a particle of charge q can be expressed as:
FE = qE (3) thus: q = mg/E (4)
In the general case, the electric field is a vector, equivalent to: E = - F(dV,dr) the potential gradient at a given point in the field. Since the field between two parallel charged plates is always uniform near the center of the plates,
E = - F(dV,dr) = F(V,d) and hence
q = mgd/V (5)
where d is the distance between plates and V is the voltage across the plates. Since m, g, and d are constant: q F(1,V).
The mass m of a sphere may be computed from the radius and density specified. The plate spacing d (4.0 mm) and the gravitational acceleration g (9.8 m/s2) are known. Thus to calculate the charge q on a particular sphere, you need only measure V using a voltmeter. When many observations are made, the resultant calculated values of q will be found to be integral multiples of a certain small value. This value is the fundamental unit (or quantum) of charge.
NOTE: Forces, electric field, and velocity are all vector quantities which, in these experiments, can only be directed up or down. All symbols for these quantities will thus refer to scalar magnitudes, not vectors, unless otherwise noted. Similarly, q will refer to only the magnitude and not the sign of a charge unless otherwise noted.
Moving particles in the absence of a field: A sphere moving through a fluid medium at a constant velocity v is subject to a viscous retarding force Fa given by Stoke's Law:
Fa = Krv (6)
where K is a constant which depends only on the fluidºs viscosity and r is the radius of the sphere. Since r is a constant in these experiments, Kr can be replaced by a constant C, and equation (6) becomes:
Fa = Cv (7)
In the absence of an electric field, a free-falling particle quickly reaches a constant terminal velocity due to the retarding force of the fluid. At the terminal velocity, Fg = Fa so:
mg = Cvg (8)
where vg is the free-fall terminal velocity.
Moving particles in an electric field: An electric field E acting on a sphere with charge q applies a force on the sphere, FE given by equation (3). If this force is applied upwards, it will oppose the force of gravity. When the sphere reaches an equilibrium velocity ve, the sum of the forces on the sphere must be zero (the forces are in equilibrium). When the field is large enough to more than overcome the force of gravity (as is always the case in these experiments), FE is greater than Fg; the forces Fa and Fg are in the opposite direction as FE, so:
FE - Fa - Fg =0 or FE = Fa + Fg (9)
Plug in equations (3), (7), and (1) to get:
qE = Cve + mg (10) or q = (Cve + mg)/E (11)
Plug in equation (8) to get: q= C/E (ve + vg). (12) Analogous equations can be derived when FE is directed downward. For plastic spheres of uniform size in a constant electric field, a change in the charge q on the sphere results only in a change in equilibrium velocity ve:
q = (C/E) ve (13)
where the signs of q, E, and ve are considered. If many experimental values of Íve are measured, they are all found to be integral multiples of a certain small value. The same must thus be true for Íq: the charge gained or lost is an exact multiple of some small charge. Thus the quantization of charge may be demonstrated even without obtaining a numerical value for the fundamental charge.
Preparation for Use: Connect a 6.3 V supply to the blue binding posts marked 6.3 V. (WARNING: Do not connect the high voltage yet!) Turn on the power supply; the lamp should light.
Connect the high voltage supply, attaching positive to the red binding post, and negative to black. Turn the voltage to maximum and set the reversing switch to the center (shorting) position. For Experiment I, attach a voltmeter across the red and black binding posts to measure the potential difference between the two plates. Withdraw the injector nozzle until the tip is just out of the field of view of the microscope. The apparatus is now ready for use.
To begin you will, squeeze the rubber bulb several times to release some spheres into the viewing chamber. View the chamber through the microscope--the fluid quickly evaporates, leaving a cloud of spheres which look like bright points of light against the dark background. Since the microscope image is inverted, falling spheres will appear to move upward. Fine adjustment of the microscope focus may be needed. Experiment 1: This is the classic Millikan experiment for determining the charge of an electron.
Squeeze the rubber bulb several times to release some spheres into the viewing chamber. Most of the spheres will be electrically charged by the friction of injection. Move the reversing switch up or down to create an electric field and observe the effect on the spheres. The upper plate is positive with the switch in the Ñupæ position and negative with the switch in the Ñdownæ position. Some spheres will move up and others down, indicating that there are both negatively and positively charged spheres. The more highly charged spheres move faster and quickly disappear from the field of view. Vary the field intensity by means of the power supply voltage control and note the effect. Keep in mind that the microscope inverts the field, and the actual direction of motion is opposite the apparent one.
With the switch in the center position, the plates are shorted and there is no electric field; the spheres fall freely under gravity. Look for occasional clumps of spheres falling more rapidly and fragments falling more slowly than single whole spheres. Do not use these in your measurements
Turn the switch back to the up position, so that the top plate will be positively charged to attract negatively charged spheres. Set the applied voltage to about 200 V. Select a single sphere that is moving slowly in the electric field and try to make it stop by carefully adjusting the voltage. Observe it long enough to make certain it is motionless, then record the voltage across the plates as measured by the voltmeter. Remember: the smaller the charge, the higher the voltage required to stop the sphere. Record the stopping voltage V for a number of spheres, trying to pick out spheres requiring the highest voltages to stop--these are the ones with the lowest charges.
Calculation of the numerical value of an elementary charge. (The following calculations use mks units.) Each plastic sphere measures 1.01 microns = 1.01 x 10-6 m in diameter, and has a density of 1.05 g/cm3 = 1050 kg/m3 (these values are obtained from the label). Then the mass of a sphere is thus:
m = (r)(4/3pr3) (14)
The plate spacing d = 4.0 x10-3 m, and g = 9.8 m/s2. Calculate the value of the constant mgd:
mgd = ??? (15)
Use this value in equation (5) to calculate the charge on each sphere. For example: if a particular sphere is stopped by a potential of 130 V, its charge must be:
q = mgd/130 volts (16)
A handy way to examine this sort of data, obtained for multiple spheres is to use a chart like the one at the right. The charge on the spheres will appear to cluster around certain values. These values are the integral multiples of the charge on an electron, which you can now determine. Experiment II. This experiment demonstrates that particles quickly reach a terminal velocity proportional to the driving force.
Inject some spheres into the viewing chamber. Use a stop clock to measure the time for a sphere to move through two graduations of the reticule (a movement of 1 mm in the chamber). Measure the velocity of a single sphere in free fall at two different parts of the field of view. The two velocities should be essentially the same, indicating that the sphere has reached its terminal velocity.
Apply an electric field between the plates and measure the velocity of a sphere in two different parts of the field of view. The two velocities should again be essentially the same. Be sure to choose a slowly moving sphere. The plate voltage should be kept constant for all measurements at about 200 V (it is not necessary to know the exact value in this experiment).
To clearly show the relationship between velocity and driving force, make three velocity measurements on each of about ten spheres: a) the velocity under free fall, (b) the velocity when the electric and gravitational fields are in opposite directions, and (c) the velocity when the electric and gravitational fields are in the same direction. The forces acting on the sphere in each of these cases can be represented by vector arrows. (Recall that the viscous force exactly balances Fg and FE at terminal velocity; the Ñdriving forceæ in each figure below thus equals the viscous force. The total force is, of course, zero.).
To avoid using data from fragments or clusters, throw out data on any sphere with a free fall time markedly different from the majority. If all spheres measured have the same mass, and the field strength is held constant, the driving force will differ only due to the charge on the sphere q.
Plot a graph of velocity versus electric force, designating downward force and velocity as negative and upward as positive. The graph of the measurements on each sphere should be a straight line. The slopes will be different because there are different charges on the spheres and hence different values of FE. Your graph (like the example to the right) should show that velocity is proportional to driving force.
Experiment lll. This experiment demonstrates that the charge on each sphere is a multiple of some small quantum of charge. By equation (13), velocity is proportional to charge in a constant field, so we can analyze velocity data to determine charge. The procedure is similar to Experiment II; if desired, data from that experiment can be used for this one.
As noted before, sphere clumps and fragments must not be used. Since vdown is proportional to FE + Fg and vup is proportional to FE + Fg, then the vector sum, vup + vdown (for a given sphere), is proportional to 2Fg and the difference, vup - vdown , is proportional to 2FE. We can use the values Of vup + vdown to screen out all but single spheres, and the values of vup - vdown for these selected spheres to determine charge quanta.
Set up the apparatus as described in Experiment II. For each of a number of slowly moving spheres, measure the velocity vdown when FE is in the same direction as Fg and vup when FE opposes Fg. Carefully record this information then tabulate it in columnar format with a column for the sum of the velocities and one for the differences. Remember that the directions of vup and vdown are opposite when adding these quantities.
Examine the vup + vdown data and disregard values which are markedly different from average; these will be fragments or clumps with a different mass than that of a typical sphere. The values that remain will demarcate the Ñselected spheresæ.
Make a bar graph of vup - vdown, for the selected spheres. The data should indicate that these values are integral multiples of some small value and, therefore, that the charges on the spheres are also integral multiples of some small value. See the discussion following Experiment I for calculation of results