A1

In this experiment you are to observe and make measurements on the diffraction patterns produced by single and double slits. Your General Physics book (Sears, Zemansky, and Young or Resnick and Halliday) will have a good elementary discussion of the phenomenon in the chapters on diffraction. There are several optics books in the library which give more complete discussions of these effects. The book, "Fundamentals of Optics", by Jenkins and White is recommended. Copies of the pertinent parts of chapters 15, 16, and 17 from Jenkins and White will be placed in the lab for local reference.

The slits are made by a process of electro-etching. They are mounted in <Picture> slide holders. Treat them gently as they are quite expensive. There are three sets of slits and combinations of slits. One slide has only single slits of various widths, a second slide has only double slits with four combinations of slit width and spacing, and a third one has a variety of multiple slits.

The diagram below shows the general arrangement of laser, slit, and screen to be used for these measurments. Following the diagram, place a slide in the holder provided and position it so that only one single slit or double slit is uniformly illuminated by the laser beam. No lenses are required. Observe the diffraction pattern on a white paper against the wall a distance L from the slit. You will need to darken the room somewhat in order to see some fine detail clearly.
<Picture>for small angles <Picture>

Equations (1) and (2) are the theoretical expressions for the intensity of the light in the diffraction patterns. They each predict that the intensity varies with angle <Picture>. Your measurements will be a series of y positions of the intensity maxima or minima. In order to compare your results with theory you need to convert the y values to angles; do this using the small-angle approximation given on the diagram above.

•1. Single slits.

For a single slit, measure the positions of 2 or 3 minima (centers of the dark bands) of its diffraction pattern. Since the center of the pattern <Picture> is not easy to locate precisely, it is recommended that you measure the distance between corresponding minima (2y on the diagram) on either side of the central peak, and then divide by 2. Also you will need to measure the distance L from the slit to the screen. Repeat these measurements for at least two of the four single slits on the slide. Be sure to record the width of the slit you are using. The equation for the light intensity of a single-slit diffraction pattern is <Picture> where b is the slit width and <Picture> is the wave length of the light. <Picture> is the intensity at the center of the pattern where <Picture>. This equation predicts minima (where the intensity I =0) when <Picture> and that occurs when <Picture> equals any integer multiple of <Picture>. Thus setting <Picture> yields <Picture> as the condition for minimum intensity. The first minimum occurs when n=1 and so on.

For each slit, use Eqn. (1) along with your data to calculate the slit width b. Compare your result with the value for b given on the slide. The wavelength of laser light is known to be 632.8 nm.

•2. Double slits.
The double-slit patterns are more complex because they contain both the double-slit and the single-slit pattern together. The theoretical intensity pattern is given by <Picture>
Notice that this equation is different from the single slit equation only by the addition of the <Picture> factor which results from the double slit. This term is a maximum (brightest spots) when <Picture> is an integral multiple of <Picture>. Thus setting <Picture> yields <Picture> as the condition for brightness, where d is the distance between the slits. Now the single-slit factor involving <Picture> also controls the intensity, so there will also be minima in locations determined by the slit width b.
For at least two double-slit combinations measure the positions of the bright, narrowly spaced spots using the same technique used to determine the single-slit minima. Use these measurements along with the known wavelength to determine the slit spacing d. Compare your results with the dimensions given on the slide. In addition, measure the position of the intensity minima which are caused by the single-slit diffraction (these minima are quite widely spaced compared to the double-slit effects). Use the data to determine the slit width b just as was done for the single slit patterns above. Compare the results to the values given on the slide. Notice that a double-slit maximum can fall on a single-slit minimum if d is an integral multiple of b. This results in what is known as a missing order in the double slit pattern.
You may wish to observe some patterns caused by other types of apertures. Multiple slits and holes which are symmetric polygons produce rather nice patterns. Feel free to try a variety of things.

 

A2

The charge carried by an electron is a fundamental constant of nature. R.A. Millikan, in 1909, was the first to show that this quantity was discrete and single valued. For this work, ant that on the photoelectric effect, he was awarded the Nobel Prize in 1923. In this adaptation of his experimental technique one can:

1. demonstrate the quantization of the charge on an electron,

2. determine the value of the elementary charge by the balancing field method, and

3. show that terminal velocity of a sphere is proportional to driving force (Stoke's Law).

The apparatus consists of (1) a viewing chamber, microscope, and illuminator for examining the movement of charged particles; (2) capacitor plates (part of the viewing chamber), switch, and binding posts for controlling the forces on the charged particles; and (3) a sphere injector for inserting particles into the chamber. The entire apparatus can be mounted on a support rod by means of a bottom hole and clamping screw; this allows the height to be adjusted to the observerOs eye level. Unlike Millikan who used oil drops, this experiment will be using small plastic spheres of known size and density. A bottle of plastic spheres suspended in solution is also included.

 

A3

The electrical charge carried by an electron is a fundamental constant of physics. During the years 1909Ð1933, 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 = m*g (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 = F*E or m*g = F*E (2)

The electric field E is defined as the force exerted on a unit charge at a given point; so the force F*E on a particle of charge q can be expressed as:

F*E = q*E (3) thus: q = m*g/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 = m*g*d/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 = K*r*v (6)

where K is a constant which depends only on the fluid Os 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 = C*v (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:

m*g = C*v*g (8)

where v*g 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:

q*E = C*v*e + m*g (10) or q = (C*v*e + m*g)/E (11)

Plug in equation (8) to get: q= C/E (v*e + v*g). (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 v*e:

q = (C/E) v*e (13)

where the signs of q, E, and ve are considered. If many experimental values of C*v*e are measured, they are all found to be integral multiples of a certain small value. The same must thus be true for C*q: 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 NupÓ position and negative with the switch in the NdownÓ 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 m*g*d:

m*g*d = ??? (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 = m*g*d/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 Ndriving forceÓ 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 Nselected spheres.

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.

 

A4

Measurement of the charge of the electron. The charge of the electron, e, is determined by measurements of the electrical, gravitational, and viscous drag forces on indivicual charged oil droplets in a replication of the original Millikan experiment.