ONE MIGHT SUPPOSE THAT DEMONSTRATIONS of advanced concepts in physics would be beyond the reach of the amateur experimenter. Three such experiments have been devised, however, by Richard E. Crandall of Reed College. One experiment demonstrates the Doppler shift of light, converting the phenomenon into sound. A second measures Planck's constant, which figures importantly in quantum mechanics. The third measures the universal gravitational constant, which does the same in Newton's theory of gravitation. In each experiment Crandall has minimized the difficulties.

Interference patterns of light are standard fare in physics classrooms and are familiar to many amateurs. Normally the pattern is seen directly by the experimenter or is scanned with a light detector. Crandall and his student Edward H. Wishnow devised a method in which the pattern can be heard.

A beam from a two-milliwatt helium-neon laser is sent through a glass slide. Part of the beam reflects from the slide and illuminates a phototransistor (Motorola MRD3052 NPN or an equivalent type) that serves as a light detector. The rest of the beam passes through the slide and then reflects from a mirror (or any thing shiny) mounted on the cone of a small loudspeaker. Part of the reflected light illuminates the phototransistor and interferes with the light arriving directly from the glass slide.

The transistor is biased (given a reference level) with a 15-volt source in line with a 30,000-ohm resistor. When the transistor is illuminated by light from the slide, it generates a voltage that is detected by a standard stereo amplifier. The amplified signal goes to a second loudspeaker that makes it audible. The other output channel of the amplifier is connected to the speaker that reflects part of the laser beam.

When the mirror moves, the signal at the transistor varies and so does the output of sound from the sound-producing speaker. Even small disturbances of the air in the room provide sufficient motion for a noticeable variation in the sound. More control of the variation is achieved by feeding a sine-wave signal into the second of the input channels of the amplifier. This signal is sent to the reflecting speaker, forcing it to oscillate steadily. The sound output from the apparatus then varies continuously.

The interference of the light at the transistor can be explained in two ways. In one view the critical factor is the difference between the distance the light travels after it is reflected from the slide and the distance it travels after it is reflected from the mirror. When the light waves emerge from the laser, they are all approximately in phase. Some of them reflect from the slide and travel to the transistor. The rest of the waves travel a different path to reach the mirror and the transistor.

The difference between the length of the path traveled by one set of waves and the length of the path traveled by the other determines the interference at the transistor. If the path lengths are such that the two sets of waves arrive in phase, the interference is constructive: the illumination of the transistor is bright and the transistor puts out a strong signal to the amplifier. If the path lengths are such that the two sets arrive out of phase, the interference is destructive: the illumination of the transistor is dark and a weaker signal is sent to the amplifier.

Suppose the first light waves arrive in phase. Also suppose the mirror moves away from the laser by a distance equal to one-fourth of the wavelength of the laser light. The light reflecting from the speaker then has an additional distance to travel that is equal to half of the wavelength of the light. (It must travel through an extra one-fourth wavelength to reach the speaker and then back again to reach the transistor.) The additional distance puts the light waves at the transistor out of phase. Thus as the cone of the reflecting speaker moves parallel to the light beam it varies the strength of the signal from the transistor.

One can hear the random motion of the mirror as it is disturbed by slight movements of air in the room. When a sine-

wave signal drives it in a more controlled oscillation, the variation in the interference at the transistor generates a repetitive pattern. The mirror moves through much more than one-fourth of the wavelength of the light, so that during each oscillation the interference at the transistor reaches many strong and weak levels.

A different but related explanation of the experiment is based on the Doppler shift of light. Wavelength is related to frequency: the number of waves that pass per second. The frequency of the light perceived by an observer can depend partly on his motion with respect to the source of the light. Assume the observer is stationary with respect to the light source. He then perceives light at a certain frequency. If the source moves toward the observer, the perceived light is shifted to a higher frequency. This shift is often called a blue shift because it is toward a higher frequency and so toward the blue end of the spectrum of visible light. If the source moves away from the observer, the perceived light is shifted to a lower frequency. This shift is often called a red shift. Both blue and red shifts are Doppler shifts.

A Doppler shift in frequency also results, of course, if the observer moves toward or away from the source. The degree of the shift depends on the relative speed of the source and the observer. The closer that speed is to the speed of light, the larger the Doppler shift is. This fact is crucial in inferring the motion of a distant astronomical object. The object's Doppler-shifted light can be measured and its speed toward or away from the earth can be calculated. In the experiment done by Crandall and Wishnow the mirror moves relative to the transistor at a speed that is low compared with the speed of light. The resulting shift in the frequency of the light is quite small. The light is actually shifted twice in frequency, once because the mirror moves relative to the laser and again because it moves relative to the transistor. In the first instance the light source is the laser and the mirror is the observer. In the second the mirror acts as a source (since it is sending out light) and the transistor is the observer.

Normally the special theory of relativity is required to calculate the Doppler shift of light. The speed of the oscillatory motions of the loudspeaker is so low compared with the speed of light, however, that an approximation can be made on the basis of classical physics. The shift in frequency is equal to twice the unshifted frequency multiplied by the ratio of the speed of the mirror to the speed of light.

The light arriving at the transistor from the glass slide is unshifted. The light from the moving mirror is shifted up or down in frequency depending on which way the mirror is moving when the light reaches it. These two signals interfere with each other at the transistor to create what is called a beat frequency. The net illumination at the transistor varies between constructive and destructive interference with a beat frequency that is equal to the difference between the unshifted frequency of the light and the shifted frequency. The experimenter hears this beating as a variation in the sound emerging from the sound-producing speaker. In other words, he hears a pattern that is related to the Doppler shift of light.

At the end of the 19th century the subject of thermal radiation presented a serious problem in physics. Every body radiates electromagnetic waves according to the temperature of its surface. Some of the radiation might be in the visible range of the electromagnetic spectrum, so that it can be seen, but every surface also emits radiation over a broader range of the spectrum. For example, a poker heated in a fireplace gives off infrared radiation that can be felt long before the surface is hot enough to glow in the visible range.

Armed with classical thermodynamics and the fresh brilliance of James Clerk Maxwell's theory of electromagnetic radiation, physicists attempted to derive an expression for the intensity of the thermal radiation at a particular frequency. Repeated efforts yielded only approximations. At low frequencies a mathematical expression derived by Lord Rayleigh approximated the experimental results, but at high frequencies the expression failed. Indeed, it predicted that the power radiated at high frequencies would be infinitely large, which was clearly wrong. This mathematical disaster was known as the ultraviolet catastrophe. Earlier Wilhelm Wien had devised an expression that worked well at high frequencies but was obviously wrong at low frequencies.

The problem was resolved in 1900 when Max Planck presented an equation that closely agreed with the experimental evidence at all frequencies. The equation was almost a product of luck, because it was not until several weeks later that Planck understood why it worked. In his theoretical model of a radiating surface he imagined that the radiation was emitted from many small oscillators. The intensity of the radiation at a given frequency was simply the combined output from a distribution of these oscillators The surprising feature of the model was that the oscillators could not have just any energy on a scale of energies. Instead each oscillator was limited to particular energy values that were integral multiples of a fundamental value given by the product of a constant (now called Planck's constant h) and the frequency f of the oscillator.

Planck had no information about the atomic nature of a radiating surface, and his imaginary oscillators certainly did not exist. With his crucial discovery, however, he was able to justify his equation for the thermal radiation as a function of frequency. It was more important that Planck's work ushered in the age of quantum mechanics. On the microscopic scale energy values were limited to certain distinct quantities that were integral multiples of hf.

Although Planck's constant is an important value in the most advanced quantum mechanics of atoms and fundamental particles, it can be measured in a surprisingly simple experiment devised by Crandall and his colleague Jean F. Delord. The apparatus consists of a color filter, a light detector and an un-frosted 60-watt tungsten-filament light bulb. Light from the filament of the bulb is filtered so that only a narrow range of frequencies reaches the detector. The intensity of the light in that range is measured for two levels of power delivered to the light bulb, and the results are inserted in an equation that yields a value for Planck's constant.

The filament of the light bulb is heated by the passage of current and emits light across a broad band of frequencies in the visible and infrared ranges. The . total power radiated from a surface was computed by Josef Stefan in 1879 as being equal to the product of four factors: (1) the surface area, (2) a constant known as Stefan's constant, (3) the absolute temperature of the surface raised to the fourth power and (4) a value known as the emissivity of the material.

At given values of surface area and temperature a material can theoretically radiate a certain maximum amount of power. Such a material is assigned an emissivity of 1 and is called a blackbody radiator. No such ideal material exists; all real surfaces radiate less power. Each surface is assigned an emissivity smaller than 1 according to how well it radiates. The filament in the light bulb employed in the experiment runs hot enough to be almost a black-body radiator. Hence Crandall and Delord have made the approximation of assigning it an emissivity of 1.

Planck's constant appears in the modern expression of Stefan's constant, so 9 that it is related to the total power radiated by the filament. It also appears in the equation devised by Planck for the intensity of the light radiated at any given frequency. Crandall and Delord perform two sets of measurements in order to get a value for Planck's constant. In each set they find the total power radiated by the filament by measuring the electric power delivered to the bulb. They also measure the intensity of the light passing through the color filter. This set of measurements is then repeated for a different amount of power delivered to the bulb. The data are inserted into an equation that is the combination of Stefan's law (the total power radiated) and the equation from Planck (the intensity of the light at a certain frequency).

The combined equation is shown in Figure 3. The equation incorporates an approximation in order to simplify the mathematics, but the accuracy of the final value of Planck's constant is hardly altered by it. The symbols I and P represent the intensity and power Crandall and Delord measure in the experiment. The equation also requires a value for the speed of light c, which is 3 X 10⁸ meters per second. The symbol f is the frequency of the light passed by the filter. Since the filter actually passes not one frequency but a narrow range of frequencies, the value for this factor is chosen as being the center of the range. If the range were narrower, less error would be introduced by that approximation.

The symbol A in the equation is the surface area of the filament. To measure it Crandall and Delord project a shadow image of the filament onto a wall. The dimensions of the image are measured and then scaled down by the magnification of the projection. You can determine the magnification easily by placing a transparent ruler near the bulb and measuring its image on the wall. The value for A in their experiment was 5.2 X 10^-5 square meter.

A Variac controls the power delivered to the bulb. To calculate the power Crandall and Delord measure the voltage across the .l-ohm resistor that is in line with the bulb. Dividing the voltage by the resistance gives the current flowing through the resistor and thus also through the filament.

Next Crandall and Delord measure the voltage across the bulb. Multiplying this voltage by the current gives them the power delivered to the bulb. Crandall says that if a Variac is not available, a 12-volt battery can be substituted, but one would then have to adjust the current with an appropriate resistor in order to deliver two different levels of power to the bulb.

The filter placed between the bulb and the photodetector is a green cellophane one bought from Corion Instruments, Inc. (73 Jeffrey Avenue, Holliston, Mass. 01746). The frequency entered in the calculation of Planck's constant was 5.3 X 10¹⁴ hertz. Similar filters can be obtained from other scientific-supply companies. Different colors can be substituted. The band pass, that is, the frequency range, of the filter should be as narrow as possible, but even an inexpensive filter will serve for determining a good value for Planck's constant.

The detection circuit consists of a phototransistor (Motorola MRD3052 or its equivalent), a 1,000-ohm resistor and a 12-volt battery. The spectral response of the phototransistor is not important because the frequency of the light illuminating the transistor does not vary. The current in the detection circuit is determined by measuring the voltage drop across the resistor. The current need not be converted into units of light intensity. Since a ratio of only two intensities is required in the equation, the units cancel.

Crandall sent me the following example of the data. With a power input of 19.3 watts to the bulb the current in the detection circuit was .81 milliampere. When the power input was changed to 36.5 watts, the current changed to 7.57 milliamperes. From these data Crandall and Delord calculated Planck's constant to be 4.9 X 10^-34 joule-second, which is; close to the accepted value of 6.6 X 10^-34 joule-second. Their value is off somewhat because of the finite band pass of the filter and because the emissivity of the filament is not exactly 1.

Of the several fundamental forces in the universe the most noticeable in everyday life is gravitation. The gravitational pull on one's body from the presence of the earth is appreciable only because of the large mass of the earth. The gravitational attraction between any pair of common objects around the house is so small that it goes unnoticed. It is overwhelmed by much larger forces such as friction.

How does one demonstrate and measure the gravitational force between two masses in a laboratory? This question was first tackled by Henry Cavendish in 1798. His apparatus consisted of a six-foot wood arm suspended horizontally by a thin wire. At each end of the arm he mounted a lead ball about two inches in diameter. The suspension was extremely sensitive to any force tending to rotate the arm about the suspension wire.

To present such a force a third lead ball was brought near one of the mounted balls. Its gravitational pull on the mounted ball exerted a force tending to rotate the arm about the suspension wire. To increase the force a fourth lead ball was brought on the other side of the ball at the other end of the arm.

Cavendish was careful to shield the apparatus from air currents. He also arranged for the third and fourth balls to be moved into position by remote control so that the mass of his own body would not interfere with the experiment. The angle of rotation was measured by monitoring a light beam reflected from a small mirror on the wire from which the beam was suspended.

With the third and fourth lead balls in position the arm rotated slightly until the gravitational pull was countered by the torque from the twisted wire. This position of equilibrium was measured Then the third and fourth balls were repositioned so that they exerted forces in the opposite direction. Again the angle of equilibrium was measured. The angular difference between the two positions of equilibrium was divided by two to give an accurate measure of the rotation in each case.

With these numbers Cavendish was able to compute the strength of the gravitational force between the pairs of lead balls at each end of the arm. From work by Isaac Newton the mathematical form of the gravitational force between two masses was already known. Its strength depends directly on the two masses attracting each other and inversely on the square- of the distance. between them. It also depends on the factor called the gravitational constant. Newton was effectively stymied on how to compute the-constant because his only demonstration of gravitation was based on at least one large object whose mass was not well known (such as the earth). The Cavendish experiment was exciting because it was based on smaller masses that could be measured easily. Cavendish computed the gravitational constant as being 6.754 X 10^-11N(m²/kg.²), close to the currently accepted value of (6.668 +/- .005) X 10^-11N(m²/kg.²). (N is newtons, m is meters, kg. is kilograms.)

The Cavendish experiment is not an ordinary demonstration in physics because it is arduous to do. The arm takes a good deal of time to settle down into its equilibrium position. Any perturbation from the environment (for example from the experimenters' moving around the laboratory) adds further to the settling time. As a result-the demonstration often takes hours. Working from a prototype built by Greg Eibel of Reed College, Crandall has devised a clever electronic version of the Cavendish experiment that takes only minutes and yields a value for the gravitational constant of (7.5 + 1.5) X 10^-11N(m²/kg.²).

Whereas the classical Cavendish experiment depends on the rotation of an arm, the experiment devised by Crandall depends on preventing such a rotation. The strength of the gravitational force is measured from the magnetic force that is required to prevent the rotation of a torsion bar when an additional mass is brought close to it. The position of the torsion bar is monitored by an optical sensor.

When the additional mass is brought near one of the masses mounted on the bar, the start of rotation by the bar triggers a servomechanism that exerts enough magnetic force on the bar to stop the rotation. By calibrating the servomeehanism Crandall-gets a measure of the gravitational pull on the mass mounted on the bar. The entire experiment can take as little as 20 minutes.

The bar is suspended by a length of piano wire that runs from an overhead mount down to a rigid rod attached to the middle of the bar. The rod is not necessary; but it helps to damp out vibrations of the wire. Masses of 10 kilograms are mounted at both ends of the bar as in the classical-apparatus. Near one end a magnet is glued into a hole extending through the bar. When the bar begins to rotate, a solenoid in the servomechanism pulls magnetically on the magnet to eliminate the rotation.

The optical sensor consists of an infrared emitter and an infrared detector positioned near the end of the bar with the magnet. The circuitry of the apparatus is shown in the illustration below. The emitter is a simple diode that radiates a small amount of heat as current from a constant-voltage source flows through it. The detector is a transistor with electrical characteristics that change when it is exposed to heat (The combined emitter and detector can be bought as an optical interrupter unit, one such being the General Electric H13A1.)

Between the emitter and the detector a vane is attached to the bar. Initially the vane shields half of the detector from the radiation. As the bar begins to rotate in one direction the detector is more exposed and the servomechanism acts to restore the bar to its initial orientation. Rotation in the other direction shields more of the detector and again brings the servomechanism into play.

The output of the detector passes through a network of two operational amplifiers and other devices. Thc system constantly compares the detector's output to a reference voltage of two volts. If the bar is in its proper position and the vane shields half of the detector from the infrared emitter, the output matches the reference voltage. If the bar rotates so- as to expose the infrared detector, the match is disrupted and the system sends current through the solenoid, generating a magnetic field that pulls on the magnet attached to the bar. The amount of current required for the correction is read indirectly from the change in the output voltage (V_o in the illustration). This output appears on a voltmeter or a stripchart recorder.

The first step in calibrating the apparatus is to calculate the torsion constant of the wire from which the bar is suspended. When the bar is rotated through an angle, the twisted wire supplies a countering torque equal to its torsion constant multiplied by the angle of rotation. To compute the torsion constant Crandall determines two numbers. With the servomechanism turned off he makes the bar swing around its equilibrium position and measures the period of swinging. This amount of time is called the free period of the bar. Next the bar's moment of inertia (with respect to the point of suspension) is computed. The torsion constant of the wire is then calculated by the formula k = I(2/P)², where k is the torsion constant, I is the moment of inertia and P is the free period.

The next step is to relate the output voltage from the servomechanism to a torque on the bar. With the servomechanism turned on the support for the suspension wire is moved through a measured angle in order to twist the wire. The bar remains in place because of the magnetic pull from the solenoid. Thus the output voltage from the circuit shifts. The voltage shift is divided by the angle of twist and the torsion constant to yield what Crandall calls the torsional gain of the apparatus. This factor (volts per unit of torque) allows him to convert any output voltage measured during an experiment into the value of the torque on the bar because of the gravitational pull on one end of the bar.

For example, when a 15-kilogram mass almost touches one of the mounted masses, the output voltage changes by about 25 millivolts. The strength of the force on the mounted mass is computed by multiplying the shift in the output voltage by half the length of the bar and dividing by the torsional gain. The gravitational constant is calculated by inserting this force into Newton's equation for gravitation, along with the distance between the center of the 15-kilogram mass and the center of the mounted mass.

To ease the introduction of an additional mass Crandall sometimes employs a rotatable plank with a 15-kilogram mass mounted at each end. It functions something like the revolving tray known as a lazy Susan. By rotating the plank in one direction one of the masses is brought to a side of the mass mounted on the bar. The bar is pulled toward the additional mass but barely moves because of the servomechanism. When the plank is rotated the other way, the other 15-kilogram mass is brought to the opposite side of the mounted mass.

The settling time for the apparatus is set by the compensation capacitor, which has a value of about one microfarad. When Crandall brings a 15-kilogram mass near one end of the bar, the shift in the output voltage: appears after only a few minutes. This speed makes Crandall's version of the Cavendish experiment ideal for a classroom demonstration. An accurate measurement of the gravitational constant calls for only 20 minutes or so of observation.

For his work on the gravitational constant Crandall won the 1981 Apparatus Competition of the American Association of Physics Teachers. This experiment and the one measuring Planck's constant will appear soon in America Journal of Physics. The experiment with the Doppler shift of light has already been published there. Crandall's colleague. Bruce Eaton contributed ideas to some of these projects.

Bibliography

LISTENING TO THE DOPPLER SHIFT OF VISIBLE LIGHT. R. E. Crandall and E. H. Wishnow in American Journal of Physics, Vol. 49, No. 5. Pages 477-478; May 1978.

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