2

3

PHYSICS LAB NOTES

FOR

WAVES

AND

MODERN PHYSICS

EXPERIMENTS

PHYSICS 39

Los Angeles Harbor College

TABLE OF CONTENTS

1.Simple and Physical Pendulums .................. 5

2.Standing Waves on Strings ...................... 9

3.Air Column Resonance ........................... 11

4.The Speed of Sound in Air ...................... 13

5.Beats .......................................... 15

6.The Visible Spectrum ........................... 17

7.Thin Films ..................................... 21

8.Diffraction and Interference: Fringe Analysis .. 23

9.The Michelson Interferometer ................... 25

10.Atomic Spectroscopy ............................ 29

11.The Photovoltaic Cell .......................... 33

12.The Hydrogen Spectrum .......................... 37

13.The Ratio e/m for Electrons .................... 39

14.The Semiconductor Diode ........................ 43

15.The Integrated Circuit: Operational Amplifier .. 47

16.Logic Gates .................................... 51

17.Radioactivity .................................. 57

18.Millikan's Oil Drop Experiment ................. 59

19.The Spectrum Analyzer .......................... 67

20.Radioactivity: Half-Life ....................... 73

The Statistics of Measurement

The Least-Squares Fit to Data

1. SIMPLE AND PHYSICAL

PENDULUMS

PURPOSE: To experimentally determine the periods of both simple pendulums and physical pendulums, and compare the results with the theoretical period.

INTRODUCTION:

The ideal simple pendulum, shown to

the right, consists of a point of mass

m, hanging on the end of an assumed

massless string or rod of length L.

The mass is allowed to swing back and θ

forth and the time required for one

complete cycle is called the period T.

In the absence of damping (frictionalL

forces), the swinging motion will

continue indefinitely. However, all

real pendulums experience damping to

some extent, and analysis shows that m

the amplitude will decrease exponentially.

mg

The torque on this system equals force times moment arm,

so τ = -mg·Lsin(θ), and its angular acceleration is

α = d2θ/dt2. For small values of θ (typically less than

about 10°), sin(θ) ~ θ when θ is measured in radians. Substituting these three equations into Newton’s Second Law

in rotational form, τ = I·θ becomes

d2θ

-mg·Lθ = I· .Eq. 1

dt2

For the simple pendulum, I = mL2, so Equation 1 becomes

g d2θ

- ·θ = .Eq. 2

L dt2

The angular frequency ω is √g/L and the period T is given by

T = 2π· .Eq. 3

How is the period of the pendulum calculated when the amplitude isn't small? Here is a sketch of the solution: Using energy considerations, the elliptic integral that gives the period is

θo dθ

T = 2π·. .

0 √sin2(θo/2) - sin2(θ/2)

The integrand is expanded in a power series and each term is integrated. Keeping three terms for the period as a function of the initial angle θo in units of radians, we have

T = 2(1 + o2 + o4).Eq. 4

Consider the case when the mass of the pendulum cannot be thought of as a point mass. This is called physical pendulum. The same expression for the period derived above (Eq. 3) for small oscillations can be used but with the moment of inertia I retained, giving

I

T = 2π· .Eq. 5

mgd

where d is the distance from the axis of rotation to the center of mass, measured perpendicularly from the axis of rotation.

APPARATUS:

Photogate timer

Clamp, table

Photogate power supply

Clamp, 90° twist

Clamp support rod

Meter stick

Support rod

Pendulum clamp

Masses, hooked

Metal spheres, hooked

Balance

Graph paper

Scissors

String

Protractor

Physical pendulum

PROCEDURE:

A.SIMPLE PENDULUM

1.Measure the period for the simple pendulum for six different lengths, each time using a small amplitude. Use lengths of 10.0 cm, 20.0 cm, 30.0 cm, 50.0 cm, 75.0 cm and

100.0 cm. Time the period using the photogate timer set to pendulum mode and 1 ms.

2.Calculate the theoretical period for each length used and compare this to the measured value.

3.Graph the square of the period as ordinates (on the vertical axis) versus the lengths as abscissas (on the horizontal axis). What value of g do you derive from the slope of your graph, based on Equation 3?

4.For this step, choose a length for the pendulum of about half a meter. Measure the period for angles of 20°, 30°, and 40°.

5.Calculate the theoretical period for each angle in units of radians using Eq. 4, and compare this to the measured value.

Use 20° = 0.34907 radians,

30° = 0.52360 radians and

40° = 0.69813 radians.

B.PHYSICAL PENDULUM

1.Measure the length and mass of your physical pendulum. Replace the pendulum clamp with the short support rod attached to the clamp with the 90° twist.

2.Determine the period for two different values of d, where d is the distance from the axis of rotation to the center of mass. Choose one axis of rotation to be near the edge of the pendulum, and the other to be near the center. Do three independent trials for each moment arm in order to get a good value for the period. Remember that we are using small amplitudes.

3.Calculate the moment of inertia of the physical pendulum. For a uniform rod of mass m and length L, the moment of inertia around the center of mass is ICM = (1/12)mL2, and the parallel-axis theorem states that I = ICM + md2.

4.Use Eq. 5 to calculate the theoretical period of the physical pendulum, and compare it to the measured value.

2. STANDING WAVES ON STRINGS

PURPOSE:

a) To determine the speed of a standing wave traveling on a string.

b) To determine the effect of tension on wave speed.

c) To determine the effect of density on wave speed.

APPARATUS:

Masses, slotted

2-meter stick

Banana wire (2)

Mass hanger

Pulley, clamp-on

Balance, electronic

Power strip

Stroboscope

Battery charger, 6V

Electric tuning fork

String, 2 thicknesses

INTRODUCTION:

Standing waves are produced by the interference of two waves. A pulse sent down the string can be reflected back. The superposition of these two waves creates a resonance state when the length of the string is L = nλ/2, with the number of antinodes being n = 1, 2, 3 … .

When standing waves are produced, a condition of resonance exists between the vibrating source and the wave on the string.

Frequency of source = Frequency of wave.

T

Speed of a wave on a string:v = fλ and v = .

μ

PROCEDURE: Tuning

ForkString Pulley

T

Fig. 1

1.Take at least two meters of string, and calculate its linear density µ from

total string mass

µ = .

total string length

2.Set up the apparatus as shown in Figure 1, with the battery charger set to 6 volts. Adjust the screw alongside the tuning fork to obtain a noticeable vibration. Accurately measure the length of the string from the tuning fork to the pulley, between 1 and 2 meters.

3.Use the strobe light to measure the frequency of the fork. The stroboscope should read close to 3000 flashes per minute. Divide by 60 to obtain the frequency in hertz.

4.Add masses to the mass holder until a single large antinode exists between the tuning fork and the pulley. Half a wave exists between the tuning fork and the pulley, so L = λ/2

and λ = 2L.

5.Record the tension in the string, T = mg, with m being the total mass and g = 9.80 m/s2.

6.Remove masses in small increments until two antinodes appear, so L = 2(λ/2) and λ = L. Calculate the tension.

7.Repeat for L = 3(λ/2), L = 4(λ/2) and L = 5(λ/2). Find the percent difference between your measured speed v = fλ and the theoretical value.

8.Repeat steps 1 - 7 for a different string.

3. AIR COLUMN RESONANCE

PURPOSE: To make use of resonance modes

in an air column to determine:

a) The speed of sound.

b) The wavelength of sound at different frequencies.

INTRODUCTION:

L1 = (1/4)λ, so L2 = (3/4)λ, so L3 = (5/4)λ, so

λ = 4·L1 λ = (4/3)·L2 λ = (4/5)·L3

n = 1 mode n = 2 mode n = 3 mode

Fig. 1

Air disturbances such as those set up by a vibrating tuning fork can be sent down a tube closed at the bottom and open at the top. Resonant modes can be detected when sound waves going down the tube combine with reflected waves going up the tube, to produce standing waves.

(2n - 1)

Resonance occurs at L = · λ ,

4

with n = 1, 2, 3 ..., and with L as the distance from the top of the plastic tube to the top of the water.

APPARATUS:

Tuning forks, 512 Hz and 1024 Hz

Thermometer

Tuning fork mallet

Beaker

Resonance tube

PROCEDURE:

1.Firmly plug the black stopper at the end of the hose into the bottom of the vertical tube, and make sure that the tube is securely held in place by the top and bottom clamps.

2.Move the reservoir cup to the bottom of the apparatus and fill the cup. Move the cup back up to the top. The water level should be about 10 cm from the top of the tube. If not, add a little more water to make it so.

3.Strike the 512-Hz tuning fork with the tuning fork mallet, and hold the vibrating fork just above the air tube. Do not touch the tube with the tuning fork.

4.Lower the water level in the tube by lowering the reservoir cup, and listen for resonance, which occurs when the sound is loudest. This occurs when the top of the tube is the antinode, created by the vibrations of the tuning fork.

5.Record the position of the water level at resonance, and repeat twice to get an average value. You should find λ ~ 0.6 meters. Calculate the experimental speed of sound in air from

ve = fλ.

6.Theoretically, the speed of sound in air is expected to be

vt = 331 + 0.607·T, with vt in meters/second, and T as the temperature of the air in degrees Celsius. Calculate the percent difference between your measured value and the theoretical value.

7.Repeat this for the n = 2 mode and the n = 3 mode. Repeat this experiment with the 1024-Hz tuning fork. You should find

λ ~ 0.3 meters.

8. When finished, dry your equipment as thoroughly as possible.

4. THE SPEED OF SOUND IN AIR

PURPOSE:

To use lissajous figures to determine the speed of sound v and to determine the wavelength of sound at different frequencies.

To learn how to use an oscilloscope to graph a voltage signal against another voltage signal, rather than voltage against time.

INTRODUCTION:

Theoretically the speed of sound is vt = 331 + 0.607·T, with vt in meters/second and T in degrees Celsius. This is derived from

B γP γRT

vt = = = .

ρ ρ M

Experimentally, ve = fλ.

Sonic Waves

INFRASONIC AUDIBLE ULTRASONIC

20 Hz to 20,000 Hz

APPARATUS:

Thermometer

Masking tape

Banana wires (4)

BNC-to-banana connectors (2)

Oscilloscope

Function generator

Resonance tube, electronic

Resonance tube battery, electronic

Resonance tube microphone, electronic

PROCEDURE:

1.Attach a BNC-to-banana wire adapter to the OUTPUT of the signal function generator, and another to the CH2 INPUT (not the CH1 INPUT) of the oscilloscope. Connect the two adapters, red to red and black to black, with banana wires. Turn on the function generator, turn the Level to 1V-10V and rotate FINE fully clockwise. Set the frequency to 2000 hertz. Turn on the oscilloscope and set MODE to CH2. If you do not get a good sine wave, ask your instructor for help.

2.Use two banana wires to connect the function generator output to the speaker input of the electronic resonance apparatus. If you do not hear a tone, ask your instructor for help.

3.Tape the tiny microphone to the tip of the narrow end of the plunger. Make sure that a battery is installed in the small black amplifier in the wire, and connect the wire to the CH1 input of the oscilloscope. Set MODE to CH1 on the oscilloscope and sing a note into the microphone. If you do not get a pattern on the oscilloscope, ask your instructor for help.

4.Insert the narrow end of the plunger into the electronic resonance tube. You should get a sine wave from the microphone when MODE is set to CH1, and a sine wave from the function generator when MODE is set to CH2. You may need to set both VOLTS/DIV dials fully clockwise. If the sine waves do not appear, ask your instructor for help.

5.Find room temperature and calculate vt.

6.Set the microphone at close proximity to the speaker. Set MODE to X-Y and DISPLAY to X-Y. Move the microphone back until an in-phase signal (the Lissajous figure "/") appears on the oscilloscope. Record the position as x(0).

7.Move the microphone back farther away from the speaker until "O" appears on the scope. Continue to move the microphone back until "\" appears on the scope. Record the position as x(λ/2).

8. Calculate λ = 2·[x(λ/2) - x(0)], then calculate ve.

9. Repeat steps 6 to 8 for frequencies of 3000 Hz, 4000 Hz and 5000 Hz.

5. BEATS

PURPOSE:

a) To determine the beat frequency of a wave.

b) To write a beat equation to describe this wave.

INTRODUCTION:

Two one-dimensional wave functions with the same amplitude and almost-identical frequencies may be described mathematically as

y1 = A·sin(2πf1t) and y2 = A·sin(2πf2t). The quantity f may be defined as the average of the two given frequencies, so

f = (f1 + f2)/2. The quantity ∆f may be defined as the difference between the two given frequencies, so ∆f = f2 - f1.

When the two waves are added together, they appear graphically as

and the function y1 + y2 may be rewritten algebraically as

y = 2A·cos(π·∆f·t)·sin(2πft).

This may be interpreted as a sine wave with its amplitude modulated by a low-frequency (∆f << f) cosine wave. One beat is defined as a section of the wave pattern between nodes of the

low-frequency ‘envelope’. Theoretically, the frequency of the beats is fb = |∆f|.

APPARATUS:

Banana wire (6)

Power strip

BNC-to-banana adapter (4)

Frequency counter

Function generator (Simpson)

Oscilloscope

Function generator (Global Specialties)

PROCEDURE:

1.Connect BNC-to-banana adapters to the OUTPUT of the Simpson function generator (Oscillator 1), the OUTPUT of the Global Specialties function generator (Oscillator 2), the CH1 INPUT of the oscilloscope and the A INPUT of the multifunction counter. Turn on all four devices.

2.On Oscillator 1, set the frequency to 5000 Hz, with the

0 dB button extended outward and the sine waveform button pushed inward. Use two banana wires to connect Oscillator 1 to the oscilloscope, red to red and black to black, and rotate the AMPLITUDE dial of Oscillator 1 to get a sine wave on the oscilloscope with an amplitude A = 0.2 volts. For a frequency of 5000 Hz, one wave should cover 4.0 divisions horizontally on the screen when the TIME/DIV dial is set to 50 µs. Ask your instructor to check your sine wave.

3.Unplug Oscillator 1 from the oscilloscope. Plug in

Oscillator 2, set at 5000 Hz to create a sine wave of amplitude

A = 0.2 volts. Ask your instructor to check your sine wave.

4.Plug both oscillators together into the CH1 INPUT of the oscilloscope, and slowly change the frequency of Oscillator 2 until beat behavior is seen on the oscilloscope. To see this clearly, set the TIME/DIV dial to .5 ms. Rotate the FINE dial of Oscillator 2 to get a clear minimum between the beats. Measure the beat period from the oscilloscope, which equals the number of divisions between minima multiplied by .5 ms. Calculate the beat frequency fb as the inverse of the beat period.

5.On the multifunction counter, press the 1s GATE TIME button and the 10 MHz button. Plug Oscillator 1 into the multifunction counter to get f1, then unplug it. Plug Oscillator 2 into the multifunction counter to get f2, then unplug it. Write down these two values, and calculate |∆f| = |f2 - f1|. Calculate the percent difference between the theoretical value ∆f and the observed value fb.

6. Repeat steps 2 to 5 for two other frequencies.

6. THE VISIBLE SPECTRUM

PURPOSE:

The wavelengths of electromagnetic waves in the visible range will be determined with a diffraction grating.

INTRODUCTION:

A diffraction grating consists of a number of closely-spaced parallel grooves ruled on a transparent surface. It is a useful device for dispersing the waves of different wavelength in a source of light. The effect of a grating is similar to a prism but exhibits greater resolving power.

The two dots in Figure 1 represent adjacent grooves on a diffraction grating, a distance d apart. When the two rays of light entering from the left strike the two grooves, each ray is scattered in all directions. A bright spot will appear on a distant screen if the two scattered rays heading toward the lower right are in phase, so that constructive interference occurs. As may be seen in Figure 1, δ is the difference in the path lengths of the two scattered rays of wavelength λ. The two rays will be in phase if δ = 0λ or 1λ or 2λ, for example. Call

n = 0, 1, 2 ... the order of the diffraction. Then constructive interference occurs if δ = nλ. From the geometry of Figure 1,

δ = d·sinθ. Equating these two, nλ = d·sinθ, so

d·sinθ

λ = –––––– .

n

Grating

Laser δ Laser L

light light

θ

Grating Ledger

d paper x θ

Laser

light

Fig. 1 Fig. 2

APPARATUS:

Window screens

Meter stick (2)

Masking tape

Grating stand and holder

Diffraction grating 500 lines/mm

Ledger paper (3 sheets)

Laboratory jack

Cardboard, large (2)

Laser

Lamp, gooseneck reading

White light source

Battery charger, 12V

PROCEDURE:

PART A: DETERMINATION OF THE GROOVE SEPARATION d

1.To determine the diffraction grating spacing d, set up the grating and helium-neon laser as shown in Figure 3. The three sheets of ledger paper can be taped to two large pieces of cardboard leaning vertically against the wall. Set the grating at exactly L = 1.000 m from the ledger paper. Orient the grating perpendicular to the beam. Measure the distance x for the orders

n = 1 and n = 2. Calculate θ from tanθ = x/L, as shown in

Figure 2. Determine an average value for the grating groove spacing d from d = nλ/sinθ. The wavelength of the laser light is 632.8 nm in air.

xleft

He-Ne Laser L

θ xcenter ~ 0

Lab Jack xright

Fig. 3

PART B: DETERMINATION OF THE WAVELENGTH RANGES FOR VISIBLE LIGHT

1.Set up the apparatus as shown in Figure 4, replacing the laser with a white light source. Adjust the lens or the filament plunger at the back to create a sharp image of the filament.

2.Record L. Record x for the boundary between each color. The distance x is measured from the center of the filament’s image to the boundary between the colors. For example, the horizontal line between violet and blue in Figure 4 is the location of xlower(Violet) = xupper(Blue).

0th order

White-light

Source

__Violet

Blue

Green

Yellow

Fig. 4 Orange

Red

3.Calculate θ from tan θ = x/L, as before. Calculate λ in units of nanometers, and calculate the percent difference between your results and the (rather arbitrary) values given below. Indigo is missing, as it is too difficult to distinguish from violet and blue in this experiment.

Colorλupperλlower

Violet400 nm425 nm

Blue425 nm490 nm

Green490 nm575 nm

Yellow575 nm585 nm

Orange585 nm650 nm

Red650 nm700 nm

7. THIN FILMS

PURPOSE: To make use of interference fringes to estimate:

a) The rate of thinning of soap films

b) The shape of soap films

INTRODUCTION:

incident ray

ray reflected from front surface,

shifted by λ/2, and displaced downward

for clarity

ray reflected from back surface,

having traveled an extra distance 2x,

and displaced downward for clarity

x

Fig. 1

A thin film of soapy water (with an index of refraction of

n = 1.33) is surrounded by air (with an index of refraction of 1.00), as shown in Figure 1. A beam of light shining on this thin film reflects from the front surface (closest to the source of light) and from the back surface as well. The wavelength of the light in air is λ, and the wavelength of the light in soapy water is λ/n.

The reflected light will appear to be bright if the wavefronts of these two reflected beams interfere with each other constructively. The light reflected off the front surface is shifted by λ/2, because air has a smaller index of refraction than the soapy water. To create constructive interference, the light reflected off the back surface must be shifted by (1/2)·(λ/n) or (3/2)·(λ/n) or (5/2)·(λ/n) and so on, as it travels the extra distance 2x. Therefore,

2x = [(2m - 1)/2]·(λ/n), with m = 1, 2, 3 ....

APPARATUS:

Buret clamp

Beaker, 600-ml

Window screens

Lens stand for soap bubble loop

Ringstand for magnifying glass

Soap, dishwasher or laundry

Slide projector

Ruler

Timer, electronic

Soap bubble loop

Sodium lamp

Magnifying glass

Projector screen

Slide projector

Loop Magnifying

glass Image

Fig. 2

PROCEDURE:

1.Set up a single apparatus for the entire class as in

Figure 2, with a large and focused image of the loop, well-illuminated by the slide projector. Place the slide projector as close as practicable to the magnifying glass, to minimize the angles of incidence and reflection. Turn on the sodium lamp, to give it several minutes to reach maximum brightness.

2.A thin, soapy film stretched across the loop will create several horizontal fringes of constructive interference, because the film is thicker on the bottom than on the top. Time how long it takes for one bright red fringe (λ ~ 650 nm) to be replaced by the adjacent bright red fringe, as the soap settles. Over this time interval, the thickness of the soap film decreases by

∆x = (λ/n)/2. Calculate the speed v = ∆x/∆t at which the front and back surfaces are approaching each other.

3.Replace the slide projector with the sodium lamp, emitting radiation of wavelength λ = 589 nm. Count how many bright fringes appear from the top to the bottom of the loop. Multiply this by (λ/n)/2 to find the difference in thickness of the film from top to bottom. Measure the inner diameter of the loop, and calculate the (extremely small) angle that the front surface of the film makes to the back surface.

8. DIFFRACTION AND INTERFERENCE:

FRINGE ANALYSIS

PURPOSE:

a) To measure slit width, wire thickness and hair thickness.

b) To measure slit separation and analyze

interference pattern and fringes.

INTRODUCTION:

Optical methods are often used today for the measurement of objects or spaces too small to be measured by micrometers. By making use of the diffraction of light waves by small objects and spaces and the interference of light rays giving rise to interference patterns, dimensions in the range of 10-7 meters can be measured.

Fringe analysis can be performed by comparing observed patterns with the intensity plots predicted with phasor addition of waves.

APPARATUS:

Lens holder on stand (2)

Masking tape

Tape measure

Ledger paper, 11” x 14”

Cardboard

Lab jack (2)

Ruler

Laser

Lamp

Diffraction wire, 30-gauge

Slits, single & multiple

Woven fiber

PROCEDURE:

PART A: SINGLE SLIT xleft

He-Ne Laser

Lxcenter ~ 0

θ

2 Lab Jacks xright

Fig. 1

1.Set up the apparatus as shown in Figure 1, for a single slit held by two lens holders. Place the paper, taped to the cardboard, as far from the laser as possible.

2.Measure L, which is the distance from the slit to the screen. Measure xright and xleft as the distances from the central dot for the first dark fringe (the first minimum). These are both positive numbers. Calculate tanθ = xaverage/L, then calculate θ.

3.Calculate the slit width a from mλ = a·sinθ, where m = 1 for the first order. Use λ = 633 nm for the He-Ne laser.

4. Repeat step 2 with the mounted wire, then calculate the wire thickness a. Use mλ = a·sinθ, where m = 1 for first order. A 30-gauge wire has a diameter of 0.010 inches.

1 inch equals exactly 2.54 centimeters.

5.Repeat step 4 for a strand of hair, taped vertically to the lens holder.

PART B: MULTIPLE SLITS

1.Set up the apparatus as shown in Figure 1, with two slits.

2.Measure L. Measure xright and xleft for the first bright fringes beside the central bright fringe.

3.Calculate d, the slit separation, from mλ = d·sinθ.

4.Repeat steps 1 and 2 for the mounted woven fiber.

5.Calculate d, the fiber separation.

9. THE MICHELSON INTERFEROMETER

INOTRODUCTION:

An interferometer is a device that splits a beam of light into two, permits the two beams to travel along separate paths, and then recombines them to form a single beam again. If the two beams are in phase, constructive interference will occur and a bright fringe will appear. If the two beams are out of phase, destructive interference will occur and a dark fringe will appear. If the path length of one beam is held constant and the path length of the other beam is slowly increased by one wavelength, the recombined beam will change from bright to dark and back to bright again.

In the diagram, a beam of light enters horizontally from the left and is split into two by the semi-silvered surface of the glass plate. Some beams in this diagram are displaced from their actual positions for clarity. The compensator is present so each beam passes through glass three times. The micrometer on the front of the interferometer moves the compensator and the mirror behind it, to shorten or lengthen the light path.

mirror

compensator

semi-silvered

surface

Light

source mirror

pin

Observer

APARATUS:

Blinds for windows

Masking tape

Paper

Lab jack (2)

Lamp

Mercury light with green filter

Michelson interferometer

Sodium lamp

PROCEDURE:

1.Place the interferometer on the center of a lab jack, with the micrometer pointed toward the observer. The interferometer should be raised so that the view through the semi-silvered glass plate is as comfortable as possible. If necessary, use one lab jack on top of another. Rotate the micrometer dial to read

0.00 mm.

2.To align the optics, rotate the screw on each of the mirrors until the two images of the vertical reference pin seen through the interferometer merge exactly to form a single image.

3.Turn on your sodium lamp, as it will take several minutes to heat up. Place the mercury light on one or two lab jacks to the left of the interferometer, to serve as the light source. A set of light and dark fringes should be visible through the interferometer. Rotate the screw on each mirror until the fringe pattern is a set of concentric circles, centered exactly on the image of the vertical reference pin. Ask your instructor to examine your fringe pattern before continuing.

4.Rotate the micrometer barrel counter-clockwise until the fringes are seen to disappear smoothly at the pin, and stop rotating when a dark fringe (caused by destructive interference) is on the image of the pin. Read the micrometer to an accuracy of 0.01 mm, then continue to rotate the micrometer barrel counter-clockwise while 50 bright fringes pass through the location of the pin, stopping at the 50th dark fringe. Read the micrometer again to get the final reading. Repeat twice, obtaining larger and larger readings. Do not rotate the micrometer barrel clockwise during step 4, as reversal may shift the micrometer screw.

5.The longer the wavelength of light, the farther the micrometer must move the mirror to pass from one dark fringe to the next. Let x represent the difference between the two micrometer readings in units of millimeters, and let λ represent the wavelength of light in units of ångstroms. Then 50λ = kx, where k is a constant for the micrometer. The wavelength of the green line of mercury is 5460.74 ångstroms. Calculate k for your interferometer.

6.Now that the value of k is known, you can accurately determine the wavelength of any light. Replace the mercury lamp with the sodium lamp and rotate the micrometer dial back to

0.00 mm. Repeat step 4 to determine the wavelength of the yellow sodium line, from λ = kx/50. If the sodium lamp is too bright, cover the beam with a piece of white paper taped to the lamp. Find the percent difference between this measurement and the accepted value of 5892.94 ångstroms.

10. ATOMIC SPECTROSCOPY

PURPOSE:

To determine the emission-line wavelengths of helium and mercury by using a diffraction grating, and to determine the wavenumber of mercury lines.

INTRODUCTION:

Atoms of an element emit a spectrum of colored lines. The wavelengths of these spectral lines depend on the atomic structure of the atoms. Excited electrons in atoms emit radiation of a specific wavelength when returning to a lower energy level as shown schematically in Figure 1.

LargeAn excited electron

voltage drops to a

lower orbit.

.

. . A photon carries away

. the excess energy

E = hf = hc/λ.

Zero

voltage

DischargeAtom

tube

Fig. 1

APPARATUS:

Clamp, buret clamp (2)

Blinds for windows

Meter stick (2)

Grating stand and holder

Ringstand (2)

Grating

Spectrum tube of helium

Spectrum tube of mercury

Spectrum tube power supply

Lab jack

Cardboard, large

He-Neon laser

Lamp

PROCEDURE:

PART A: DETERMINATION OF THE GROOVE SEPARATION d

xleft

He-Ne Laser L

θ xcenter ~ 0

Lab Jack xright

Fig. 2

1.Set up the grating an helium-neon laser, as shown in

Figure 2. Place the grating at approximately one meter from the

cardboard. Measure xleft and xright, and calculate xaverage as

xaverage = |xright - xleft|/2, for the orders n = 1 and n = 2. From geometry, θ = arctan(xaverage/L) is the angular displacement of the spectral line. Determine an average value for the groove spacing d from your data, from d = nλ/sinθ.

PART B: DETERMINATION OF THE WAVELENGTHS OF THE SPECTRAL LINES

Meter stick

Emission

tube

Diffraction

grating

Fig. 3

1.Set up apparatus as shown in Figure 3, with a helium emission tube.

2.Determine the distance L from the diffraction grating to the center of the meter stick. Place the glowing helium emission tube just behind the horizontal meter stick.

3.Measure xleft as the position on the meter stick of a bright emission line as seen through the grating. Measure xright, and calculate xaverage, θ and λ = d·sinθ (for n = 1). The known emission-line wavelengths are given in the tables below.

4.Repeat step 3 for three other emission lines.

5.Repeat steps 3 and 4 for the mercury emission tube.

6.For each of your measured wavelengths of the mercury emission lines, calculate the wavenumber k. This is the number of waves in 2π meters.

Helium

λ (nm)Color

447.1 violet

471.3blue

492.2 blue-green

501.6 green

587.6 yellow

667.8 bright red

Mercury

λ (nm)Color

404.7 faint violet

435.8 blue

546.1 green

578.2 yellow-orange

11. THE PHOTOVOLTAIC CELL

(THE SOLAR CELL)

PURPOSE:

To observe current production by photons (photoelectric effect), to determine the stopping potential Vs and to calculate the velocity of the electrons.

APPARATUS:

Blinds for windows

Rheostat, 2.8-ohm

Graph paper

Multimeter, BK (ammeter)

Multimeter, Extech (voltmeter)

Lab jack

Spade lug (3)

Battery, 1.5-volt

French curve

Meter stick

Ringstand with holder

Banana wire (7)

Solar cell (photovoltaic cell)

Lamp, 7-watt mounted

INTRODUCTION:

The photovoltaic cell consists of a light-sensitive diode made of n-type (electron donor) and p-type (electron acceptor) doped semiconductors. Photons incident on this photodiode excite electrons and cause them to break free. The electrons move in one direction, thereby constituting a current flow.

photonsphoton dislodged electron

with speed v

hole

n-type .

p-type

I

Let Ep represent the energy of the photon absorbed by the electron.

Let Eg represent the band gap energy.

Let Ek represent the kinetic energy of the dislodged electron.

From the law of conservation of energy, Ep = Eg + Ek

If a back voltage V is adjusted to drop thecurrent to 0 amperes, the energy gained by __________the electron as kinetic energy Ek must equal the energy lost as the electron crosses Vs, called the stopping potential. That is,

Vs(1/2)mv2 = qVs. Therefore, the speed of the

dislodged electron is

current = 0 A 2qVs

v = .

m

PROCEDURE:

PART A:

1. Shine the lamp fully onto the solar cell held upright on the ringstand as seen in Figure 1, and connect the solar cell to the ammeter set at 200 mA_ _ _. Measure the current I from the solar cell at various distances d from the lamp, measured to the nearest centimeter. If the reading is negative, reverse the terminals. Calculate I·d2, converted to units of A·m2. If the solar cell is a linear device, this value should be approximately constant, as the light intensity should decrease as 1/d2.

Ammeter

+ -

Solar

cell

Fig. 1

PART B:

light

+

solar cell V A

-

///////////The bottom half of this circuit acts as a variable voltage source, with

V = 0 volts on the right side of the rheostat and V = 1.5 volts on the left

1.5 voltsside, counter-acting the current from the solar cell.

Fig. 2

1.Set up the equipment as shown in Figure 2, starting with the tap placed close to the negative end. A large, constant amount of light should shine on the solar cell for Part B. Set the voltmeter to 2 V_ _ _.

2.Change V by moving the tap on top of the rheostat approximately 4 centimeters at a time, recording I and V at each step. The current should change sign during this process. If it does not, reverse the wires on the solar cell. Plot I against V, and use the French curve to plot a smooth line through your data points.

3.For I = 0 from the graph, the back voltage V is the stopping potential Vs. Calculate the speed of the electron dislodged by the photon. For the electron, the magnitude of the charge is q = 1.60 × 10-19 C and its mass is m = 9.11 × 10-31 kg.

12. THE HYDROGEN SPECTRUM

PURPOSE:

(a) To determine the energy of emitted radiation from

an electronic transition in the hydrogen atom.

(b) To determine the Rydberg constant R

(c) To determine Planck's constant h.

APPARATUS:

Blinds

Hydrogen spectrum tube

Spectrum tube power supply

Lab jack

Power strip

Lamp

Spectrometer

Transmission grating

screw

collimator grating telescope eyepiece

hydrogen

stage

Fig. 1

PROCEDURE:

1.Set up the apparatus as shown in Figure 1, which is a top view of the apparatus after the metal cap has been removed. Do not touch the surface of the diffraction grating, as it is finely ruled with 600 grooves per millimeter. This make the spacing between adjacent grooves equal to d = 1.667 × 10-6 meters. 1 ångstrom = 10-10 meters.

2.Line up the hydrogen spectrum tube, the collimator, the grating and the telescope. The grating should be exactly perpendicular to the line of sight through the telescope and collimator, and in the center of the circular stage.

3.Adjust the eyepiece until the crosshairs are in focus. Adjust the eyepiece until the slit beside the screw is also in focus. The crosshairs should be rotated to be parallel to the slit, and should be as sharply focused as the slit. Replace the metal cap onto the stage.

4.Adjust the slit size to a narrow slit, and set the spectrometer to exactly zero degrees.

5.Move the telescope arm to approximately 15°, to locate the first spectral line.

6.Record θBlue, θTurquoise and θRed. Use the vernier to get as accurate an angular measurement as you can.

7.Calculate λ = d·sinθ.

8.Calculate the percent differences from the known values

of λ. You should be within one percent.

9.When finished, turn off the spectrum tube power supply and wait a few minutes for the tube to cool down before removing it.

13. THE RATIO e/m FOR ELECTRONS

INTRODUCTION:

The Helmholtz coil apparatus consists of two coils of wire, each of radius R and with N loops in each coil, separated by the same distance R along their axis of symmetry. When an electric current I passes through both coils, the magnetic field generated at the center of the apparatus is essentially constant over a significant volume, and has a magnitude B, with

64μo2N2I2

B2 = .

125R2

Electrons are accelerated across a voltage V from the filament to the plate. Voltage = energy/charge by definition, so the energy of each electron of charge e is eV. From the law of conservation of energy, this must equal the kinetic energy of the electron, so

mv2 = eV ,

where m is the mass of the electron and v is its speed. The force on the electron as it moves perpendicular to the magnetic field is evB, which must equal the centripetal force mv2/r, where r is the radius of the beam, so

evB = mv2/r .

Combining these two equations,

e 2V

= .

m B2r2

APPARATUS:

Rheostat, 5-ohm

Multimeter, green Extech

Power strip

Multimeter, blue BK Precision (2)

Spade lug (2)

DC Regulated Power Supply

Metric ruler

Lamp, reading

Banana wire (5 black, 9 red)

hp Power Supply

Helmholtz coil apparatus

Helmholtz coil circuit diagram

Potentiometer, 1000 ohm

PROCEDURE:

1.Set up the equipment as shown in the diagram, without the black or red wires connected. Rotate the coarse dial on the hp Power Supply fully counter-clockwise, and make sure both of its switches are flipped from “ON” to off. Make sure the switch on the DC Regulated Power Supply is off, and rotate its voltage dial fully counter-clockwise. Find the radius R of the coils, and record the number of loops, printed on the base of each coil.

2.Connect the red and black wires. Push the tap of the rheostat as far from the connection to the blue AC ammeter as it can go. Turn on the three multimeters, and set their dials and buttons as shown in the diagram. Have your instructor check the wiring before flipping on the three power switches. Close the curtains, and turn off the lights.

3. Slowly rotate the coarse dial of the hp Power Supply until the green voltmeter reads approximately 80 volts. Slowly move the tap on the rheostat, heating the filament until a blue beam appears in the Helmholtz-coil tube. The blue AC ammeter should read between 0.1 and 0.5 A. Do not let it go beyond 1.0 A. Rotate the potentiometer dial to charge the grid, until the beam is focused. Rotate the dial of the DC Regulated Power Supply fully clockwise, to create the magnetic field.

4.Record the DC current reading on the blue DC ammeter, which measures the electric current creating the magnetic field. Rotate the voltage dial of the DC Regulated Power Supply until the beam curves enough to illuminate the outermost ring. When the ring is brightest, write down the green DC voltmeter reading. This is the voltage V that is accelerating the electrons.

5.Repeat Procedure 4 for the next ring.

6.Reverse the wires on the DC Regulated Power Supply, black to red and red to black. Repeat Procedures 4 and 5. This reversal of the coil’s magnetic field will partly cancel out the effect of the Earth’s magnetic field.

7.Flip the three power switches to off, and turn off

the multimeters. For each set of readings, calculate B,

e/m and the percent difference from the accepted value of

e/m = 1.75882 × 1011 C/kg. In the calculation of B, use

μo = 4π × 10-7 T·m/A.

8.In the Los Angeles area, the Earth’s magnetic field is

4.8 × 10-5 teslas. How much effect would you expect it to have on your experiment?

9.When an electron accelerates through 100 volts, it gains a kinetic energy of 100 electron-volts = 1.6 × 10-17 joules. Given that the mass of the electron is 9.109 × 10-31 kg, calculate v, β and γ to six significant digits. Given that your voltages (and electron speeds) are less than this, are relativistic effects likely to be important in this experiment?

14. THE SEMICONDUCTOR DIODE

PURPOSE:

To view the half wave and full wave rectification

of an AC signal.

INTRODUCTION:

Energy

Planck and Bohr

n = 3

n = 2Quantum Theory

n = 1

Semiconductors Conduction Band

Energy Bands

- - - - - Fermi Level

Energy Energy

p-type and

n-type Donor

semiconductors

Acceptor

Diode p-type n-type |

Symbol

APPARATUS:

Banana wire (8)

Power strip

Adapter, 2-3 prong electric plug

Alligator clip (2)

BNC-to-banana connector (2)

Straightedge

Capacitor, 10 microfarad

Rectifier diode, 1N4005

Resistor, 1000 ohm (mounted)

Function generator

Oscilloscope~ |

Light-emitting diode (yellow)

PROCEDURE:

Oscillo-

scope

Fig. 1

PART A: HALF-WAVE RECTIFICATION

1.Attach a BNC-to-banana connector to the CH1 input of the oscilloscope, set CH1’s VOLTS/DIV to 0.5 set the TIME/DIV dial to .5 ms and set the switch above the CH1 INPUT to DC. Plug the function generator into the 2-3 prong adapter, which should then be plugged into the power strip. Attach a BNC-to-banana connector to the OUTPUT of the function generator, set the frequency to 1 kHz and connect it to the oscilloscope (black port to black port). Rotate the function generator’s amplitude dial to obtain a sinusoidal wave with an amplitude of 2.0 volts, and rotate the frequency dial to obtain 5.0 cosine waves across the display grid.

2.Set up the half-wave rectifier circuit as shown in Figure 1, with the ground (black) port of the oscilloscope connected to the ground (black) port of the function generator. Adjust the function generator’s DC offset to obtain half-wave rectification.

3.Sketch the resultant waveform, labeling the axes.

Do not change the function generator’s settings until

Part C, Activity 2.

Oscillo-Oscillo-

Scopescope

~ ~

Fig. 2Fig. 3

PART B: FULL-WAVE RECTIFICATION

1.Set up the full-wave rectifier circuit as shown in

Figure 2.

2.Sketch the resultant waveform.

3.Set up the full-wave filtered rectifier circuit as shown in Figure 3.

4.Sketch the resultant waveform. You should find that the circuit has converted AC to DC.

PART C: HALF-WAVE RECTIFICATION, LIGHT-EMITTING DIODE

1.Set up the half-wave rectifier circuit with a light-emitting diode as shown in Figure 1. Sketch the resultant waveform.

2.Set the generator to maximum amplitude and the frequency to

1.0 Hz. Watch the diode as you increase the frequency to 10 Hz, and watch how the oscilloscope responds as well.

15. THE INTEGRATED CIRCUIT:

OPERATIONAL AMPLIFIER

PURPOSE:

To use an IC op-amp 741 to amplify and invert an electrical signal. It represents many circuit elements and does the work of a complex circuit with many resistors, capacitors, diodes and transistors.

INTRODUCTION:

One of the applications of quantum physics is the integrated circuit. An example of such a device is the operational amplifier 741.

8 pins, dual in-line package (DIP)

APPARATUS:

Multimeter, BK Precision

Power strip

Alligator clips (2)

Adapter, 2-prong to 3-prong

BNC-to-banana connector (3)

Power supply, DC regulated

BNC cable

Banana wires (12)

Power supply, hp Model 711A

Integrated circuit connector

Resistor, 1000-ohm (mounted)

Function generator

Oscilloscope

Resistor, 4,700-ohm

Resistor, 10,000-ohm

Resistor, 15,000-ohm

Resistor, 22,000-ohm

Integrated circuit op-amp 741

PROCEDURE:

1.Connect the BNC-to-banana adapters to CH1 and CH2 of the oscilloscope, and to the OUTPUT of the function generator. Set the oscilloscope MODE to CHOP, set the SOURCE switch on the lower-right of the oscilloscope to EXT, and use a BNC cable to connect its TRIG INPUT to the TTL (transistor-transistor logic

= 5 V square wave) output of the function generator. Plug the operational amplifier into the integrated circuit connector. The tiny circle on top of the operational amplifier is closest to pin 1, and should be placed in the port labeled as 1 on the blue tape of the integrated circuit connector. Plug the function generator into the 2-prong-to-3-prong adapter, and plug this and the other three devices into the power strip, which itself should be plugged into the floor power outlet. Flip up both ON switches of the hp Model 711A power supply, set the multimeter to 1000 V_ _ _ and plug it into the power supply’s + and - DC voltage terminals. Adjust its Coarse and Fine dials to give a voltmeter reading of 12.0 volts. Do the same for the DC regulated power supply. Remove the multimeter and both wires. Do not turn off or adjust the power supplies for the rest of the experiment.

2.To provide a signal to the circuit, follow the upper diagram on the next page to connect the function generator set at 1000 Hz to the resistors, the op-amp and CH2 of the oscilloscope. Be sure that the oscilloscope grounds, the function generator ground and pin 3 are connected together. Rotate the DC OFFSET of the function generator fully counter-clockwise to OFF, and use the multimeter set at 2V~ and the AMPLITUDE dial of the function generator to set Vrms at approximately 0.0707 ± 0.001 volts. This will set the amplitude of the sine wave to approximately 0.10 volts. Remove the multimeter. Double-check that CH1 of the oscilloscope is connected directly to the function generator OUTPUT. Figures 1, 2 and 3 illustrate how the op-amp’s input, feedback and output have now been created.

3.To provide power to the circuit, follow the lower diagram on the next page by plugging the DC regulated power supply (in the black casing) + terminal into pin 7 and its - terminal into

pin 3. Plug the hp model 711A power supply (in the brown casing)

+ terminal into pin 3 and its - terminal into pin 4. Notice that

V = +12 volts for pin 7, and V = -12 volts for pin 4 because

pin 3 is grounded (V = 0 volts).

4.Turn on the oscilloscope. You should see two bright, focused sinusoidal waves. The voltage amplitude of each wave can be determined by counting the number of vertical divisions (accurate to one-tenth of a division), and multiplying by the VOLTS/DIV setting.

5. Draw the oscilloscope output for various frequencies and resistances, labeling the axes. Notice that the output signal is inverted (upside-down compared to the input), and that this circuit takes a small input voltage and converts it into a large output voltage.

6.Verify that the voltage amplification occurs with a ratio:

Vfeedback R2

= , at three different frequencies.

Vinput R1

Procedure 2: To amplify and invert the signal.

CH1 +

R1 = 1 kΩR2 = 10 kΩ

(mounted)

2

+ Function

~ generator6 +

-

3 CH2 of oscilloscope

CH1 - -

Procedure 3: To provide DC power to the op-amp.

18

27

12-volt36

hp Model 711A45 12-volt

power supply DC regulated

power supply

R1 = 1 kΩ

18

27

36

45

Fig. 1: Input signal from the

function generator through R1 = 1 kΩ

R2 = 10 kΩ

18

27

36

45

Fig. 2: Feedback through R2 = 10 kΩ,

the ratio R2/R1 giving the output gain

18

27

36

45To CH2 of

oscilloscope

Fig. 3: The output of the op-amp

connected to the oscilloscope

16. LOGIC GATES

PURPOSE:

The purpose of this laboratory exercise is to verify the logic of NAND and NOR gates using Boolean algebra.

INTRODUCTION:

Binary logic elements are called gates. They are represented by symbols which stand for the binary logic that they obey. The gates that we will use have outputs that are either "on" or "off". The "on" state corresponds to a "1", a "high" or about 3.5 to 5 volts. The "off" state corresponds to a "0", a "low" or about 0 to 2 V. The logic of states that are either on or off has a basis in the branch of mathematics called Boolean algebra which was formulated by George Boole in the mid-1800's, long before logic circuits came into being. The following is a listing of the rules of Boolean algebra.

RuleDisjunctive (OR)Conjunctive (AND)

1.Associative(A + B) + C (AB)C = A(BC)

= A + B + C

2.Commutative A + B = B + A AB = BA

3.Distributive (A + B)C = AC + BC AB + C

= (A + C)(B + C)

4.A + 0 = A A1 = A

_ _

5.A + A = 1 AA = 0

6.A + 1 = 1 A0 = 0

7.A + A = A AA = A

8.De Morgan's_____ _ __ _ _____

Theorem A + B = A BA B = A + B

__

9.Complement 1 = 0 0 = 1

10.Absorption

Rule and_ _____ _

its dualA + (A + B) = AA(A + B) = A

As an example of the use of Boolean algebra in digital electronics, consider the logic of a NAND-gate:

_

AB = C NAND-gate.

The bar over the C corresponds to negation, and is the opposite state of C. _

The output C of the NAND-gate depends on the two input states A and B. If one performs logical multiplication of the possible input states and then negates the result, we then have what is known as a truth table.____________________

_

The truth table for A B C C |

a NAND-gate is shown 0 0 0 1 |

to the right. 0 1 0 1 |

1 0 0 1 |

1 1 1 0 |

A NOR-gate corresponds to the negation of logical addition:

_

A + B = CNOR-gate.

_______________

The truth table for A B C |

a NOR-gate is shown 0 0 1 |

to the right. 0 1 0 |

1 0 0 |

1 1 0 |

APPARATUS:

Power supply

Banana wires

Multimeter (voltmeter)

Alligator clips

Wires for breadboard

Integrated circuit connector

NAND gate, #7400 quad two-input (5)

NOR gate, #7402 quad two-input

Resistor, 1000 ohm

PROCEDURE:

1.Mount a #7400 quad two-input NAND-gate on your breadboard. Use an LED connected to the gate output to determine its state. Be sure to put a 1000 Ω pull-up resistor from ground to the ground leg of the LED. Verify the truth table for the NAND-gate.

2. Study the truth table for a NAND-gate and then determine how you would configure an inverter. An inverter is a logic gate which negates an output. Configure an inverter and verify its truth table.

Inverter

___________

_

The truth table for A A |

an inverter is shown 0 1 |

to the right. 1 0 |

3.Mount a #7402 quad two-input NOR-gate on your breadboard. Use an LED connected to the gate output to determine its state. As before, be sure to put a 1000 Ω pull-up resistor from ground to the ground leg of the LED. Verify the truth table for the NOR-gate.

4.Set up the

arrangement of

NAND-gates shown

to the right.

This arrangement is called an exclusive OR-gate or XOR-gate and follows the logic:

_ _

AB + AB = C XOR-gate

Determine the output states for the possible binary inputs. Measure E, F, and G with a voltmeter. Use the same LED for the output C.

Pin numbers of the 7400 IC and the 7402 IC:

Two-input NAND-gate

Two-input NOR-gate 7402

QUESTIONS:

1. Prove De Morgan’s Theorem using Venn diagrams, on the back of the data sheet.

A B A B

______ _

A + BA B

2.Consider yourself to have had the misfortune of being in a mythological two-dimensional place called ‘Flatland’ and further suppose that you wanted to transmit two logical states across Flatland such that they become exchanged. Show by using Boolean algebra that the following arrangement of XOR-gates could accomplish this task, on the back of the data sheet.

17. RADIOACTIVITY

PURPOSE:

To determine the shielding effects of various materials (lead, aluminum and cardboard) to β (beta) emission.

APPARATUS (2 for entire class):

Vernier caliper

Aluminum sheets

Cardboard sheets

Lead sheets

Forceps

Geiger counter

Geiger tube

Thallium-204, radioactive source

CAUTION:

Do not put anything on the window of the tube.

This experiment uses high voltage (800 volts).

Turn off the power when finished.

PROCEDURE:

PART A: Background Radiation

1.Plug the Geiger tube into the back of the Geiger counter. Set the Geiger tube to face vertically downward. Place the plastic sheet with the central depression into the second-highest slot of the Geiger tube. The highest slot will hold the shielding material in PART B.

2.Turn on the Geiger counter.

3.Set preset time dial to 1 minute.

4.Adjust the voltage to 800 V.

5.Push in STOP, RESET and COUNT.

6.Record ten 1-minute counts with no radioactive source present, to measure the background radiation count. Take the average and the standard deviation of these ten counts. Calculate the square root of the average. Theoretically, this square root should be a good approximation of the standard deviation.

PART B: Shielding

1.Use forceps to place the disk containing Thallium-204 on the plastic sheet. Record counts for one minute of time.

2.Shield the Thallium-204 sample with 1 lead sheet in the highest slot. Record Counts for 1 minute of time. Subtract the average of the background radiation count to obtain the corrected counts per minute.

3.Repeat step #2 with 2 lead sheets.

4.Repeat steps #2 and #3 with aluminum sheets.

5.Repeat steps #2 and #3 with cardboard sheets.

6.Write a paragraph on the back of your data sheet analyzing your results.

18. MILLIKAN’S OIL-DROP EXPERIMENT

APPARATUS:

Blinds for windows

Circular level

Multimeter, BK Precision

Power strip

Timer, electronic

Lamp

Shims

Ring stand (thick-stemmed)

Banana wire (6)

hp model 711A power supply

Millikan apparatus

INTRODUCTION:

Electric charge is not infinitely divisible, but exists as multiples of a fundamental unit of charge, e = 1.602 × 10-19 coulombs. This unit of charge is very small, and the huge number of these tiny charges needed in most electrical measurements makes the quantized nature of electrical charge unnoticeable.

In this experiment, tiny plastic spheres are injected between two metal plates of opposite electric charge. Millikan used a fine mist of oil drops, but oil drops have a wide variety of sizes, complicating the analysis. The tiny spheres used in this experiment are all roughly the same size and mass. When these spheres are sprayed between the plates, many of them will have an excess or deficit of a few electrons, and the amount of charge can be measured by watching the spheres through a microscope. If this experiment is done accurately, it

will be found that the charges on the spheres are

integral multiples of e.

Each sphere must be measured with and without y axis

the electric field, to allow the calculation of two

different quantities; the radius r of the sphere

and the charge q of the sphere. The first measurement f1

to be taken is the amount of time t1 the sphere takes

to fall through a distance y as seen through a W

microscope, when there is no electric field. The force

diagram shows two forces; the downward force of the

sphere’s weight W, and the upward force of air resistance

f1 (in the direction opposite to the downward velocity).

The sphere quickly reaches terminal velocity, so Newton’s

second law gives:

∑ F = 0

which becomes

f1 - W = 0 .

Equations for the force of air resistance and weight give

6πrηv1

- mg = 0, where

1 + b/Pr

r is the radius of the sphere,

η = 1.51 × 10-5 kg/(m·s) is the viscosity of air,

v1 is the terminal velocity of the sphere as it falls,

b = 0.0082 N/m is a constant,

P = 1.013 × 105 pascals is the atmospheric pressure,

m is the mass of the sphere and

g = 9.80 m/s2 is the acceleration of gravity.

The density of the sphere is ρ = 1050 kg/m3. Substituting

m = density × volume = ρ × (4/3)πr3 and

v1 = distance traveled/time taken = y/t1

and rearranging to solve for r gives

_________________ __

r = √9ηy/2ρg(1 + b/Pr) / √t1 .

The value of y equals 0.001 meters in this experiment. The term

‘1 + b/Pr’ is approximately the same for all of the spheres, which have an average radius of 5.0 × 10-7 meters. If this value is substituted into the ‘1 + b/Pr’ term, the equation becomes

2.384 × 10-6

r = ,(1) √t1

where r is in meters and t1 is in seconds. y axis

The second measurement to be taken is the amount

of time t2 that the sphere takes to rise through the

same distance y as seen through a microscope, when qE

there is an electric field E pulling the sphere upward.

The force diagram shows three forces; the downward force

of the sphere’s weight W, the downward force of air

resistance f2 (in the opposite direction to the upward f2 W

velocity), and the upward force qE that the electric

field exerts on the few excess electrons (or protons)

that the sphere contains. The sphere quickly reaches

terminal velocity, so Newton’s second law gives:

∑ F = 0, which becomes

qE - f2 - W = 0 .

Equations for the force of air resistance and weight give

6πrηv2

qE - - mg = 0.

1 + b/Pr

The electric field between the two parallel plates can be replaced by E = V/d, where V is the voltage between the two plates and d is their separation in meters. In this experiment, V will be set at 150 volts. The velocity v2 can be replaced by y/t2, and the mass m can be replaced by ρ × (4/3)πr3. Rearranging to solve for q gives

6πηy r 4πρg

q = × d × + × d × r3 .

V(1 + b/Pr) t2 3V

The ‘1 + b/Pr’ term can again be considered to be of constant value, and the equation can be rewritten as

r

q = (1.63313 × 10-9 × d) × + (287.35 × d) × r3 . (2)

t2

The terms in parentheses in this equation will be constants for your experiment, once you determine your value of d. The values of d are listed below.

Equipment d Equipment d Equipment d

Number (meters) Number (meters) Number (meters)

434568 0.00346 434573 0.00330 434578 0.00329

434569 0.00349 434574 0.00351 434579 0.00344

434570 0.00374 434575 0.00336 434580 0.00343

434571 0.00351 434576 0.00366 434581 0.00359

434572 0.00406 434577 0.00332 434583 0.00363

PROCEDURE:

1.Write down the LACCD equipment number of the Millikan apparatus, and the value of the plate separation d listed on the previous page. Fit the ring stand through the hole in the middle of the apparatus, lower it until the microscope is at a comfortable eye level, then rotate the screw underneath the front of the apparatus clockwise until the apparatus is firmly clamped. Make sure that the ring stand is not too close to the edge of the table.

2.Set the power supply’s voltage to zero by turning the two D.C. voltage adjustment dials fully counter-clockwise, and flip both switches to the off (down) position. Connect the 6.3-volt terminals of the Millikan apparatus to the upper and lower

6.3-volt terminals on the extreme left side of the power supply. Connect the High-Voltage terminals of the apparatus to the High-Voltage terminals (the + and - terminals) on the right side of the power supply. Connect these same High-Voltage terminals to the COM terminal and right-side terminal of the multimeter, dialed to 1000 V _ _ _ . Ask your instructor to inspect the wiring, then flip on the left-side switch on the power supply, which turns on the light bulb.

3.Unscrew the three screws above the black circular plate, and remove the upper plate. Push the nozzle in as far as it will go. Look through the microscope, and adjust the black eyepiece (by pulling it in and out) so that the reference scale is in sharp focus, and rotate the entire microscope tube so that the lines of the scale are horizontal. Turn the focus knob (to the immediate right of the microscope) until the nozzle is in focus.

4. If the light bulb does not shine directly on the tip of the nozzle, ask your instructor to rotate the light bulb base with a pair of needle-nose pliers. Retract the nozzle and use a paper towel to wipe clean the upper and lower black plates, as well as the window that the microscope looks through. Place the circular level on the lower plate, and put shims under the ring stand feet so that the circular level shows the lower plate to be horizontal. Do not move the ring stand for the rest of the experiment. Remove the circular level, and reinsert the nozzle so that its tip is just outside the field of view of the microscope. Return the upper plate to its correct position, attach the short wire to the central screw of the upper plate and return the three screws to their positions. Flip on the right-side switch of the power supply. Unscrew the black cap of the reservoir connected to the nozzle, and add some latex sphere solution if the end of the bulb attachment is not already immersed in fluid. Reassemble the reservoir. Rotate the two D.C. voltage adjustment dials on the power supply until the voltmeter reads 150 volts. Warning! 150 volts can give you a nasty shock. Keep away from the power supply terminals and the top of the Millikan apparatus.

5. Set the plate-polarization switch on the right side of the apparatus to the ‘Plates Shorted’ position, give the bulb a short, quick squeeze and look for the spheres through the microscope. The spheres are falling slowly downward due to the force of gravity, but since the microscope inverts the field of view, they appear to drift upward. If no spheres appear, ask the instructor to unclog the nozzle by unscrewing it from the rubber bulb, washing it out and inserting an unbent paper clip into the nozzle. Choose one sphere, and flip the plate polarization switch up or down so that the sphere reverses direction. You will be timing how long each sphere takes to travel 1.00 millimeter (from reference lines 1 to 3, or 2 to 4, or 3 to 5 etc.) Ignore fast-moving spheres, as they have too many charges on them to give accurate results. When the chosen sphere has moved below the lower reference line, flip the plate-polarization switch to the neutral position, and time how long it takes to pass from the lower reference line to the upper reference line, to an accuracy of one-tenth of a second. This will be the t1 measurement, which should be between 10 and 30 seconds. If not, choose another sphere. Flip the plate-polarization switch up or down so that the sphere reverses direction, and time how long it takes to pass from the upper reference line to the lower reference line. This will be the t2 measurement, which should be more than 3 seconds. If not, choose another sphere.

6.Repeat step 5 until you have filled out the data sheets.

Check the voltmeter occasionally to make sure it is still reading 150 volts. Make sure that each of your lab partners takes some of these measurements. While data is being collected, one person in the lab group should be calculating r and q by using Equations 1 and 2 for each sphere.

7.Place a ‘+’ on the number line for the charge of each sphere. You should find that these group around multiples of 1.602 × 10-19 coulombs. Label the groups ‘1’, ‘2’, ‘3’ and so on for the number of elemental charges present, and place this number as N beside each q on the data sheet. For each value of N, write the average of the q values above its group of data points on the number line. Divide each average q value by N, and take the average of these values as your best value of the elemental charge e.

8.When finished, turn off the power supply switches and the voltmeter, and dismantle the wiring. Remove the upper plate of the apparatus and wipe the upper and lower plates with a paper towel. Return the upper plate to its correct position and, the short wire to the top of the upper plate and return the three screws to their positions. Pour the unused solution back into the solution bottle, then wash and dry the reservoir before reattaching it to the apparatus.

Millikan’s Oil-Drop Experiment

Spacings Inside Capacitor

Equipment x1 x2 x3 x4 x5 d

Number(mm)(mm)(mm)(mm)(mm) (m)

4345681.576.676.561.5613.560.00346

4345691.656.646.721.6713.530.00349

4345701.596.626.251.5613.460.00374

4345711.666.716.651.6713.540.00351

4345721.576.286.251.5713.450.00406

4345731.586.696.611.5613.460.00330

4345741.556.586.551.5813.510.00351

4345751.586.686.591.5713.480.00336

4345761.576.716.281.6013.480.00366

4345771.576.696.601.5613.480.00332

4345781.586.746.571.5513.470.00329

4345791.566.646.581.6513.450.00344

4345801.666.676.631.5813.490.00343

4345811.576.706.441.5813.580.00359

4345831.656.706.541.6413.580.00363

Average->1.5946.6486.5211.59313.5010.003519

x1

x2

x5 d

x3

x4

19. THE SPECTRUM ANALYZER

APPARATUS:

Graph paper

Power strip

BNC-to-banana adapter (2)

Banana wires (4)

Function generator, Simpson 420

Multimeter, DigiTec

AM/FM radio

Attenuator

Spectrum analyzer, HP 8591A

Oscilloscope

PROCEDURE:

The Simpson 420 Function Generator

This device creates a changing voltage with an adjustable frequency, amplitude and shape. Attach a BNC-to-banana adapter to the OUTPUT of the function generator and use two banana wires to connect it to the DigiTec multimeter, dialed to 20V and with the DCV button pushed in. Make sure the GND side of the adapter is plugged into the negative input (LO). On the function generator set the range at X.1 Hz and the frequency dial at 1, to get an output with a frequency of 0.1 Hz. Set the AMPLITUDE dial half-way between MIN. and MAX. and the AMPLITUDE button pushed in to get an extra 30 decibels of amplitude. Set the waveform to sine wave. The DC offset should be OFF.

1.Observe how the voltage changes. Time this to see how long it takes to go through one cycle. Explain this result.

2.Set the signal generator to f = 1 Hz. Explain this result.

3.Set the signal generator to f = 1 kHz. Explain this result.

4.Set the multimeter to AC volts (ACV). Even though the frequency is too high for the DC multimeter setting to follow, the AC setting does give a reading. Vary the amplitude dial on the function generator to confirm that the multimeter reading is genuine. This is the root-mean-square voltage Vrms. Adjust the function generator amplitude to give an output of exactly

0.2 volts.

5.Place a 2X, a 5X or a 10X attenuator in the circuit. What is the new voltage? What is the percent difference between the measured voltage and the theoretical voltage?

The Hitachi V550-B Oscilloscope

1.Connect a BNC-to-banana adapter to the CH1 INPUT of the oscilloscope, and use two banana wires to connect it to the function generator, negative to negative. Set the function generator to 0.1 Hz. Set the Input switch (just above the Input) to DC. The Volts/div dial adjusts the vertical scale; its inner dial should be completely clockwise for the results to be calibrated accurately. Set the outer dial to .1 Volts/div. Find the Power On knob and turn it on. Adjust the Intensity and the Focus to get a thin, easy-to-see line. Turn off the Scale Illumination if it is on. We will be using Channel 1 (not 2) and Trace A (not B).

An oscilloscope is a device that graphs the instantaneous voltage (y-axis) as a function of time (x-axis). It is in effect a voltmeter that measures voltages that are rapidly changing but periodic. Set the multimeter to read DC Volts and see how the oscilloscope behavior compares to the multimeter readings. There are many different oscilloscope settings that can be adjusted for different circumstances. You have looked at six already; now look at the others.

2.Below the screen are the channel-1 vertical adjustments. Set the left-hand (Input) switch to GND (ground) and use the Position switch to place the V=0 volts line in the middle of the screen. The Mode switch should be set to Ch. 1. Now set the input switch back to DC. Measure the amplitude (one-half of the peak-to-peak amplitude) with the multimeter and the oscilloscope. Assuming the oscilloscope is completely accurate, what is the percent difference of the observed value of the multimeter reading?

3.The CH2 or Y dials will not be used, because no input is being received. This is also true for B TRIG, DLY TIME MULT and CAL, leaving only the A TRIG controls and the Time/div controls to be explained.

4.The A Trig Mode, Coupling and Source switches should all be set to the far left. The Level dial is used to stabilize the trace, and you may have to adjust this frequently.

Change the frequency on the function generator to 1 kHz. On the oscilloscope, set the input switch to AC and the Time/div dial to 1 msec. This dial determines how much time it takes the sweep to cross one division horizontally. Does the pattern on the screen match what you would expect from the two dial settings? Explain.

5.Adjust the signal generator’s amplitude so that the oscilloscope shows an amplitude of 0.2 volts. You may have to adjust the Position dial to get the signal to be symmetric. What does the multimeter read? What is the percent difference between the voltmeter reading V (the root-mean-square voltage) and the reading you would predict from the oscilloscope amplitude Vmax? Theoretically, Vrms = Vmax/√2 for a sinusoid.

6.Repeat step 5, with the signal generator generating a square wave. Explain why Vrms = Vmax for a square wave.

7.Repeat step 5, with the signal generator generating a triangular wave. Clearly and carefully show your calculation for the theoretical value of Vrms.

HP 8590 Series Spectrum Analyzer

1.The spectrum analyzer is a device that takes a periodic function and finds out how much amplitude it has at each wavelength, just as a prism does when it breaks visible light into a spectrum. You will compare the observed signal amplitudes at certain wavelengths with the theoretical amplitudes.

Suppose that a function v(t) is periodic with a period of

2π so that v(t) = v(t + 2π), and is symmetric around the y-axis so that v(t) = v(-t). Fourier analysis shows that the function can be written as a sum of cosine functions:

v(t) = a1.cos(1.t) + a3.cos(3.t) + a5.cos(5.t) + ... ,

with the values of an depending on the shape of v(t).

A triangular wave with an amplitude of 0.25 volts and a period of 2π can be written as

v(t) = 2.(1.π)-2cos(t) + 2.(3.π)-2cos(3t) + 2.(5.π)-2cos(5t) + ...

A square wave with an amplitude of 0.25 volts and a period of 2π can be written as

v(t) = (1.π)-1cos(t) + (-3.π)-1cos(3t) + (5.π)-1cos(5t) + ...

A cosine wave with an amplitude of 0.25 volts and a period of 2π can be written as

v(t) = 0.25cos(t) with no other terms (of course!).

2.Write down the theoretical values of a1, a3, a5, a7 and a9 on the data sheet for these three waves, assuming each wave has an amplitude of 0.25 volts.

3.Press LINE to turn the spectrum analyzer on, and use a BNC-to-banana adapter on its INPUT to connect it to the function generator (set to give a 50 kHz triangular wave) and the oscilloscope.

4.Press the FREQUENCY button. Five options will appear on the right-hand side of the screen, with buttons offscreen to their right. Press the START FREQ button, and key in 0 kHz with the DATA buttons. Press the STOP FREQ button, and key in 500 kHz. The display now shows the spectrum between 0 kHz and 500 kHz.

5.Press the AMPLITUDE button. If the SCALE is set to LIN, press the button to set it to LOG. The display seems to show many overtones (9 peaks should be visible) but that is because this is now a logarithmic plot; a decrease of one division is a factor-of-ten decrease in strength. Adjust the frequency dial on the function generator to make these peaks line up with the vertical grid lines. The highest peak is at 50 kHz, as you would expect.

6.Press the SCALE button to make the scale linear. Find the MARKER section of the front panel, and press PEAK SEARCH. A small diamond ◊ will appear above one of the peaks. Use the NEXT PK RIGHT or NEXT PK LEFT buttons to move it to the peak at 50 kHz. If the diamond does not move, the peak is too small. Rotate the large white dial counter-clockwise until the peak becomes large, and try again.

7.Notice that the frequency of the peak is listed on the upper right of the screen. Just below this is the voltage of the peak. Press PEAK SEARCH again, and adjust the amplitude of the function generator until the voltage reading is as close as possible to 202.6 mV (= 0.2026 volts = a1).

8.Use the NEXT PK RIGHT or NEXT PK LEFT buttons to move the diamond to the next peak, at 150 kHz. Again, if the diamond does not move, the peak is too small. Rotate the large white dial counter-clockwise until the peak becomes large, and try again. Read the value of a3, then a5, a7 and a9. Read Vmax from the oscilloscope.

9.Repeat steps 7 and 8 for the square wave, with

a1 = 318.3 mV.

10.Repeat steps 7 and 8 for the cosine wave, with

a1 = 250.0 mV.

11.For the triangular wave data, plot a graph of an for

n = 1, 3, 5, 7 and 9 as a function of n. Do this for both the theoretical and observed values, on the same graph. How closely do the theoretical and observed values seem to match?

12.Repeat step 11 for the square wave. The a3 term in the equation is negative; since the spectrum analyzer measures the absolute value of this term, take its absolute value.

13.Disconnect the function generator from the other equipment and run a wire 1 meter long horizontally from the positive output. Turn off the oscilloscope and the multimeter and put them away. Set the function generator to create a cosine wave with as large an amplitude as possible, with a frequency of 50 kHz. This will function as a radio transmitter.

Run a wire 1 meter long horizontally from the positive input of the spectrum analyzer, and place it parallel to the radio transmitter one meter away. Do not allow them to come into contact. This will function as a radio antenna.

14.Locate the signal on the spectrum analyzer, and measure its amplitude. The antenna will be more sensitive if a student holds on to the antenna’s metal end. Repeat for distances of 2 meters, 5 meters, 7.5 meters and 10 meters.

Graph the results and find a formula relating distance and amplitude. You may find that the formula does not match the results for distances that are comparable to the length of the wire.

AM and FM Radio Signals

1.Turn off the function generator, and connect several wires in series to create a very long radio antenna for the spectrum analyzer. Examine the spectrum between 600 kHz and 1600 kHz. This is the AM band. Select one of the strongest signals and measure its frequency. Locate the station on the radio. How closely does its announced frequency match its actual frequency?

2.Find a talk show on the FM band, which is between 90 MHz and 108 MHz. Locate the talk show’s signal on the spectrum analyzer, and set the stop frequency to 0.5 MHz above the start frequency. How does the shape of the spectrum analyzer profile change when the station broadcasts an audio signal, compared to when it is silent?

20. RADIOACTIVITY: HALF-LIFE

PURPOSE: To determine the half-life of Ba-137m.

INTRODUCTION:

All radioactive substances experience a decline in activity (rate of emission of nuclear radiation) with time. This decline in activity is characterized by the half-life T1/2 of the substance. The half-life is defined as the time required for the activity of the substance to decline by one-half.

In the present experiment an ion-exchange column (referred to as a isogenerator) is used which has Cs-137 loaded into it.

Cs-137 is the long-lived parent isotope (half-life = 30 years) and β-decays to the daughter particle Ba-137m, an excited state of Ba-137. The “m” stands for “metastable”. The subsequent

β-decay of Ba-137m to Ba-137 has a short half-life of 2.55 minutes that can be found by monitoring the activity of Ba-137m over a period of time with a Geiger counter.

APPARATUS:

Beaker, 150-mL

Buret clamp

Graph paper

Timer, electronic

Ring stand

Isogenerator

Planchet

Geiger counter with 6-second interval

Geiger tube

Distilled water

PROCEDURE:

1.Turn on the Geiger counter and reset it to zero. Make sure that the operating voltage is set to the plateau voltage (about 800 V). Set the count time dial to 0.1 minute (= 6 sec).

2.Place the isogenerator far from the Geiger counter. When you are ready to begin counting, place the planchet under the isogenerator and turn the white plastic handle vertically and elute 10 drops, then turn the handle sideways to stop the flow.

3.The counting procedure is as follows:

a) Slide the planchet under the Geiger counter tube, as close to the tube as possible.

b) Simultaneously start the Geiger counter and the timer. This is time t = 0 s (the timer will run continuously).

c) After the 6-second counting period has elapsed, record the counts in the data table. Now reset the Geiger counter.

d) At time t = 30 s, start counting for another six seconds. As before, record the data and reset the Geiger counter.

e) Repeat the above procedure at 30 second intervals to

t = 720 s. Now multiply all of these six-second counts by 10 (by adding a zero to your Observed Activity) to create Observed Activity in units of counts per minute (CPM).

f) Do not touch the sample. Leave it where it is for about

25 minutes. Then take a count of the sample activity for

5 minutes, and divide this by 5 to get the Correction to Observed Activity.

4.The CPM from step 3f is the background count plus any of the Cs-137 that might have eluted with the Ba-137m. The idea here is that after ten half-lives there should be essentially no Ba-137m left in the sample.

5.This CPM from step 4 is to be subtracted from each of the sample counts you made, in order to get the Corrected Observed Activity.

6.The law of radioactive decay says that the decline in activity of a sample is a decreasing exponential function of time:

R = Ro·e-λt .

where Ro is the initial activity and R is the activity at any given time.

Taking the natural logarithm of both sides of the above equation, we have:

ln(R/Ro) = -λt .

A plot of ln(Ro/R) vs. time should yield a straight line, the slope of which is equal to λ. The half-life can be determined from T1/2 = ln(2)/λ.

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