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i University of Northern British Columbia Physics Program Physics 206 Laboratory Manual Winter 2012 ii Contents 1 Measuring Wavelengths of Light Waves Using A Transmission Diffraction Grating 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Measuring Mass-to-Charge Ratio e/m 2.1 Introduction . . . . . . . . . . . . . . 2.2 Experimental Procedure . . . . . . . 2.3 Supplementary Notes . . . . . . . . . 3 Michelson’s Interferometer 3.1 Introduction . . . . . . . . . . 3.2 Experimental Procedure . . . 3.2.1 Setup and alignment of 3.2.2 Measurements: . . . . of . . . . . . the Electron 11 . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . . . . . . the Interferometer: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 20 20 21 4 Determination of the Elementary Charge e Using Millikan Apparatus 23 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Photoelectric Effect 31 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Radioactivity 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Geiger-Mueller Counter . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Random Nature of Radioactive Decay . . . . . . . . . . . . . . . . . . . . 6.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Determination of the Operating Voltage of the G-M Tube: . . . . . . . . 6.2.2 Statistics of Counting Experiments (Random Nature of Radioactive Decay) iii 37 37 37 39 40 40 42 iv 7 Radioactivity II 7.1 Introduction . . . . . . . . . . . . . . 7.1.1 Inverse Square Law . . . . . . 7.1.2 Absorption of Radiation . . . 7.2 Experimental Procedure . . . . . . . 7.2.1 Inverse Square Law: . . . . . 7.2.2 Linear Absorption coefficient: CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Error Analysis A.1 Measurements and Experimental Errors . . . . A.1.1 Systematic Errors: . . . . . . . . . . . A.1.2 Instrumental Uncertainties: . . . . . . A.1.3 Statistical Errors: . . . . . . . . . . . . A.2 Combining Statistical and Instrumental Errors A.3 Propagation of Errors . . . . . . . . . . . . . . A.3.1 Linear functions: . . . . . . . . . . . . A.3.2 Addition and subtraction: . . . . . . . A.3.3 Multiplication and division: . . . . . . A.3.4 Exponents: . . . . . . . . . . . . . . . A.3.5 Other frequently used functions: . . . . A.4 Percent Difference: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 45 45 46 46 47 . . . . . . . . . . . . 49 49 49 50 51 52 53 54 54 54 54 55 55 List of Figures 1.1 1.2 Diffraction of light through a diffraction grating. . . . . . . . . . . . . . . . . . . Diffraction Grating spectroscope. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 e/m apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Circuit diagram for the e/m tube. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Circuit diagram for Holmholtz coils. . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Basic components of Michelson’s interferometer. . . . . . . . . . . . . . . . . . . 20 4.1 4.2 Circuit diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Various forces acting on the oil drop. . . . . . . . . . . . . . . . . . . . . . . . . 27 5.1 5.2 5.4 Circuit diagram for a photo cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage vs current produced by light beams with the same frequency and different intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage vs current produced by light beams with the same frequency and different intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Work functions of various photo cathodes. . . . . . . . . . . . . . . . . . . . . . 6.1 6.2 Circuit Diagram for a Geiger Mueller Counter. . . . . . . . . . . . . . . . . . . . 37 Voltage vs count rate of a G-M tube. . . . . . . . . . . . . . . . . . . . . . . . . 38 7.1 7.2 Geometry of a point source and a G-M tube. . . . . . . . . . . . . . . . . . . . . 46 Absorption of radiation in a slab. . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 1 6 7 32 33 35 36 Introduction to Physics Labs Lab Report Outline: Physics 2?? section L? date Your Name Student # Lab Partner’s Name Lab #, Lab Title The best laboratory report is the shortest intelligible report containing ALL the necessary information. It should be divided into sections as described below. Object - One or two sentences describing the aim of the experiment. (In ink) Theory - The theory on which the experiment is based. This usually includes an equation which is to be verified. Be sure to clearly define the variables used and explain the significance of the equation. (In ink) Apparatus - A brief description of the important parts and how they are related. In all but a few cases, a diagram is essential, artistic talent is not necessary but it should be neat (use a ruler) and clearly labelled. (All writing in ink, the diagram itself may be in pencil.) Procedure - Describes the steps of the experiment. In particular, remember to state what quantities are measured, how many times and what is done with the measurements (are they tabulated, graphed, substituted into an equation, etc.). Then explain what sort of analysis will be performed. (In ink) Data - Show all your measured values and calculated results in tables. Show a detailed sample calculation for each different calculation required in the lab. Graphs should be included on the page following the related data table. (If experimental errors or uncertainties are to be taken into account, the calculations should also be shown in this section.) Include a comparison of experimental and theoretical results. (The graphs may be filled out in pencil but everything else must be in ink.) Discussion and Conclusion - What have you proven in this lab? Support all statements with the relevant data. Was your objective achieved? Did your experiment agree with the theory? With the equation? To what accuracy? Explain why or why not. Identify and discuss possible sources of error. (In ink) Introduction to Physics Labs 3 Lab Rules: Please read the following rules carefully as each student is expected to be aware of and to abide by them. They have been implemented to ensure fair treatment of all the students. Missed Labs: A student may miss a lab without penalty if due to illness(a doctor’s note is required) or emergency. As well, if an absence is expected due to a UNBC related time conflict (academic or varsity), arrangements to make up the lab can be made with the Senior Laboratory Instructor in advance. Students will not be allowed to attend sections, other than the one they are registered in, without the express permission of the Senior Laboratory Instructor. Failure to attend a lab for any other reason will result in a zero (0) being given for that lab. Late Labs: Labs will be due Tuesday at 3:00pm the week following the lab, after which labs will be considered late. Late labs will be accepted up until 1 week from the beginning of the lab. Students will be allowed 1 late lab without penalty after which all subsequent late labs will be given a mark of zero (0). Any labs not received within one week will no longer be accepted for grading and given a mark of zero (0). Completing Labs: It is unlikely that these lab reports can be completed within the scheduled lab time. Given that the lab time is Friday in the late afternoon it is highly recommended you come prepared prior to the lab. The due date is Tuesday so as to allow adequate time for questions on the Monday. Please allot sufficient time over the weekend to work on the lab in case you need to ask questions on Monday. Conduct: Students are expected to treat each other and the instructor with proper respect at all times and horseplay will not be tolerated. This is for your comfort and safety, as well as your fellow students’. Part of your lab mark will be dependent on your lab conduct. 4 Introduction to Physics Labs Plagiarism: Plagiarism is strictly forbidden! Copying sections from the lab manual or another person’s report will result in severe penalties. Some students find it helpful to read a section, close the lab manual and then think about what they have read before beginning to write out what they understand in their own words. Use Ink: The lab report, except for diagrams and graphs, must be written in ink, this includes filling out tables. If your make a mistake simply cross it out with a single, straight line and write the correction below. Professional scientists use this method to ensure that they have an accurate record of their work and so that they do not discard data which could prove to be useful after all. For this reason also, you should not have “rough” and “good” copies of your work. Neatly record everything as you go and hand it all in. If extensive mistakes are made when filling out a table, mark a line through it and rewrite the entire table. Lab Presentation: Marks will be given for presentation, therefore neatness and following the outline is important. It makes your lab easier to understand and your report will be evaluated accordingly. It is very likely that, if the person marking your report has to search for information or results, or is unable to read what you have written, your mark will be less than it could be! Contact Information: If your lab instructor is unavailable and you require extra help, or if you need to speak about the labs for any reason, the Senior Laboratory Instructor, Jon Schneider, is available by email: [email protected] or phone: 250.960.5708. Office hours are posted on the physics lab webpage www.unbc.ca/physics/labs.html. Supplies needed: Clear plastic 30 cm ruler Metric graph paper (1 mm divisions) Paper duotang Lined paper Calculator Pen, pencil and eraser Experiment 1 Measuring Wavelengths of Light Waves Using A Transmission Diffraction Grating 1.1 Introduction In one Physics 202 experiment the prism spectrometer had to be calibrated in terms of known wavelengths before we were able to determine unknown wavelengths of light experimentally. How, then, are the wavelengths of spectral lines or colors initially determined? This is most commonly done with a diffraction grating, a simple device that allows for the study of spectra and the measurement of wavelengths. By replacing the prism with a diffraction grating, a prism spectrometer becomes a grating spectrometer. When using a diffraction grating, the angle(s) at which the incident beam is diffracted relate directly to the wavelength(s) of the light. In this experiment the properties of a transmission grating will be investigated and the wavelengths of several spectral lines will be determined. A diffraction grating consists of a piece of metal or glass with a very large number of evenly spaced parallel lines or grooves. This gives two types of gratings: reflection gratings and transmission gratings. Reflection gratings are ruled on polished metal surfaces; light is reflected from the unruled areas, which act as a row of “slits.” Transmission gratings are ruled on glass, and the unruled slit areas transmit incident light. The transmission type is used in this experiment. Common laboratory gratings have 300 grooves per mm and 600 grooves per mm (about 7500 grooves per in. and 15,000 grooves per in.), and are pressed plastic replicas mounted on glass. Glass originals are very expensive. 6 EXPERIMENT 1 Figure 1.1: Diffraction of light through a diffraction grating. Diffraction refers to the “bending” or deviation of waves around sharp edges or corners. The slits of a grating give rise to diffraction, and the diffracted light interferes so as to set up interference patterns (Figure (1.1)). Complete constructive interference of the waves occurs when the phase or path difference is equal to one wavelength, and the first-order maximum occurs for d sin(θ1 ) = λ (1.1) where d is the grating constant or distance between the grating lines, θ1 the angle the rays are diffracted from the incident direction, and d sin θ1 the path difference between adjacent rays. The grating constant is given by: d= 1 N (1.2) where N is the number of lines or grooves per unit length (usually per millimeter or per inch) of the grating. A second-order maximum occurs when d sin θ2 = 2λ, and so on, so that in general we may write: d sin(θn ) = nλ n = 1, 2, 3, .... (1.3) where n is the order of the image maximum. The interference is symmetric on either side of an undeviated and undiffracted central maximum of the slit image, so the angle between symmetric image orders is 2θn (Figure (1.2)). MEASURING WAVELENGTHS OF LIGHT WAVES USING A TRANSMISSION DIFFRACTION GRATING 7 Figure 1.2: Diffraction Grating spectroscope. In practice, only the first few orders are easily observed, with the number of orders depending on the grating constant. If the incident light is other than monochromatic, each order corresponds to a spectrum. (That is, the grating spreads the light out into a spectrum.) As can be seen from Equation (1.1), since d is constant, each wavelength (color) is deviated by a slightly different angle, so that the component wavelengths are separated into a spectrum. Each diffraction order in this case corresponds to a spectrum order. (The colorful displays seen on compact disks [CDs] result from diffraction.) 1.2 Experimental Procedure 1. Review the general operation of a spectrometer if necessary. Record the number of lines per metre of your diffraction grating . Mount the grating on the spectrometer table with the grating ruling parallel to the collimator slit and the plane of the grating perpendicular to the collimator axis. DETERMINATION OF THE WAVELENGTH RANGE OF THE VISIBLE SPECTRUM 2. Mount an incandescent light source in front of the collimator slit. Move the spectrometer telescope into the line of the slit of the collimator and focus the cross hairs on the central slit image. You will want to adjust the scale so that the angle reading is zero at this point. Notice that this central maximum or “zeroth”-order image does not depend on the wavelength of light, so that a white image is observed. Then move the telescope to either side of the incident beam and observe the first- and second-order spectra. Record which one is spread out more in you laboratory report. 8 EXPERIMENT 1 3. Focus the cross hairs on the blue (violet) end of the first-order spectrum at the position where you judge the spectrum just becomes visible. Record the divided circle readings (to the nearest minute of arc). 4. Move the telescope to the other (red) end of the spectrum and record the divided circle reading of its visible limit. 5. Repeat this procedure for the first-order spectrum on the opposite side of the central maximum. The angular difference between the respective readings corresponds to an angle of 2θ (Figure (1.2)). 6. Compute the grating constant d in millimeters and with the experimentally measured θ0 s compute the wavelengths of red and blue ends to define the range of the visible spectrum in nanometers and angstrom units. DETERMINATION OF THE WAVELENGTHS OF SPECTRAL LINES 7. Mount the mercury discharge tube in its power supply holder and place in front of the collimator slit. Caution: Work very carefully, as the discharge tube operates at high voltage and you could receive an electrical shock. Turn on the power supply and observe the first- and second-order mercury line spectra on both sides of the central image. 8. Because some of the lines are brighter than others and the weaker lines are difficult to observe in the second-order spectra, the wavelengths of only the brightest lines will be determined. Find the listing of the mercury spectral lines in Table (1.1), and record the color and wavelength. Then, beginning with either first-order spectra, set the telescope cross hairs on each of the four brightest spectral lines and record the divided circle readings (read to the nearest minute of arc). Repeat the readings for the first-order spectrum on the opposite side of the central image and find the average angle and its error for each line. 9. Repeat the measurement procedure for the four lines in the second-order spectra and, using Equation (1.1), compute the wavelength of each of the lines for both orders of spectra. Compare with the accepted values by computing the percent error of your measurements in each case. Which order gives you more accurate results? Note: In the second-order spectra, two yellow lines, a doublet, may be observed. Make certain that you choose the appropriate line. (Hint: See the wavelengths of the yellow lines Table (1.1). Which is closer to the red end of the spectrum?) MEASURING WAVELENGTHS OF LIGHT WAVES USING A TRANSMISSION DIFFRACTION GRATING Table 1.1: Wavelength of visible spectral lines of Helium and Mercury. Wavelength Element (nm) (Å) Helium 388.9 3889 396.5 3965 402.6 4026 438.8 4388 447.1 4471 471.3 4713 492.2 4922 501.5 5015 587.6 5876 667.8 6678 706.5 7065 Mercury 404.7 4047 407.8 4078 435.8 4358 491.6 4916 546.1 5461 577.0 5770 579.0 5790 690.7 6907 Color Relative Intensity Violet 1000 Violet 50 Violet 70 Blue-Violet 30 Dark Blue 100 Blue 40 Blue-Green 50 Green 100 Yellow 1000 Red 100 Red 70 Violet 300 Violet 150 Blue 500 Blue-Green 50 Green 2000 Yellow 200 Yellow 1000 Red 125 9 10 EXPERIMENT 1 Experiment 2 Measuring Mass-to-Charge Ratio e/m of the Electron 2.1 Introduction When a charged particle such as an electron moves in a magnetic field in a direction at right angles to the field, it is acted on by a force, the value of which is given by f = Bev (2.1) where B is the magnetic-flux density in webers/meter2 (W b/m2 ), e is the charge on the electron in coulombs (C), and v is the velocity of the electron in meters/sec (m/s). This force causes the particle to move in a circle in a plane perpendicular to the magnetic field. The radius of this circle is such that the required centripetal force is furnished by the force exerted on the particle by the magnetic field. Therefore mv 2 = Bev r (2.2) where m is the mass of electron in kg, and r is the radius of circle in m, i.e. 1/2 D. If the velocity of the electron is due to its being accelerated through a potential difference V , it has, due to its velocity, a kinetic energy of 1 2 mv = eV 2 (2.3) where V is the accelerating potential in volts. Substituting the value of v from Equation (2.2) into Equation (2.3), we get: e 2V = 2 2 m B r (2.4) 12 EXPERIMENT 2 Thus when the accelerating potential, the flux density of the magnetic field, and the radius of the circular path described by the electron beam are known, the value of e/m can be computed, and is given in C/kg by Equation (2.4) if V is in volts, B is in W b/m2 , and r is in m. The magnetic field which causes the electron beam to move in a circular path is produced by Helmholtz coils. The magnetic field produced by Helmholtz coils can be expressed, in terms of the current going through the coils and the dimensions of the coils, as: 8µ◦ N I B= √ 125a (2.5) where: N is the number of turns of wire in each coil, I is the current through the coils in amperes (A), a is the mean radius of the coil in meters, µ◦ is the permeability of empty space, which is 4π × 10−7 W b/A · m. Substituting Equation (2.5) in Equation (2.4) gives 3.91 a2 V e = ( 2 · 2) 2 2 m µ◦ N I r e a2 V = (2.47 × 1012 2 ) 2 2 m N I r (2.6) Equation (2.6) is the working equation for this apparatus. The quantity within parentheses is a constant for any given pair of Helmholtz coils. The value of r, the radius of the electron beam, can be varied by changing either the accelerating potential or the Helmholtz field current. For any given set of values, the value of e/m can be computed. The apparatus used in this experiment is composed of a vacuum tube where an accelerated electron beam is produced. The tube is placed between the Helmholtz coils. The magnetic field produced by the coils bends the electron beam. The beam of electrons in the tube is produced by an electron gun composed of a straight filament surrounded by a coaxial anode containing a single axial slit. See Figure (2.1A and 2.1B) for schematic diagrams of the tube. Electrons emitted from the heated filament F are accelerated by the potential difference applied between F and the anode C. Some of the electrons come out as a narrow beam through the slit S in the side of C. When electrons of sufficiently high kinetic energy (10.4 electron volts or more) pass through the mercury vapor in the tube, a fraction of the atoms will be ionized. On recombination of these ions with stray electrons, the mercury spectrum is emitted with its characteristic blue color. Since recombination with emission of light occurs very near the point where ionization took place, the path of the beam of electrons is visible as the electrons travel through the mercury vapor. MEASURING MASS-TO-CHARGE RATIO E/M OF THE ELECTRON Figure 2.1: e/m apparatus. 13 14 EXPERIMENT 2 Figure (2.1A) is a sectional view of the tube and filament assembly. Figure (2.1B) is a detailed section of the filament assembly at right angles to Figure (2.1A). Where: “A” Five cross bars attached to staff wire “B” Typical path of beam of electrons “C” Cylindrical anode “D” Distance from filament to far side of each of the cross bars “E” Lead wire and support for anode “F” Filament “G” Lead wires and supports for filament “L” Insulating plugs “S” Slit in cylindrical anode The magnetic field of the Helmholtz Coils causes the stream of electrons to move in a circular path, the radius of which decreases as the magnetic field increases. By proper control of the magnetic field, the sharp outer edge of the beam can be made to coincide with any one of five bars spaced at known distances from the filament. Each coil of the pair of Helmholtz Coils has 72 turns of copper wire with a resistance of approximately one ohm. The approximate mean radius of the coil is 33 centimeters. The coils are supported in a frame which can be adjusted with reference to the dip angle so that their magnetic field will be parallel to the earth’s magnetic field but oppositely directed. A graduated scale indicates the angle of tilt. A safety device prevents the coils from accidentally dropping back to the horizontal. Wooden seats with retaining straps cradle the tube in a central plane midway between the two coils. 2.2 Experimental Procedure 1. Place the tube in its support in the center of the frame. The end of the tube from which the leads extend should be adjacent to that part of the frame on which connectors are mounted. Secure the tube in place with the straps. Orient the Helmholtz Coils so that the tube will have its long axis in a magnetic northsouth direction, as determined by a compass. Measure the magnetic inclination at this location with a good dip needle. Tip the coils up until the plane of the coils makes an MEASURING MASS-TO-CHARGE RATIO E/M OF THE ELECTRON 15 angle with the horizontal equal to the complement of the dip angle. The axis of the coils should now be parallel to the earth’s magnetic field. Figure 2.2: Circuit diagram for the e/m tube. 2. Make electrical connections to the tube as shown in Figure (2.2). Make electrical connections to the Helmholtz Coils as shown in Figure (2.3). 3. The filament of the tube has the proper electron emission when carrying 2.5 to 4.5 amperes. Exceeding 4.5 amperes will materially shorten the tube life. Good results can be obtained using an accelerating voltage of 22.5 to 45 volts. An electron emission current of 5 to 10 mA produces a visible beam. For maximum filament life, operate the tube with the minimum filament current necessary for proper electron emission. It is good practice to apply about 25 volts accelerating potential to the anode before heating the filament. Always start with a low filament current and carefully increase it until the proper electron emission is obtained. Since the electron beam appears suddenly, it is advisable to increase the filament current slowly to avoid overheating and possibly burning out the filament. Apply 20 to 30 volts accelerating potential to the tube. With maximum resistance in the filament circuit, close the switch in this circuit. Gradually decrease this resistance until the milliammeter indicates 5 to 10 mA and the electron beam strikes the tube wall. Rotate the tube in its cradle until the electron beam is horizontal and the bars extend upward from the staff wire. Note that the electron beam is deflected slightly toward the base of the tube by the earth’s magnetic field. 16 EXPERIMENT 2 Figure 2.3: Circuit diagram for Holmholtz coils. Table 2.1: Distance of various cross bars to the filament of the e/m tube. Crossbar Number Distance to Filament (D), meters 1 0.065 2 0.078 3 0.090 4 0.103 5 0.115 4. Once the anode voltage has stabilized send a small current through the Helmholtz Coils. If the magnetic field of the coils straightens the beam or curves it in the opposite direction, the coils are correctly connected. If this field increases the deflection toward the base, the current is flowing through the coils in the wrong direction, and the connections must be reversed. If no effect is observed on the beam, the field due to one coil is opposing that of the other, and connections to one coil must be reversed. Make any necessary changes and increase the field current sufficiently to cause the beam to describe a circular path without striking the wall of the tube. The apparatus should now be ready for obtaining the necessary data to compute the value of e/m. In the manufacture of the tubes, the crossbars have been attached to the staff wire, and this assembly attached to the filament assembly in precise fixtures which accurately control the distances between the several crossbars and the filament. Distances given are to the far side of the crossbar. These distances are the diameters of the several circles which the electron beam will describe. Following the prescribed procedure, heat the filament until the electron beam is visible. Allow a few minutes for warm-up and temperature equilibrium. MEASURING MASS-TO-CHARGE RATIO E/M OF THE ELECTRON 17 Table 2.2: Data table. Crossbar No. D r V I1 I2 I e/m 5. Close the circuit comprising the Helmholtz Coils and adjust the value of the current until the electron beam is straight. One method is to adjust the beam until it is aligned with a straight edge held as close to the tube as possible. Another method is as follows. Increase the current through the coils until the beam shows a slight curvature as compared with the straight edge and record the value of the current. Then decrease the current until the beam shows an equal curvature in the opposite direction and record the current. The average of these values will be the value of the current required to straighten the beam. This adjustment should be made carefully and accurately. When the beam is straight, the magnetic field of the coils just equals the earth’s magnetic field. Record the value of this current as I1 in a table similar to Table (2.2) The value of I1 is dependent only on the accelerating voltage, V , and the earth’s magnetic field. It is constant unless V is changed, in which case I1 must be redetermined. 6. Increase the field current until the electron beam describes a circle. Adjust its value until the sharp outside edge of the beam strikes the outside edge of a crossbar. Record this value as I2 in the table. The outside edge of the beam is used because it is determined by the electrons with the greatest velocity. The electrons leaving the negative end of the filament fall through the greatest potential difference between filament and anode, and have the greatest velocity. It is this potential difference which the voltmeter measures. 7. Determine and record the field current required to cause the electron beam to strike each of the other bars. Subtract I1 from each value of I2 and record as I. This is the value of the current to be used in Equation (2.6) for computing e/m. Compute e/m for each value of I. 8. Repeat the above using a different value of accelerating voltage. 9. Calculate the average e/m ratio and it’s related error. Compare this result with the accepted value of e/m, which is 1.76 × 1011 C/kg. 18 EXPERIMENT 2 2.3 Supplementary Notes • By subtracting the field current just necessary to overcome the earth’s field from the total field current required to curve the beam into a given circular path, the effect of the earth’s field on the determination of e/m is eliminated, and there is no need to know or otherwise determine the earth’s magnetic field. However, if desired, the value of the earth’s magnetic field may be computed using the value of I1 in Equation (2.5). • If the magnetic field of the Helmholtz Coils is increased sufficiently to cause the electron beam to describe a circle of small radius, then by rotating the tube slightly, the beam will spiral either upward or downward. Reversing the filament current will reverse the direction of the spiral (the spiraling of the beam is caused by the magnetic effect of the filament current). A reversing switch in the filament circuit facilitates this demonstration. • The action of this tube and Helmholtz Coils can be used in explaining the principle of a mass spectrometer. A consideration of Equation (2.6) shows that, for charged particles carrying the same charge but differing in mass, the radius of the circular path into which they are caused to move by the magnetic field differs for particles of different masses. Knowing the value of the charge and measuring the radius of the circular path, the mass of particles can be determined. This is essentially what is done in a mass spectrometer. Experiment 3 Michelson’s Interferometer 3.1 Introduction An interferometer is a device that can be used to measure wavelength, length or change in length with great accuracy by means of interference fringes. In this experiment we will use an interferometer which was devised and built by A. A. Michelson in 1881. This interferometer and variations of it are still in use. The interferometer is shown schematically in Figure (3.1). Light from the source falls on mirror M which is half-silvered i.e. it is a beam splitter. This mirror has a silver coating thick enough to transmit half the incident light and to reflect the other half. At M the light, then, splits into two waves. One is transmitted to mirror M1 and the other is reflected to mirror M2 . The two mirrors M1 and M2 reflect the light waves back along their directions of incidence and the waves will then arrive at the observation screen. The two waves are coherent since they originated from the same source. An interference pattern will then form at the observation screen. The two light waves travel through two different paths. If the mirrors M1 and M2 are exactly perpendicular to each other, then the path difference between the two waves arriving at the observation screen is simply d2 − d1 . As a result a circular fringe pattern will form on the observation screen due to small changes in the angle of incidence of the light waves emanating from different points on the source. If M2 moves backward or forward, the effect is to change the path difference d2 − d1 . If M2 moves a distance λ/2 then the path difference will change by λ and a bright fringe will then become a dark one and a dark fringe becomes a bright one. In other words, for every fringe change from bright to dark (or from dark to bright) the moving mirror moves a distance of λ/2 of the light used. If now M2 moves a distance dm and as a result N fringes changed from bright fringes to dark fringes, then: λ= 2dm N (3.1) 20 EXPERIMENT 3 If the wavelength λ of the light used is known one then can measure distances as small as the wavelength. Figure 3.1: Basic components of Michelson’s interferometer. 3.2 Experimental Procedure In this experiment we are going to measure the wavelength of a He-Ne laser using Michelson Interferometer. 3.2.1 Setup and alignment of the Interferometer: 1. Set the interferometer on a lab table with the micrometer knob pointing toward you. Position the laser to the left of the base. 2. Place the mirror (M1 ) that is mounted on right angle bracket in the recessed hole in the interferometer base, and secure with a thumbscrew. 3. Place the laser on the lab table. Adjust the laser until the laser beam is approximately parallel with the top of the interferometer base and strikes the mirror in the center. To check that the beam is parallel with the base, place a piece of paper in the beam path, with the edge of paper flush against the base. mark the height of the beam on the paper. Using the piece of paper, check that the beam is at the same height at both ends of the MICHELSON’S INTERFEROMETER 21 interferometer base. 4. Adjust the laser until the beam is reflected from the mirror right back to the laser aperture. This can be done by gently tapping the rear end of the laser transverse to the axis of the beam. 5. Mount the beam splitter (M ), the second mirror (M2 ), and the observation screen on the interferometer base as shown in Figure (3.1). 6. Position the beam splitter at a 45◦ angle to the laser beam so that the beam is reflected to mirror M2 . Adjust the angle of the beam splitter as needed so that the reflected beam hits mirror M2 near its center. 7. There should now be two sets of bright dots on the observation screen; one set comes from the fixed mirror and the other comes from the movable mirror. Each set of dots should include a bright dot with two or more dots of lesser brightness (due to multiple reflections). Adjust the angle of the beam splitter again until the two sets of dots are as close together as possible, then tighten the thumbscrew to secure the beam splitter. 8. Using the knobs on the back of mirror M2 , adjust the tilt of mirror M2 until the two sets of dots on the observation screen coincide. 9. Place the 18 mm (focal length) lens in front of the laser as shown in Figure (3.1), and adjust its position until the diverging beam is centered on the beam splitter. You should now see circular fringes on the observation screen. If not, carefully adjust the tilt of mirror M2 until the fringes appear. 3.2.2 Measurements: 10. Adjust the micrometer knob to a medium reading (approximately 50 µm). In this position the relationship between the micrometer reading and the mirror M2 movement is nearly linear. 11. Turn the micrometer one full turn counterclockwise. Continue turning counterclockwise until the zero on the knob is aligned with the index mark. Record the micrometer reading. When you reverse the direction in which you turn the micrometer knob, there is a small amount of slack before the mirror begins to move. This is called mechanical backlash, and is present in all mechanical system involving reversals in direction of motion. Beginning 22 EXPERIMENT 3 with a full counterclockwise turn, and then turning only counterclockwise when counting fringes, eliminates errors due to backlash. 12. Adjust the position of the observation screen so that one of the marks on the millimeter scale is aligned with one of the fringes in the interference pattern. It is easier to count the fringes if the reference mark is one or two fringes away from the center of the pattern. 13. Rotate the micrometer knob slowly counterclockwise. Count the fringes as they pass the reference mark. Continue until some predetermined number of fringes have passed the reference mark (count at least 20 fringes). As you finish your count the fringes should be in the same position with respect to the reference mark as they were when you started to count. Read and record the final reading of the micrometer dial, the difference between this and the original reading is the distance, dm , the amount the mirror M2 moved toward the beam splitter. Each small division on the micrometer knob corresponds to one µm (10−6 m) of mirror movement. 14. Record N , the number of fringe transitions counted. 15. Repeat steps 10 through 14 several times, (a minimum of five times) recording the results each time. 16. For every set of measurement, calculate the wavelength of the laser using Equation (3.1), then average your results. 17. Compare the measured value of the wavelength λ to the known value of 632.8 nm by finding the percent difference. Experiment 4 Determination of the Elementary Charge e Using Millikan Apparatus 4.1 Introduction The charge on an electron is a fundamental quantity of physics. This charge is the smallest amount of charge in nature. However, the quark theory of matter, which has been gaining strong acceptance during the past decade or so, stipulates that quarks have charges of 13 e and − 32 e where e is the negative charge of the electron. There is experimental evidence that particles like the neutron and the proton are composed of smaller ones. But no one has ever observed a free quark. So far, then, the smallest charge which can be observed is the electronic charge e. In 1909 Robert A. Millikan reported the “oil drop experiment”. The experiment was designed to demonstrate that electric charge is “quantized” and to measure the quantum of the electric charge e. In the experiment small drops of oil were placed between two charged horizontal parallel plates which subject the drops to a combination of electrical, gravitational and viscous forces. The drops were small enough to have acquired random charges equal to small integral multiples of the quantum of charge. Analyzing the motion of the drops under the influence of an electric field allowed Millikan to determine the charge on an electron. The size and mass of oil drops vary, making Millikan’s original experiment complex. In this experiment we will use small plastic spheres of known uniform size and density to simplify the analysis. Occasionally you might observe a fragment of a sphere or a cluster of several spheres. However, these can be quickly recognized by their size and velocity compared to most observed spheres, and thus can be discarded. If the mass of a sphere under observation is m, then the gravitational force on it is given by: Fg = mg (4.1) 24 EXPERIMENT 4 If the electric field E is varied until the electric force FE on the sphere is upward and equals the downward gravitational force Fg , the sphere will remain stationary. The viscous force due to the friction between the sphere and the air is, in this case, zero because there is no net motion, so we get: Fg = FE or mg = FE (4.2) The electric force on the sphere is given by: FE = qE (4.3) thus: mg (4.4) E The electric field between two charged parallel plates is uniform in the central region, then: q= E=− dV V = dr d Using Equation (4.4) we get: mgd V where d is the distance between the plates and V is the voltage across them. q= (4.5) Since m, g and d are constants then: q∝ 1 V The mass m of the sphere may be calculated from the radius and density specified. The plate spacing d (4.0 mm) and the acceleration due to gravity g (9.8 m/s2 ) are known. Thus to calculate the charge q on a particular sphere, you need only measure V using a voltmeter. When many observations are made, the resultant calculated values of q will be found to be an integral multiple of a certain small value. This value is the fundamental unit of charge. A sphere moving through a fluid medium at a constant velocity v is subject to a viscous retarding force Fa , given by Stoke’s law: Fa = Krv (4.6) where K is a constant which depends only on the fluid’s viscosity and r is radius of the sphere. Since r is a constant in these experiments, Kr can be replaced by another constant C, and Equation (4.6) becomes: Fa = Cv (4.7) DETERMINATION OF THE ELEMENTARY CHARGE E USING MILLIKAN APPARATUS 25 In the absence of an electric field, a free-falling particle quickly reaches a constant terminal velocity due to the retarding force of the fluid. At the terminal velocity, Fg = Fa , so: mg = Cvg (4.8) where vg is the free fall terminal velocity. An electric field E acting on a sphere with charge q applies a force, FE , given by Equation (4.3). If this force is applied upwards, it will oppose the force of gravity. When the sphere reaches equilibrium velocity ve , the sum of the forces on the sphere must be zero. When the field is large enough to more than overcome gravity, FE is greater than Fg ; the forces Fa and Fg are in opposite directions as FE , so: FE − Fa − Fg = 0 or FE = Fa + Fg (4.9) Using Equations (4.3), (4.7) and (4.1) we get: qE = Cve + mg (4.10) or q= Cve + mg E (4.11) Using Equation (4.8) we get: C (ve + vg ) (4.12) E A similar equation can be derived when FE is directed downward. For a sphere of uniform size in a constant electric field, a change in the charge q results only in a change in the equilibrium velocity ve , i.e.: q= C ∆ve (4.13) E where the signs of q, E, and ve are considered. If many experimental values of ∆ve are measured, they are all found to be integral multiples of a certain small value. The same must be true for ∆q: the charge gained or lost is an exact multiple of some small charge. ∆q = 4.2 Experimental Procedure PART I: THE CLASSICAL MILLIKAN EXPERIMENT FOR DETERMINING THE CHARGE OF AN ELECTRON 1. Squeeze the rubber bulb several times to release some spheres into the viewing chamber. Most spheres will be electrically charged by the friction of the injection. Move the connections in Figure (4.1) to the up (upper plate +ve and lower plate -ve) or down (upper plate 26 EXPERIMENT 4 -ve and lower plate +ve) positions and observe the effect on the spheres. Some spheres will move up and others down, indicating that there are both negatively and positively charged spheres.The more highly charged spheres move faster and quickly disappear from the field of view. Vary the field intensity by means of the power supply voltage control and comment on the effect. Keep in mind that the microscope inverts the field, and the actual direction of motion is opposite the apparent one. Figure 4.1: Circuit diagram. With the connection in the center position, the plates are shorted and there is no electric field; the spheres fall freely under gravity. Look for occasional clumps of spheres falling more rapidly and fragments falling slower than single whole spheres. Do not use these in your measurements. 2. Turn the connection back to the up position, so that the top plate will be positively charged to attract negatively charged spheres. Set the applied voltage to about 200 V . Select a single sphere that is moving slowly in the electric field and try to make it stop by carefully adjusting the voltage. Observe it long enough to make certain it is motionless, then record the voltage across the plates. Record the stopping voltages for at least ten spheres, try to pick out spheres requiring the highest voltage to stop; these are the ones with the lowest charges. 3. Using the spheres radius, density, plate separation (4.0 × 10−3 m) and stopping voltage for each sphere, calculate and record the charge on the each sphere using Equation (4.5). 4. Graph the experimentally determined values of q versus the sphere number. The charges on the spheres will appear to cluster around certain values. These values are integral multiples of the charge of the electron which can then be determined. 5. Draw horizontal lines which intersect the y-axis at values of e, 2e, 3e, etc. In the conclusion comment on the clustering of your data in relation to these lines. DETERMINATION OF THE ELEMENTARY CHARGE E USING MILLIKAN APPARATUS 27 PART II: TERMINAL VELOCITY AND DRIVING FORCE Figure 4.2: Various forces acting on the oil drop. 6. To clearly show the relationship between velocity and driving force, you will need to make three velocity measurements on each of a minimum of ten spheres (you can keep them in your field of view by switching the electric field) under the following conditions (see Figure (4.2)): (a) The velocity under free fall, (b) The velocity when the electric and gravitational fields are in opposite directions, and (c) The velocity when the electric and gravitational fields are in the same direction. At terminal velocity the viscous force exactly balances Fg and FE ; the “driving force” in each part of Figure (4.2) equals the viscous force. The total force is zero. To avoid a mixture of both positively and negatively charged particles be sure to use only the spheres that move downward when the top plate is positive. 7. Make sure there is no electric field applied to the apparatus and inject some spheres into the viewing chamber. Choose a slowly moving sphere in free fall (remember the microscope inverts the field). Use a stop watch to measure the time for the sphere to move through two graduations of the reticule (a movement of 1 mm in the chamber) in two different parts of the field of view. The two times should essentially be the same, indicating that the sphere has reached terminal velocity. 8. Calculate the velocity values for the sphere and record your data in a table similar to Table (4.1). The designation “up” and “down” refer to actual direction of motion (opposite the apparent direction). 9. For the same sphere measure its velocity in two different parts of the field of view with an electric field applied between the plates. The two velocities should again be essentially the same. Note its actual direction of motion. The plate voltage should be kept constant for all measurements at about 200 V (it is not necessary to know the exact value in this experiment). 28 EXPERIMENT 4 Table 4.1: Data Table I. Sphere Free Fall Up Down Time (s) v (mm/s) Time (s) vup (mm/s) Time (s) vdown (mm/s) 1 2 3 4 5 6 7 8 9 10 10. Watching the same sphere reverse the electric field applied between the plates. The sphere should reverse direction. Again measure and record the sphere’s velocity in two different parts of the field of view, as well as its actual direction of motion. 11. To avoid using data from fragments or clusters, throw out data on any sphere with a free fall time markedly different from the majority. If all spheres have the same mass, and the field strength is held constant, the driving force will differ only due to the charge on the sphere q. 12. Plot a graph of the velocity versus electric force (constant values), designating downward force and velocity as negative and upward as positive. The graph of the measurements on each sphere should be a straight line. That is, the three points for each sphere should fall on a straight line. The slopes will be different (although the values for the free fall velocity should be similar) because the charges on each sphere are different, and hence different values of FE . The graph should show that the velocity is proportional to the driving force. PART III: QUANTIZATION OF CHARGE 13. As noted before, sphere clumps and fragments must not be used. Since vdown is proportional to FE + Fg and vup is proportional to FE − Fg , then the vector sum, ~vup + ~vdown for a given sphere, is proportional to 2Fg and the difference, ~vup − ~vdown , is proportional to 2FE . We can then use the values of ~vup + ~vdown to screen out all but single spheres, and the values of ~vup − ~vdown for the selected spheres to determine charge quanta. 14. Using the data gathered in Part II, calculate ~vup + ~vdown and ~vup − ~vdown and fill in Data Table II. Remember that the direction of v~up and vdown ~ are opposite when adding them. 15. Examine the v~up + vdown ~ column and discard spheres for which the value is markedly different from the average. DETERMINATION OF THE ELEMENTARY CHARGE E USING MILLIKAN APPARATUS 29 Table 4.2: Data Table II. Sphere Up Down Time (s) vup (mm/s) Time (s) vdown (mm/s) ~vup + ~vdown ~vup − ~vdown 1 2 3 4 5 6 7 8 9 10 16. Make a bar graph of ~vup − ~vdown for the selected spheres. The data should indicate that these values are integral multiples of some small value and, therefore, that the charge on the spheres are also integral multiples of some small value. See Part I for calculation of the results. 30 EXPERIMENT 4 Experiment 5 Photoelectric Effect 5.1 Introduction The photoelectric effect was discovered by Heinrich Hertz in 1887 during the course of experiments; the primary intent of which was confirmation of Maxwell’s theoretical prediction (1864) of the existence of electromagnetic waves produced by oscillating electric currents. The photoelectric effect is but one of several processes by which electrons may be removed from the surface of a substance (we shall refer here to metal surfaces in particular; since the effects were first observed in metals and still are most easily observed in them). These effects are: 1. Thermionic emission: Heating of a metal, which gives thermal energy to the electrons and effectively boil them from the surface. 2. Secondary Emission: Transfer of kinetic energy from particle that strike the surface to the electrons in the metal. 3. Field Emission: Extraction of electrons from the metal by a strong external electric field. 4. Photoelectric Emission: Described below. The photoelectric effect occurs when electromagnetic radiation shines on a clean metal surface and electrons are released from the surface. The valence electrons in a metal are free to move about through its interior but are bound to the metal as a whole. It is such relatively free electrons with which we shall be concerned here. We can most simply describe the photoelectric effect as a light beam supplying electrons with an amount of energy that equals or exceeds the energy which binds the electrons to the surface and allowing those electrons to escape. A more detailed description of the photoelectric effect requires a knowledge, based on experiment, of how the several variables involved in photoelectric emission are related to one another. These variables are the frequency ν of the light, the intensity I of the light beam, the photoelectric current i, the electron’s kinetic energy 21 m◦ v 2 , and the chemical identity of the surface from which the electrons emerge. The emergent electrons are called photoelectrons. The minimum 32 EXPERIMENT 5 extra kinetic energy that allows electrons to escape the material is called the work function φ. The work function is the minimum binding energy of the electron to the material. The work functions of alkali metals are smaller than those of other metals. Figure 5.1: Circuit diagram for a photo cell. Experiments carried out using an apparatus similar to the one in Figure (5.1), showed that incident light falling on the “photocathode” ejects electrons. Some of the electrons will travel toward the “anode” where a negative or a positive V is applied. If the voltage of the anode is positive; the electrons will accelerated toward the anode. On the other hand, if the anode voltage is negative; the electrons will be retarded. The stream of electrons traveling from the photocathode to the anode form an electric current I in the circuit which is measured by the ammeter. This current is called “photocurrent”. Those experiments established the following facts about the photoelectric effect: 1. One can measure the kinetic energy of the photoelectrons by applying negative voltage to the anode. The negative voltage that prevents the electrons from reaching the anode is a measure of the kinetic energy of the photoelectrons and is called the “stopping potential” or “stopping voltage”. The kinetic energies or the stopping potentials of the photoelectrons are independent of the intensity of the incident light. However, the number of electrons emitted depends on the intensity as shown in Figure (5.2). 2. The maximum kinetic energy of the emitted photoelectrons, from a given material, depends only on the frequency of the incident light. In other words, the stopping potential depends on the frequency ν and not on the intensity as shown in Figure (5.3). 3. The smaller the work function φ of the photocathode material the smaller the minimum frequency required to eject photoelectrons, no matter how high the intensity. Figure (5.4), shows the minimum frequency ν◦ for different materials. PHOTOELECTRIC EFFECT 33 Figure 5.2: Voltage vs current produced by light beams with the same frequency and different intensities. 4. For a constant voltage applied to the anode, and a constant light frequency the photoelectron current is directly proportional to the intensity of the incident light. 5. The photoelectrons are emitted almost instantly after illuminating the photocathode independent of the intensity of the incident light. It is possible to use a classical theory to explain the emission of electrons from metals using electromagnetic radiation. However, such a theory could not predict all the experimental facts mentioned above except # 4. In 1905, Einstein was able to use the concept of light quanta introduced by Max Planck in 1900 to fully explain the photoelectric effect and was able to predict all the experimental facts of the phenomena. For a detailed explanation of Einstein’s quantum theory of the photoelectric effect, see any book on “Modern Physics”. We give here the important equations. From Einstein’s quantum theory: E = hν (5.1) λν = c (5.2) where E is the energy of the incident photon, ν is the photon’s frequency, λ is the photon’s wavelength and h is Planck’s constant. 34 EXPERIMENT 5 hν = φ + K (5.3) where K is the kinetic energy of the photoelectron after emission and φ is the work function or the energy required to take the electron away from the metal. Since K = 12 mv 2 then: 1 (5.4) hν = φ + mv 2 2 If V◦ is the potential required to stop all photoelectrons from reaching the anode then we get: 1 eV◦ = mv 2 2 (5.5) 1 2 mv = eV◦ = hν − φ 2 (5.6) Substitute this into (3.4) to get: 5.2 Experimental Procedure The apparatus used in this experiment is a “photo cell” composed of a glass container under vacuum and has a photocathode and anode. The photo cell is housed in a box which also contains a variable voltage supply and an ammeter to measure the photocurrent. The photocurrent produced by the cell itself is extremely small (of the order of several nanoamperes). One needs to amplify such currents to be able to detect them with reasonable accuracy. The current from the photo cell is then introduced to an amplifier before detection. In order to minimize the effects of any stray voltages on the measurements, the amplifier is built inside the box and only few centimeters away from the cell’s base. The apparatus is built by “Daedalon Corporation” and is called “EP-05”. The apparatus includes three filters (Red, Green, and Blue) to provide spectral separation from wide spectrum light sources. In this experiment you will use a mercury lamp as a light source. 1. Set up the EP-05 on a table such that the aperture in front of the photo cell faces the light source. 2. Connect a voltmeter to the red and black banana jacks on the top of the EP-05 box. They are connected across the photo cell and measure the voltage across the tube. 3. Cover the aperture with a black piece of paper, card board, or metal to set the zero adjustment on the amplifier. Do not use your hand; it is not opaque enough. Turn on the amplifier and adjust the “Zero Adjust” knob until the meter reads zero. It is a good practice to check the zero adjustment frequently during the measurements to correct for any drifts. 4. Turn the “Voltage Adjust” knob to its counter clockwise limit. The voltmeter should read zero or very close to it. PHOTOELECTRIC EFFECT 35 Figure 5.3: Voltage vs current produced by light beams with the same frequency and different intensities. 5. Turn on the mercury lamp. Place the filter which transmits the shortest wavelength over the aperture. 6. Move the EP-05 until the radiation is striking the center of the photo cell. The reading on the output meter is helpful in making the adjustment.The radiation intensity should be adjusted so that the meter is approximately 10 on the scale. The intensity of the radiation at the cell can be changed by changing the distance between the EP-05 box and the light source. The amplifier gain is quite high (1 mA on the meter is 5 × 10−9 A of photocurrent), so if the meter goes off scale, the photo cell would not be harmed. The meter would not be harmed either; the amplifier limits the current delivered to it. 7. Note that the output current is a function of the negative voltage applied to the anode. As the voltage increases, fewer and fewer electrons have enough energy to reach the anode and the current drops. The critical point is the voltage at which the current falls to zero. 8. Measure the voltage for zero current five times, resetting the zero after each measurements. Be careful not to pass the zero current value. The current remains at zero for stopping voltage higher than the critical value. The value you need is when the current just reaches zero. Then calculate the average stopping voltage. 9. Place the filter which transmits the next wavelength over the aperture. Repeat steps 7 and 8 You will find that the stopping voltage of this light is greater than that for the previous wavelength. 36 EXPERIMENT 5 Figure 5.4: Work functions of various photo cathodes. 10. Repeat step 9 until you have used all the filters. 11. Use your data and Equation (5.6) to graphically find Planck’s constant h and the work function φ of the material of the photocathode used in the photo cell. Experiment 6 Radioactivity 6.1 Introduction Figure 6.1: Circuit Diagram for a Geiger Mueller Counter. In this experiment you will study some of the characteristics of radioactive decay. You will also learn how to measure and count the radiation produced by a radioactive source. We will use a radioactive source which emits radiation such as gamma rays and/or charged particles like β particles (electrons) or α particles. The radiation is detected and counted by a device called Geiger-Mueller counter. It is not possible, however, to measure the energy of these particles using this simple device. The Geiger-Mueller counter used in this experiment operates under computer control. The computer will allow you to set up the counter and control the counting procedure. The computer accumulates the data and displays the results in a graph form. 6.1.1 The Geiger-Mueller Counter The Geiger-Mueller counter, normally known as G-M tube, is an insulated; gas filled tube which has a conducting hollow cylinder (the cathode) concentric with a conducting wire (the anode) as shown in Figure (6.1). The cylinder is kept at a lower potential than the central wire. When the particles of the radiation enter the counter; they interact with the atoms of the gas and 38 EXPERIMENT 6 produce some initial ionization. The ionization process produces ion pairs each consisting of a positive ion and an electron. If there is no voltage across the counter, then, the electrons and positive ions will recombine to form neutral atoms and nothing happens further. If, however, the potential difference between the electrodes of the counter increases, the electrons will be swept away toward the central wire and the positive ions toward the negative cylinder. As a result of the motion of the ion pairs some charge will develop on the capacitor in the circuit. This charge will change the potential difference across the resistor. This change is detected and amplified by the amplifier and then counted by a counter (not shown in Figure (6.1)). Figure 6.2: Voltage vs count rate of a G-M tube. Figure (6.2) shows the relationship between the G-M tube voltage and the number of counts per unit time. When the voltage is below certain value (tube threshold) the voltage is not enough to sweep the ion pairs. Above threshold the voltage is high enough such that the ion pairs will accelerate and produce secondary ion pairs which in turn get swept by the voltage to the electrodes. Within a certain range of voltage above threshold, every particle enters the tube will cause enough ionization in the tube and get detected, irrespective of its energy. The count rate in this region is then independent of the applied voltage. This region is called the plateau. Raising the applied voltage above this region will cause a very large amount of secondary ionization and the gas in the tube simply breaks down. The voltage region below the plateau can be different from the one shown in Figure (6.2). The tube behavior in this region varies with the design and the intended use of the tube. RADIOACTIVITY 39 The G-M tube should then operate at a voltage somewhere in the plateau region. The first part of this experiment is to reproduce the curve in Figure (6.2) and determine the operating voltage of the G-M tube. 6.1.2 Random Nature of Radioactive Decay The laws governing the radioactive decay process were established experimentally by Rutherford and Soddy and states that the activity of a radioactive sample decays exponentially in time, and in a random fashion. Let us assume that λ is the probability that a nucleus decays per unit time. In a sample of N such nuclei, the mean number of nuclei dN decaying in a time interval dt is : dN = −λN dt (6.1) λ is called the decay constant and depends only on the particular decay process of a particular nucleus. If N is very large, which is normally the case even in a small amount of radioactive material, then the change in N can be considered as continuous. Integrating Equation (6.1) we get: N (t) = N (0)e−λt (6.2) Where N (0) is the number of nuclei at t = 0. The decrease in radioactivity is then controlled by the decay constant λ. It is customary to use the inverse of λ, τm = 1 λ (6.3) which is called the mean life time. The time interval τm is the time required for the sample to decay to 1/e of its initial activity. One can then define the half lif e, T1/2 , which is the time it takes the sample to decay to one-half its original activity, thus T1/2 1 = e−λT1/2 = e− τm 2 (6.4) T1/2 = τm ln(2) (6.5) this gives Consider now the number of decays, n, undergone by a radioactive source in a time interval ∆t. If ∆t is short compared to the half-life of the source, then, the total activity of the source could be considered as constant. If we measure n several times we would observe fluctuations in its value from measurement to measurement. This is because n is related to the probability 40 EXPERIMENT 6 of decay. It can be shown that the probability of observing n decays in a time interval ∆t is given by Poisson distribution: P (n, ∆t) = µn −µ e n! (6.6) where µ is the average number of counts in the interval ∆t. The standard deviation of the Poisson distribution is: σ= √ µ (6.7) The standard deviation σ is, in general, a measure of the spread of the data around its mean value. However, in case of Poisson distribution it is also a measure of the uncertainty associated with our experimental attempts to determine “true” values. 6.2 6.2.1 Experimental Procedure Determination of the Operating Voltage of the G-M Tube: 1. Turn off computer power. 2. Connect the RS-232C cable from the GMX box to the serial port of the computer. 3. Connect the G-M tube to the GMX box with the coaxial cable that is terminated with MHV connector. 4. Turn on the power to the GMX box, then turn the computer on. 5. After the computer boots and you get the DOS prompt, type “cd windows” this will make the windows directory the current directory. 6. Type “win” to start the windows program. 7. The program which controls the GMX box is called “gmx.exe”. The full path to this program is “c:/windows/gmx/gmx.exe”. This means that the program resides in a folder called “gmx” which is in a folder called “windows” which is on the hard drive called the “c” drive. 8. To reach program “gmx.exe” you locate the icon “Main”. Use the mouse to drag the pointer to the “Main” icon and click the mouse button to open the “Main” window. 9. Locate the icon of the “File Manager” and click on it to open its window. RADIOACTIVITY 41 Table 6.1: Summary of the gmx program commands and the keyboard keys used to invoke them. Key/Keys F1 F3 F4 F5 F6 F7 F8 Up/Dn Arrow Plus/Minus (Numeric Keypad) Ctrl F2 Function Acquire toggle Set preset (non-acquire mode only) Draw Grid toggle Select Vertical Scale (Linear or Log) Load a Binary File (non-acquire mode only) Save a Binary File (non-acquire mode only) Select any Linear Vertical Scale Multiplier at any time except when in acquire mode Vertical Scale Multiplier/Divider in steps of 2 Increment/Decrement volts in steps of 25 (non-acquire mode only). Has different meaning in other experiments Erase Spectrum 10. The “File Manager” window is split into two parts. Click on the image of the yellow folder on the left half called “windows”. The contents of the “windows” folder will then be displayed on the right half of the “File Manager” window. 11. Click on the folder named gmx. An image for folder gmx will appear below the image of the “windows” folder on the right side. The contents of the gmx folder will be displayed on the right side, where you will find “gmx.exe”. 12. Click on “gmx.exe” to start the program. An introductory screen appears. Press any key to continue. 13. The Geiger Plateau Experiment screen will be loaded. 14. The program requires certain information to set the experiment. The keyboard is the only way to interact and control the experiment. Table (6.1) lists command associated with certain keys. 15. Press “Alt E” (the “alt” key and the “e” key simultaneously) and then the arrow keys to select the experiment. 16. Type a vertical scale multiplier (e.g. 200) in response to “Set Vertical scale Multiplier (10 to 500000) :” With this setting the scale can show a maximum count of (200 × 10) = 2000 without going off scale. 17. Type a preset time (in seconds) (e.g. 10) in response to “Enter preset time (10 to 600) :”. The preset time is the time the program use to accumulate data before stopping to take 42 EXPERIMENT 6 another measurement. Since the uncertainty in counting experiments is the square root of the count, one then should select a time long enough to accumulate certain number of counts which has an error of about 5% or less. 18. Type an initial tube voltage of 250V in response to “Select initial voltage (10 to 1200) :”. 19. Place Co60 source under the G-M tube and press F 1 to start acquire. Data will be acquired at 25 volt interval until the maximum voltage of 1200 is reached or F 1 key is pressed. 20. During acquire, counts are plotted inside the data window. If counts are out of range for the given scale, dots are plotted near the upper boundary of the data window. You can always press Up/Down arrow keys to adjust the vertical scale. You can also press F 4 to draw/erase grids on the screen. 21. You can save your data on a disk as a binary file or as an ASCII file, for later reference or for printing on another computer with a printer. Press “Alt F” and select “Save a Binary File” or “ASCII File Save” and enter the file name. There is no need for an extension to the file name. The extension is added automatically to the file name depending on the type of the file. If you save a file under the name “geiger” then its actual name will be “geiger.spm” for binary files and “geiger.asc” for ASCII files. 22. From the plot of counts vs tube voltage choose an operating voltage which close to lower end of the plateau region. Record this value in your lab report and use it in all subsequent experiments. 6.2.2 Statistics of Counting Experiments (Random Nature of Radioactive Decay) 1. Clear the data of the previous experiment using Ctrl F2. 2. Press “Alt E” and then the arrow keys to select the experiment. 3. Type a preset time (in seconds) (e.g. 5) in response to “Enter preset time (10 to 600) :”. See step 17 in the previous experiment. 4. Type an initial tube voltage in response to “Select initial voltage (10 to 1200) :”. Use the operating voltage determined in the plateau experiment. 5. Enter total number of readings; at least 50 readings should be taken in this experiment. 6. This experiment does not use any vertical or horizontal scales. So, Up/down keys will have no effect. 7. Place the Co60 in the lowest ’shelf’ under the G-M tube and press F 1 to start. RADIOACTIVITY 43 8. Individual counts are displayed after each measurement. At the end of the acquire process you will have a list of the all the measurements as well as (avg - n), (avg - n)2 and the standard deviation σ. 9. Divide your data into some 10 or 15 ranges. Plot the frequency of the data (the number of data point within a range) vs middle value of each range. 10. Use the average of all the data and the middle value of each range to calculate the Poisson distribution using Equation (6.6) and the standard deviation using Equation (6.7). Plot the calculated Poisson distribution to denote the standard deviation on the same graph as the data. Does the data fit the Poisson distribution? 44 EXPERIMENT 6 Experiment 7 Radioactivity II 7.1 Introduction In this part of the radioactivity experiment we will investigate the effect of distance upon the intensity of radiation. We will also investigate the absorption of radiation in matter. 7.1.1 Inverse Square Law Radiation is emitted from the source in all directions. However, only a small fraction of the emitted radiation actually enters the detector, the G-M tube. This fraction is a function of the distance d from the source (assumed to be point source) to the G-M tube, see Figure (7.1). If the source is moved further away from the tube, the amount of radiation entering the tube should decrease with the square of the distance if the inverse square law applies. As an α or a β source is moved away from the tube, the activity will decrease more rapidly than predicted by the inverse square law as these particles are easily absorbed in air. 7.1.2 Absorption of Radiation The absorption of radiation in matter is governed by an exponential decay law. To demonstrate this, assume that radiation of intensity I0 is incident on a slab of thickness x. The change in intensity as the radiation passes through the slab is proportional to the thickness and to the incident intensity, see Figure (7.2). ∆I = −µI0 x (7.1) where the constant of proportionally µ is called the “absorption coefficient”. If the energy of the incident radiation is constant, then µ is independent of x and the integration of Equation (7.1) yields: 46 EXPERIMENT 7 Figure 7.1: Geometry of a point source and a G-M tube. I = I◦ e−µx (7.2) where I and I◦ are the intensity of the radiation after passing through a thickness x of the material, and the initial intensity of the radiation, respectively. 7.2 7.2.1 Experimental Procedure Inverse Square Law: 1. Determine the operating voltage of the G-M tube as outlined in Experiment 6. 2. Press “Alt E” and then the arrow keys to select the experiment. 3. Type a vertical scale multiplier (e.g. 200) in response to “Set Vertical scale Multiplier (10 to 500000) :” With this setting the scale can show a maximum count of (200 × 10) = 2000 without going off scale. 4. Type a preset time (in seconds) (e.g. 10) in response to “Enter preset time (10 to 600) :”. The preset time is the time the program use to accumulate data before stopping to take another measurement. Since the uncertainty in counting experiments is the square root of the count, one then should select a time long enough to accumulate certain number of counts which has an error of about 5% or less. RADIOACTIVITY II 47 Figure 7.2: Absorption of radiation in a slab. 5. Select a tube voltage that is on the plateau of the G-M tube. 6. Initial distance selected by default is “distance = (distance × 1)”. This is displayed near the right end corner of the bottom of the screen. You can select different distances by pressing “PLUS/MINUS” keys on the numeric keypad. 7. Press F 1 to start. Total count is plotted at the end of each preset interval. Counting stops at the end of each preset interval to allow time to place the source at a different distance. To continue counting with the same distance, press F 1 again. Counts are plotted in the same location. You may have a minimum of 10 readings for each distance. Move the source to the next position and press F 1 again to start acquiring data. 8. Repeat step 7 until all desired positions have been counted and save your data. 9. Plot a graph of the number of counts vs. 1 . d2 10. In the conclusion explain whether the inverse square law applies. 7.2.2 Linear Absorption coefficient: 11. Select the experiment. 12. Enter the necessary parameters to set up the experiment. 48 EXPERIMENT 7 13. Place the absorber in between the G-M tube and the source. 14. Press F 1 to start. Total count is plotted at the end of each preset interval. Counting stops at the end of each preset interval to allow time to change the absorber. To continue counting with the same absorber, press F 1 again. Counts are plotted in the same location. You may have a minimum of 10 readings for each absorber. Change the absorber and press F 1 again to start acquiring data. 15. Repeat step 14. until all desired absorbers have been tested and save your data. 16. Tabulate the absorber thicknesses in cm by dividing the given mg/cm2 for each slab by the known density of the metal (ρAl =2700 mg/cm3 and ρP b =11360 mg/cm3 ). 17. Graph lnI vs. x (convert x values to m) for the results of each of the materials. 18. Find the absorption coefficients for the different materials from their respective graphs. Appendix A Error Analysis A.1 Measurements and Experimental Errors The word “error” normally means “mistake”. In physical measurements, however, error means “uncertainty”. We measure quantities like temperature, electric current, distance, time, speed, etc ... using measuring devices like thermometers, ammeters, tape measures, watches etc. When a measurement of a certain quantity is performed, it is important to know how accurate or precise this measurement is. That is, a quantitative evaluation of how close we think this measurement is to the “true” value of the quantity must be established. An experimenter must learn how to assess the uncertainties or errors associated with a physical measurement so that he/she can convey a true picture of just how accurately a certain quantity has been measured. In general there are three possible sources or types of experimental errors: systematic, instrumental, and statistical. The following are guidelines which are commonly followed in defining and estimating these different forms of experimental errors. A.1.1 Systematic Errors: These errors occur repeatedly (systematically) every time a measurement is made and in general would be present to the same extent in each measurement. They arise because of miscalibration of a measuring device, ignoring some physical assumptions that should be taken into account when using an instrument, or simply misreading a measuring device. An example of a systematic error due to miscalibration is the recording of time at which an event occurs with a watch that is running late; every event will then be recorded late. Another example is reading an electric current with a meter that is not zeroed properly. If the needle of the meter points below zero when there is no current, then the reading of any current will be systematically less than the true current. As an example of a systematic error due to ignoring some physical assumptions in a measurement, consider an experiment to measure the speed of sound in air by measuring the time it takes sound (from a gun shot, for example) to travel a known distance. If there is a wind 50 APPENDIX A blowing in the same direction as the direction of travel of the sound then the measured speed of sound would be larger than the speed of sound in still air, no matter how many times we repeated the measurement. Clearly this systematic error is not due to any miscalibration or misreading of the measuring devices, but due to ignoring some physical factors that could influence the measurement. An example of a systematic error due to misreading an instrument is parallax error. Parallax error is an optical error that occurs when a laboratory meter or similar instrument is read from one side rather than straight on. The needle of the meter would then appear to point to a reading on the side opposite to the side the meter is being looked at. There is no general rule for the estimation of systematic errors. Each experimental situation has to be investigated individually for the possible sources and magnitudes of systematic errors. Care should be taken to reduce as much as possible the occurrence of systematic errors in a measurement. This could be done by making sure that all instruments are calibrated and read properly. The experimenter should be aware of all physical factors that could influence his/her measurement and attempt to compensate for or at least estimate the magnitude of such influences. A widely used technique for the elimination of systematic errors is to repeat the measurement in such a way that the systematic error adds to the measured quantity in one case and subtracts from it in another. In the example about measuring the speed of sound given above, if the measurement was repeated with the position of the source and detector of the sound interchanged, the wind speed would diminish the sound speed in this case, instead of adding to it. The effect of the wind speed will thus cancel when the average of the two measurements is taken. A.1.2 Instrumental Uncertainties: There is a limit to the precision with which a physical quantity can be measured with a given instrument. The precision of the instrument (which we call here the instrumental uncertainty) will depend on the physical principles on which the instrument works and how well the instrument is designed and built. Unless there is reason to believe otherwise, the precision of an instrument is taken to be the smallest readable scale division of the instrument. In this course we will report the instrumental uncertainty as one half of the smallest readable scale division. Thus if the smallest division on a meter stick is 1 mm, we report the length of an object whose edge falls between the 250 mm and 251 mm marks as 250.5±0.5 mm. If the edge of the object falls much closer to the 250 mm mark than to the 251 mm mark we may report its length as 250.0±0.5 mm. Even though we might be able to estimate the reading of a meter to, let’s say, 1 of the smallest division we will still use one half of the smallest division as our instrumental 10 uncertainty. Our assumption will be that had the manufacturer of the instrument thought that 1 his/her device had a precision of 10 of the smallest division, then he/she would have told us so explicitly or else would have subdivided the reading into more divisions. For meters that give out a digital reading, we will take the instrumental uncertainty again to be one half of the least ERROR ANALYSIS 51 significant digit that the instrument reads. Thus if a digital meter reads 2.54 mA, we will take the instrumental error to be ±0.005 mA, unless the specification sheet of the instrument says otherwise. A.1.3 Statistical Errors: Several measurements of the same physical quantity may give rise to a number of different results. This random or statistical difference between results arises from the many small and unpredictable disturbances that can influence a measurement. For example, suppose we perform an experiment to measure the time it takes a coin to fall from a height of two meters to the floor. If the instrumental uncertainty in the stop watch we use to measure the time is 1 second, repetitive measurement of the fall time will reveal different results. very small, say 100 There are many unpredictable factors that influence the time measurement. For example, the reaction time of the person starting and stopping the watch might be slightly different on different trials; depending on how the coin is released, it might undergo a different number of flips as it falls each different time; the movement of air in the room could be different for the different tries, leading to a different air resistance. There could be many more such small influences on the measured fall time. These influences are random and hard to predict; sometimes they increase the measured fall time and sometimes they decrease it. Clearly the best estimate of the “true” fall time is obtained by taking a large number of measurements of the time, say t1 , t2 , t3 , ...tN , and then calculating the average or arithmetic mean of these values. The arithmetic mean, t is defined as: t= N (t1 + t2 + t3 + ... + tN ) 1X ti = N N i=0 (A.1) where N is the total number of measurements. To indicate the uncertainty or the spread in these measured values about the mean, a quantity called the standard deviation is generally computed. The standard deviation of a set of measurements is defined as: s σsd = 1 [(t1 − t)2 + (t2 − t)2 + (t3 − t)2 + ... + (tN − t)2 ] N or σsd = v u u t N 1 X ∆t2 N i=1 i (A.2) where ∆ti = (ti − t)) is the deviation and N is the number of measurements. The standard deviation is a measure of the spread in the measurements ti . It is a good estimate of the error or uncertainty in each measurement ti . The mean, however, is expected to have a smaller 52 APPENDIX A uncertainty than each individual measurement. We will show in the next section that the uncertainty in the mean is given by: σsd δt = √ N (A.3) According to the above equation, if four measurements are made, the uncertainty in the mean will be two times smaller than that of each individual measurement; if sixteen measurements are made the uncertainty in the mean will be four times smaller than that of an individual measurement and so on; the large the number of measurements, the smaller the uncertainty in the mean. Most scientific calculators nowadays have functions that calculate averages and standard deviations. Please take the time to learn how to use these statistical functions on your calculator. Significance of the standard deviation: If a large number of measurements of an observable are made, and the average and standard deviation calculated, then 68% of the measurements will fall between the average minus the standard deviation and the average plus the standard deviation. For example, suppose the time it takes a coin to fall from a height of two meters was measured one hundred times and the average found to be 0.6389 s and the standard deviation 0.10 s. Then approximately 68% of those measurements will be between 0.5389 s (0.6389-0.10) and 0.7389 s (0.6389+0.10), and of the remaining 32 measurements approximately 16 will be larger than 0.7389 s and 16 will be less than 0.5389 s. In this example ). the statistical uncertainty in the mean will be 0.010 s ( √0.10 100 A comment about significant figures: The least significant figure used in reporting a result should be at the same decimal place as the uncertainty. Also, it is sufficient in most cases to report the uncertainty itself to one significant figure. Hence in the example given above, the mean should be reported as 0.64±0.01 s. A.2 Combining Statistical and Instrumental Errors In the following discussion we will assume that care has been taken to eliminate all gross systematic errors and that the remaining systematic errors are much smaller than the instrumental or statistical ones. Let IE stand for the instrumental error in a measurement and SE for the statistical error, then the total error, TE, is given by: TE = q (IE)2 + (SE)2 (A.4) In a good number of cases, either the IE or the SE will be dominant and hence the total error will be approximately equal to the dominant error. If either error is less than one half of the other then it can be ignored. For example, if, in a length measurement, SE = 1 mm and IE = 0.5 mm, then we can ignore the IE and say TE = SE = 1 mm. If we had done the exact calculation (i.e. calculated TE from the above expression) we would have obtained TE = 1.1 mm. ERROR ANALYSIS 53 Since it is sufficient to report the error to one significant figure, we see that the approximation we made in ignoring the IE is good enough. Equation (3) suggests that the uncertainty in the mean gets smaller as the number of measurements is made larger. Can we make the uncertainty in the mean as small as we want by simply repeating the measurement the requisite number of times? The answer is no; repeating a measurement reduces the statistical error, not the total error. Once the statistical error is less than the instrumental error, repeating the measurement does not buy us anything, since at that stage the total error becomes dominated by the IE. In fact, you will encounter many situations in the lab where it will not be necessary to repeat the measurement at all because the IE error is the dominant one. These cases will usually be obvious. If you are not sure then repeat the measurement three or four times. If you get the same result every time then the IE is the dominant error and there is no need to repeat the measurement further. A.3 Propagation of Errors Very often in experimental physics a quantity gets “measured” indirectly by computing its value from other directly measured quantities. For example, the density of a given material might be determined by measuring the mass m and volume V of a specimen of the material . If the uncertainty in m is δm and the uncertainty and then calculating the density as ρ = m V is V is δV , what is the uncertainty in the density δρ? We will not give a proof here, but the answer is given by: δρ = s 2 δm ρ m + 2 δV V More generally, if N independent quantities x1 , x2 , x3 , ... are measured and their errors δx1 , δx2 , δx3 , ... determined, and if we wish to compute a function of these quantities y = f (x1 , x2 , x3 , ..., xN ) then the error on y is: s σy = v u N uX ∂y ∂y ∂y ∂y [ δx1 ]2 + [ δx2 ]2 + [ δx3 ]2 + ... = t [ δxi ]2 ∂x1 ∂x2 ∂x3 ∂x i i=1 (A.5) ∂y Here ∂x is the partial derivative of f with respect to xi . Note that the x0i s do not have to be i measurements of the same observable; x1 could be a length, x2 a time, x3 a mass and so forth. It makes the notation easier to call them x0i s rather than L,t, m and so forth. Applying the above formula to the most common functions encountered in the lab, the following equations can be used in error analysis. 54 A.3.1 APPENDIX A Linear functions: y = Ax (A.6) where A is a constant (i.e. has no error associated with it), then δy = Aδx A.3.2 (A.7) Addition and subtraction: z =x±y (A.8) then δz = q δx2 + δy 2 (A.9) Clearly if z = Ax ± By (A.10) q (A.11) where A and B are constants, then δz = A.3.3 (Aδx)2 + (Bδy)2 Multiplication and division: z = xy or z= x y (A.12) then we can show that: δz = z v u 2 u δx t x 2 δy + y (A.13) Note: This method should not be used in cases including exponents. A.3.4 Exponents: z = xn (A.14) where n is a constant, then we can similarly show that: nδx δz = z x (A.15) ERROR ANALYSIS 55 For the more general case where z = Axn y m (A.16) where A, n, m are constants, the error would be given by: v u 2 2 δz u nδx mδy + = t z x y (A.17) Examples on the application of the above formula follow: r 2 δx z = Axy δz z = z = Ax2 δz z = 2 δx x √ z=A x δz z = 12 δx x δz z = z= A x A.3.5 x 2 + δy y δx x Other frequently used functions: y = Asin(Bx) δy = ABcos(Bx)δx y = ex δy = ex δx y = ln x δy = x1 δx Often, the uncertainty in a quantity y is not reported as the absolute uncertainty δy but rather as a percentage uncertainty, 100%× δy . For example, if the length of an object is measured to be y 0.3 25.6 mm, with an uncertainty of 0.3 mm, then the percentage uncertainty is 1% (100% × 25.6 ). A.4 Percent Difference: Calculating percent difference when comparing results is often useful. The appropriate formula is: P ercent Dif f erence = |T heoretical V alue − Experimental Result| · 100% |T heoretical V alue| (A.18)