PTL e/m and electron mass
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Physics Teaching Lab – Mass of the Electron (DRAFT)
A. Pedagogy and this exercise 1. The e/m of the Electron Apparatus is designed to enable the student to determine the ratio of the charge of an electron to the mass of an electron. For the purpose of the following high school student exercise, this will be simplified to finding the mass of an electron. This allows for easy motivation and connection to the principles of the mass spectrometer and particle accelerators. It provides the student with a straightforward, recognizable goal dealing with a single value – what is the mass of an electron? While advanced physics students possess the maturity and sophistication to handle the combination of the classic charge to mass ratio determination, first year Regents level students can more readily relate to the simpler goal of measuring a mass value. All of the same principles are involved in both determinations, but the first year student may find it easier to focus on a single quantity – the mass. 2. Relatively inexpensive apparatus for measuring the mass of an electron (or the e/m ratio), such as the National Electronics 6E5 tube, has been available for nearly 50 years. However, physics teachers have moved away from this experiment because of the need for two separate circuits (one to power the heating element and one to vary the current in the solenoid used to produce the magnetic field) and the mediocre results. The best that can be hoped for with this equipment is an electron mass value within one order of magnitude. The e/m of the Electron Apparatus used in the PTL exercise is much easier to use as both the accelerating voltage and the magnetic field current can be adjusted and measured by internal circuitry. The results are satisfying with typical values within 10% of the accepted value. Unfortunately, the cost of the apparatus makes it prohibitive for most schools to purchase class sets for student use. The Physics Teaching Lab provides a venue for high school physics classes to have the opportunity to perform this experiment in groups of 2-3 students and obtain satisfying results. 3. This exercise addresses the following New York State Learning Standards for Mathematics, Science, and Technology: a. Standard 1 – Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions. b. Standard 2 – Students will access, generate, process, and transfer information, using appropriate technologies. c. Standard 4 – Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science. d. Standard 6 – Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning. 4. This exercise addresses the following Performance Indicators from the Physical Setting/Physics Core Curriculum: a. 4.1j - Energy may be stored in electric or magnetic fields. This energy may be transferred through conductors or space and may be converted to other forms of energy. b. 4.1k - Moving electric charges produce magnetic fields. The relative motion between a conductor and a magnetic field may produce a potential difference in the conductor. c. 4.1n - A circuit is a closed path in which a current can exist. d. 5.1k - According to Newton’s Second Law, an unbalanced force causes a mass to accelerate. e. 5.1n - Centripetal force is the net force, which produces centripetal acceleration. In uniform circular motion, the centripetal force is perpendicular to the tangential velocity.
B. Order of activities 1. Group session a. Motivation and demos b. Discussion of concepts and formula derivation 2. Individual lab team (2-3 students) work 3. Group session - tie up discussion 4. Optional – accelerator fundamentals and tour of tandem Van de Graff facility
C. Motivation – What is the mass of a carbon atom? 1. Look at the Periodic Table (http://www.chemicalelements.com/show/mass.html). a. According to this source, the atomic mass of carbon is 12.0107 universal mass units (u). However, which carbon atom does this represent? b. Clicking on the “C” box in the Periodic Table link given above shows Basic Information about carbon including an atomic mass of 12.0107 u and five different isotopes, or forms of carbon with different numbers of neutrons. c. The atomic mass given on this chart and other similar tables is the average mass of atoms of the element calculated using the relative abundance of naturally occurring isotopes of the element. That is, the chart value is a weighted average of all the naturally occurring varieties of atoms of that specific element. This value is sometimes referred to as the “atomic weight,” while the term “atomic mass” may be reserved for the mass of a specific individual isotope. 2. Look at a Table of Isotopic Masses (http://www.chem.ualberta.ca/~massspec/atomic_mass_abund.pdf) a. Scroll down to carbon. Three isotopes are shown: Carbon-12 with mass 12.000000 u Carbon-13 with mass 13.003355 u Carbon-14 with mass 14.003242 u b. How are these masses measured? Do scientists place a single atom on an atomic sized balance and read the mass value off directly? No, a mass spectrometer is used. 3. Mass spectrometer (http://hyperphysics.phy-astr.gsu.edu/Hbase/magnetic/maspec.html) a. Atoms are first ionized and then accelerated by an electric field. The speed of the particle can be determined from the charge of the ion and the voltage associated with the electric field (qV = mv2/2). b. A velocity selector, using both an electric field and a magnetic field, sorts out particles with the desired speed and allows them to pass into a region that is influenced only by a magnetic field. c. The charged ions moving perpendicular to the magnetic field result in a magnetic force on the ion that is perpendicular to the velocity vector. This unbalanced force at right angles to the velocity causes a change in direction but not a change in speed. Consequently, the ion is deflected into a circular path. d. Since the magnetic force (Fm) is causing the ion to move in a circle, it is also a centripetal force (Fc). Setting Fm = Fc leads to qvB = mv2/r. e. This gives two equations, qV = mv2/2 and qvB = mv2/r, with two unknowns, v and m. Combining the equations results in the elimination of the speed (v) and a means of calculating mass (m) using the ion charge (q), accelerating voltage (V), magnetic field strength (B), and path radius (r). f. Since the mass spectrometer only works with ions, to obtain the final mass of the atom, the mass of the missing electron(s) must be added to the ion mass. That requires knowing the mass of an electron. The same process using electric and magnetic fields can be used to measure the mass of an electron.
C. Demos 1. Equipment – bar magnets, old CRT television set, old analog CRT oscilloscope, small suspended bar magnets on a frame that surrounds the Helmholtz coil apparatus, magnaprobe 2. Bring a bar magnet near the television set. Use both poles and hold the magnet in different positions near the back of the tube. Have students write observations in the Pre-Lab Packet. 3. Focus the electron beam of the oscilloscope to a point hitting the approximate center of the screen. Bring a bar magnet near the back end of the oscilloscope tube. Use both poles and move the magnet toward and away from the tube. Do this with the magnet above the tube, on the right side, and on the left side. Relate the motion of the electron beam to the magnetic force between the moving electrons and the magnetic field of the bar magnet. Use the third hand rule for the magnetic deflecting force to predict the motion of the beam. Discuss the relationship Fm = qvB in determining the magnitude of the force on the electron. Have students write observations in the Pre-Lab Packet. 4. Place the frame holding the suspended bar magnets over the Helmholtz coil apparatus. Turn on the current to the coil. Have students write observations in the Pre-Lab Packet. 5. Bring a strong bar magnet near the electron beam. In the proper location, the magnet will cause the electron beam to miss the absorbing electrode and follow a spiral or corkscrew path. Have students write observations in the Pre-Lab Packet. 6. Show the magnaprobe and describe its function. Hold the magnaprobe near a bar magnet and move it around the magnet noting the change in orientation of the north pole of the magnaprobe. Indicate that the magnaprobe can be used to determine the shape of the magnetic field of the Helmholtz coil.
D. Concepts and derivation (in the Pre-lab Packet) 1. Darken the room and turn on the Helmholtz coil equipment. Increase the voltage to produce a visible electron beam. Increase the coil current to show the deflection of the electrons. 2. Electrons are produced by indirectly heating a cathode, which is connected to a high voltage electrical source. When the cathode is heated, electrons leave the cathode. This is called thermionic emission and is analogous to water molecules transitioning from the liquid phase to the gas phase when water boils. 2. An electric field is used to accelerate the electrons. The increase in kinetic energy of an electron is equal to the work done by the electric field and can be used to develop an expression for electron speed (v): W = qV, KE = mv2/2, and W = KE so qV = mv2/2 and v2 = 2qV/m This expression contains two unknowns, the mass (m) and speed (v) of the electron. We want to determine the mass of the electron, but we have no way to measure the speed of the electron directly. We need another equation containing both these unknowns to eliminate the speed. 3. The magnetic field of the Helmholtz coil produces a magnetic force on a moving electron that is perpendicular to the velocity of the electron. This results in the circular motion of the electron beam so the magnetic force (Fm) is the centripetal force (Fc). Setting these two forces equal to each other allows us to derive a second equation containing v and m: Fm = Fc qvB = mv2/r qB = mv/r v = qrB/m v2 = q2r2B2/m2 4. Using the two expressions for v2 and letting q = e (1.60 x 10-19 C, the value of the elementary charge), we can derive a formula for the mass of the electron: 2qV/m = q2r2B2/m2 2eV/m = e2r2B2/m2 m = er2B2/2V 5. Note the similarities to the formulas from the mass spectrometer. 6. The value of the magnetic field strength can be calculated from this formula: B = (m0NI/a)(4/5)3/2 Where m0 (vacuum permeability) = 4p x 10-7 Tm/A N (number of turns of wire) = 130 I = current value a = average radius of Helmholtz coil
For AP-C classes, include the derivation of the formula for the magnetic field of the Helmholtz coil using the Biot-Savart Law.
E. The experiment (in the Pre-Lab Packet) 1. Purpose – to examine the magnetic field associated with the Helmholtz coil and determine the mass of a single electron. 2. Equipment – the Helmholtz coil apparatus, a meter stick, a magnaprobe 3. Procedure a. Be sure the power switch is off, and then plug in the apparatus. Turn on the power switch. The unit performs a 30-second self-test, and then displays “000” on both meters. A further 5-10 minute warm up is recommended before taking careful measurements. b. The wire windings of the Helmholtz coil are not perfect circles so several measurements must be made to determine the average radius (a) of the coil. With the lights on, use the meter stick to measure the inner diameter and the outer diameter in four different locations – horizontally, vertically, lower left to upper right, and lower right to upper left. Do this for both coils. Enter the measurements in meters into an Excel spreadsheet and calculate the average radius. This can be done during the warm up period. c. With the lights on, turn the Current Adjust control up to 3.00 A. Use the magnaprobe to map the magnetic field. Show the location and direction of the magnetic field lines on the diagram. The diagram is a top view showing the horizontal cross-section of the circular wire windings. From the direction of the magnetic field lines, determine the direction of the current around the wires. Then turn the current back to zero.
d. With the lights out, turn the Voltage Adjust control up to 200 V. The blue beam produced by the electrons interacting with the low-pressure helium gas should be directed straight down. e. Turn the Current Adjust control up until the beam forms a complete circle within the glass envelope. How is the radius of the beam affected if the voltage is increased while the current remains constant? Explain why this happens. How is the radius of the beam affected if the current is increased while the voltage remains constant? Explain why this happens. f. Bring the magnaprobe magnet near to the electron beam circle. Describe the effect on the beam. g. Begin taking data. Adjust the voltage to 200 V and the current to produce a circle. The diameter of the beam can be read from the internal centimeter scale. The gradations and numerals are illuminated by the collision of electrons with the scale. The scale is marked every half-centimeter. The current should be adjusted until the beam is centered on one of the gradation marks. Measure the diameter for seven different current values. Record both the diameter and current. h. Raise the accelerating voltage to 300 V and repeat the measurements of step g. i. Repeat step g for two additional accelerating voltages. j. When all the data has been collected, switch off the apparatus. 4. Analysis a. Enter the data into a separate sheet of the Excel spreadsheet. Group the data according to the voltage values. b. Add a line in the data chart for magnetic field strength in teslas (T) and calculate it from the current and average coil radius. c. Determine the beam radius in meters. d. Direct calculation option (based on the Instruction Manual procedure) (1) Start with the 200 V trials. (2) Add a line for mass of the electron. (3) Write a formula to calculate the mass for the first data entry. (4) Drag across the row to calculate the mass for each data entry. (5) Repeat for all voltage trials. (6) Average all of the mass values and perform a percent error calculation. e. Graphical determination option (1) Start with the 200 V trials. Copy the data to a new sheet. (2) Plot the beam radius (r) vs. the magnetic field strength (B). [Instructor note – This graph has a generally hyperbolic shape. If students are to “discover” the next step, they will need to be familiar with modifying the independent variable on nonlinear graphs to produce straight lines.] (3) Modify the graph by manipulating the independent variable. The idea is to produce a straight line through the origin. Plot the beam radius (r) vs. the reciprocal of the magnetic field strength (1/B). [Instructor note – This graph should be linear with slope equaling the square root of 2Vm/e. Rearranging this expression allows one to solve for the mass, m.] (4) Add a trend line to the graph, have it pass through the origin, and include the equation of the line plus the statistics. Adjust the equation to view two decimal places. Use the slope to calculate the mass of the electron. (5) Plot the beam radius (r) vs. the reciprocal of the magnetic field strength (1/B) for the other voltage trials. Calculate the mass from the slope for each. (6) Average all of the mass values and perform a percent error calculation. f. Option – accounting for the Earth’s magnetic field (1) An extra step is needed in the Procedure section. The coil must be turned so that the field of the coil is parallel with the Earth’s magnetic field. Place a compass between the two coils on the top of the unit’s case. Turn the case until the compass needle is perpendicular to the plane of the coil. Remove the compass and take data as previously indicated in the Procedure. (2) In the following discussion, BT = the total magnetic field, BC = the magnetic field of the Helmholtz coil, and BE = the magnetic field of the Earth. (3) BT = BC +/- BE and from the derivation in step D-4, BT = (2Vm/e)1/2(1/r). Solving for BC and substituting gives BC = (2Vm/e)1/2(1/r) +/- BE. This is the equation of a straight line with slope equal to (2Vm/e)1/2 and y-intercept equal to +/-BE. (4) Plot BC vs. 1/r. Add a trend line to the graph, do NOT have it pass through the origin, and include the equation of the line plus the statistics. Adjust the equation to view two decimal places. Use the slope to calculate the mass of the electron. Read off the y-intercept to determine the value of the Earth’s magnetic field. [Instructor note – the BE value actually includes the Earth’s magnetic field plus any stray magnetic fields in the vicinity of the apparatus. It would be prudent to move all bar magnets and magnaprobes as far from the equipment as possible.]
F. Post-lab discussion 1. Discuss the results for each group and the sources of error. 2. Between 1895 and 1897, J.J. Thomson performed similar experiments. In fact, Thomson’s experiments led him to announce the existence of the electron in 1897, and he was credited with its discovery. Unfortunately, he could not determine the mass of the electron. Since the electron was unknown up to this time, its properties were also unknown. Its charge, which we used to calculate its mass, was not measured until Millikan’s oil drop experiment in 1909. So what did Thomson actually measure? It was the charge to mass ratio of the electron! Our apparatus also measured this ratio (e/m), and that is the value that is classically measured when this experiment is performed. We added the known value of the charge carried by electron to determine its mass.
Sample data (June 14, 2008) for Mass of Electron
References:
1. Instruction Manual for EP-20 e/m of the Electron Apparatus
1. Ratio of Electron Charge to Mass Lab (Northwestern University)
