OBJECTIVE The e/m Ratio of the Electron To study the behavior of a charged particle in the presence of a potential difference. To study the behavior of a charged particle moving in a magnetic field. To understand the principle behind a magnetic solenoid. To experimentally determine the charge per mass ratio (e/m) for the electron. INTRODUCTION The mass of an electron is 9.11 x 10-31 kg, and its charge is 1.60 x 10-19 C. These quantities are too small to directly measure, even if you were somehow able to isolate a single electron in the laboratory. However, using principles of electromagnetism, indirect measurements of the charge and the mass of the electron can be accomplished. These indirect measurements will be accomplished by taking advantage of quantities that are directly measurable in the laboratory setting. These direct quantities will be derived from knowledge of the velocity of an electron moving in a magnetic field. APPARATUS (1) 6 VAC power supply, (1) 0-1000 VDC power supply, (1) 0-5 ampere DC power supply, (1) e/m apparatus, (1) e/m solenoid, (1) reversing switch, (1) digital multimeter, (1) metric ruler, and (1) lab marker. THEORY An electron of mass m [kg] and electric charge e [C] that travels with a velocity v [m/s] in a direction perpendicular to a magnetic field B [Tesla = T] will experience a centripetal force F [N] that is mutually perpendicular to both the velocity and magnetic field, expressed by: F = e v Equation 1 B This centripetal force causes the electron's path to always be at right angles to the direction of its motion. Therefore, the electron is forced to travel in a circular path of radius R [m]. The force on the electron, moving in a circular orbit, is thus given by: m v F = R Equation The e/m Ratio of the Electron - Page 1
Combining Equation 1 and Equation yields: e v m v B= R Equation 3 Solving Equation 3 for the velocity yields: R e v= m Equation 4 B In the case of electrostatics, the initial speed v of an electron is acquired by accelerating the electron through an electric potential V [V]. Therefore, by the work-energy theorem, the electron will gain kinetic energy of 1/mv [J]. At the same time, it loses potential energy of ev [J]. This gives the equality: m v = e Equation 5 Substituting Equation 4 into the velocity term of Equation 5, and rearranging the terms, yields: e m = B Equation 6 Therefore, the quantity e/m can be indirectly measured by the determination of V (the accelerating potential for the electron), B (the strength of the magnetic field the electron is deflected by), and R (the radius of curvature of the electron's path in the magnetic field). In this experiment, the electrons are produced and accelerated by a cathode ray vacuum tube from an old oscilloscope (we used one for our Speed of Sound in Air laboratory). The emission of electrons comes from a hot metal (usually tungsten) cathode that is so heated by passing current though its wires. The flow of the electrons from wires is caused by thermal vibrational energy overcoming the electrostatic forces restraining the charge carriers. The oscilloscope has been modified such that its four deflecting plates along with its second and third anode grids are all connected to the positive side of the same DC power supply. Additionally, the tube's first anode grid and its cathode grid are connected to the negative side of the aforementioned DC power supply. In the cathode ray tube, the electrons are emitted by a hot tungsten filament and accelerated by a potential difference V [V] between the cathode and the anode inside the tube. V R V The e/m Ratio of the Electron - Page
This potential difference, typically several hundred volts, causes the electron to accelerate toward and hit the inside screen of the tube. The electrons, striking this surface, cause a chemical coating on the inside face to fluoresce, thereby, causing a spot to appear on the tube's face. The magnetic field is supplied by a large solenoid. The Coil / Solenoid: A solenoid is the scientific name for a current-carrying coil of wire that acts like a magnet when a current passes through its wires. The more familiar expression for the µ N I magnetic field through the center of a solenoid is B = o. However, this equation L only works for a ROUND coil which is long and narrow. The solenoid used in this experiment is, however, a rectangle. This coil is positioned so that the center is located at the position of the accelerating anode, mentioned earlier. The plane of the coil is situated so that it is parallel to and on axis with the length of the cathode ray tube. This positions the coil such that the electron path is also within the plane of the coil. The magnetic field of this rectangular coil is based on the Biot-Savart Law; relating the magnetic fields to the currents which are their sources. Finding the magnetic field resulting from a current distribution involves the vector sum of all of the associated fields. We can imagine a rectangle as being constructed from four straight sections of currentcarrying wire. Figure 1 Shown is a current-carrying wire of known length. We're interested in the magnetic field at a point P that is a distance z from the wire. If we set up a symmetry along the center of the length of current-carrying wire, that the Biot-Savart Law tells us that the magnetic field at a point along this perpendicular bisector is: Where: sin θθ = BB = μμ ooii ππππ LL LL + zz sin θθ = LL (LL + 4zz ) The e/m Ratio of the Electron - Page 3
Therefore, BB = μμ ooii ππππ LL (LL + 4zz ) Equation 7 Where, μ o [Tm/A] is the permeability of free space (4π x 10-7 Tm/A), I [A] is the current in the coil, L [m] is the length of one side of the rectangular coil, z [m] is the distance from the wire to the center of the rectangle, and B [T] is the magnetic field of the current-carrying wire segment. Since Equation 7 is only for a single side, we must use it four times in order to get the total magnetic field of the coil (B TOTAL = B 1 + B + B 3 + B 4 ); opposite sides of the rectangular coil having the same magnetic field. Additionally, the coil is made of more than one loop of current-carrying wire. As such, the final equation for the total magnetic field can be multiplied by the number of loops (N [#]) in order to find this new, amplified, magnetic field: N * B TOTAL = B SOLENOID. L 1 z 1 L z z Figure z 1 BB SSSSSSSSSSSSSSSS = NNμμ ooii ππ LL 1 zz 1 (LL 1 + 4zz 1 ) + LL zz (LL + 4zz ) Equation 8 The e/m Ratio of the Electron - Page 4
The Path Trajectory: Again, the electron emerges from the filament and is accelerated to the screen of the cathode ray tube at a certain position. Thus, when the trajectory is deflected by the magnetic field, the electron appears at a new location on the screen. The distance between the off-magnetic-field location and the on-magnetic-field location is proportional to the radius of curvature of the electron's circular path. The problem, however, is to determine this radius since what you observe on the screen is a deflected distance and not the actual radius of the electron's path. Figure 3, below, is a diagram that illustrates the measurements involved in this calculation. Figure 3 Position A corresponds to the exit location of the anode, which is the position where the electrons are accelerated from. Position C corresponds to the location where the electrons strike the screen without a magnetic field present. Position B corresponds to the location where the electrons strike the screen with the magnetic field turned on. Distance d is the deflection distance between the off-magnetic-field path and the onmagnetic-field path. Distance X is the distance from the anode exit to the screen. The center of the radius of the curved electron path is denoted with the letter O. Thus, the line OB equals the radius R. The line connecting position B with position E is given as an approximate distance equal to X. The line EB is not equal to X, because the actual screen of the cathode ray tube is curved. Therefore, EB is actually a fraction shorter than X. Based on geometry, we have the relation OE = R - d. Based on the right triangle EBO, and Pythagorean theorem, we have that (R - d) + X = R. Solving for the radius of curvature R, we have: X R= + d d The e/m Ratio of the Electron - Page 5 Equation 9
Thus, by knowing the distance X, between the accelerating anode (a constant), and the deflection distance d (easily measured), the radius of curvature R of the electron's path can be easily calculated. The deflection distance is basically limited by the size of the oscilloscope screen which has a radius of approximately d = 0.0670m. EXPERIMENTAL SETUP The circuits, as shown in the schematic diagrams below, will be used for the experimental procedures. Figure 4 **The ground is wired into the tube socket. No additional power supply connections needed! Figure 5 The e/m Ratio of the Electron - Page 6
In the wiring for the cathode ray tube, the connections between the negative DC, positive DC, and AC are done at the power supply. The ground is wired inside the tube's socket connection. Note that the voltmeter is in parallel with the tube's power supply. You will also note, in Figure 5 for the magnetic coil connections, that a reversing switch is used. The current may enter and traverse around the coil in either a clockwise or a counterclockwise direction. The affect of this is to change the direction of the magnetic field at the center of the coil from either perpendicularly into the plane of the coil (clockwise current) or perpendicularly out of the plane of the coil (counterclockwise current). The reversing switch (also known as a current reversing switch in the context of this experiment) serves to accomplish this directional change. Assemble the equipment but DO NOT ENGAGE ANY OF THE POWER SUPPLIES UNTIL YOUR WIRING HAS BEEN APPROVED! SERIOUS DAMAGE TO YOU AND THE EQUIPMENT COULD RESULT IF THIS STEP IS NOT ADHERED TO. EXPERIMENTAL PROCEDURE a) Slowly turn up the DC power to the tube. As you approach 500 VDC, you should begin to see a spot appear on the screen. You may have to go slightly beyond 500 VDC to see the spot engage, but afterwards, the power should be turned down to 500 VDC and the spot should remain. b) Note that all of the associated constants for the experiment are already indicated/entered into the Excel data table. These include, the plate voltage V for each set of trials, the current I passing through the coil for each voltage trial, the length of each coil segment L 1 & L, the distance from each wire segment to the center of the coil z 1 & z, the number of turns on the coil N, the distance from the anode to tube face X, and the accepted value of the e/m ratio for an electron. c) Mark, with the lab marker, one EDGE (not the middle) of the central spot. Beginning with a coil current of 0.30 A, close the reversing switch in one direction. The spot will shift to one side from its original location. Measure the distance between the two spot locations (original marked line and new location). Be sure to measure to the SAME SIDE of the spot that was originally marked! Record the current I and the deflection distance d #1 of the electron. Before Sharpie line drawn on tube face d#1 After Electron spot The e/m Ratio of the Electron - Page 7
d) Move the reversing switch to its other position and retake the measurement of the electron's deflection distance d #. Record this value and continue to use the lab marker to mark the deflected spot locations. This average will provide a greater accuracy in the deflection distance measurement used in the calculations. e) Additional readings, taken at the 500 VDC starting plate voltage V, are to be done at intervals corresponding to an increase in the current in 0.30 A steps up to 1.50 A. This yields five readings at this voltage. f) Upon completion of the measurements at the 500 VDC starting plate voltage V, increase the plate voltage V by 100 V and repeat the procedure of current readings from 0.30 A to 1.50 A. g) Continue to increase the voltage in 100 V intervals, while repeating the five current measurements, up to 900 V. DO NOT INCREASE THE VOLTAGE HIGHER THAN 1000 VDC. THIS WILL RESULT IN THE DESTRUCTION OF THE MULTIMETER USED TO MEASURE THE PLATE VOLTAGE; AND POSSIBLE DAMAGE TO THE TUBE! COVER PAGE REPORT ITEMS (To be submitted and stapled in the order indicated below) (-5 points if this is not done properly) Lab Title Each lab group member's first and last name printed clearly Group Color Date DATA (worth up to 15 points) Data tables are available as a downloadable Excel file DATA ANALYSIS (worth up to 15 points) The FOUR required sample calculations, to be shown on a separate sheet of paper in your laboratory report (NOT on the data table sheets themselves), are highlighted in yellow on the downloadable Excel data table spreadsheet The e/m Ratio of the Electron - Page 8
Some useful calculation information: The Percentage Error: In several laboratory settings the true value of a quantity being measured is known. This comparison is normally done by a percentage error calculation, where the accepted value of the quantity is compared to the experimental result. The Percentage Difference: In other experimental settings where a known value is not known, a comparison may still be useful to evaluate the effectiveness of the experiment based on the comparison of two experimental results. This will not tell you anything as to the accuracy of the experiment but will give you an indication of how repeatable (precise) the experiment is. GRAPHS (worth up to 10 points) Graph the values of two times the voltage {V} on the y-axis and the average {B R }, for each corresponding voltage, on the x-axis; you'll have five total data points. Indicate the best-fit-straight-line on this graph. o In addition, record the slope and y-intercept. GRAPH ANALYSIS (worth up to 5 points) On the graph itself, explain the quality of the data plotted in relation to the bestfit-straight-line indicated. What is the slope of this graph supposed to represent? CONCLUSION (worth up to 35 points) See the Physics Laboratory Report Expectations document for detailed information related to each of the four questions indicated below. 1. What was the lab designed to show?. What were your results? 3. How do the results support (or not support) what the lab was supposed to show? o Be sure to clearly justify the experimental results in terms of the data collected and the theory based from the equations. For example, you might want to say something like: "From equation # I'd expect that the will because. Based on the data I collected I found that ; thereby." 4. What are some reasons that the results were not perfect? The e/m Ratio of the Electron - Page 9
QUESTIONS (worth up to 10 points) 1) Suppose that protons were accelerated to the screen instead of electrons. How would this affect the experiment...specifically? The "charge/mass" ratio The deflection direction The deflection distance (d) The slope of the graph ) Given: EEEEEEEEEEEEEEcc CChaaaaaaaa (ee): CCCCCCCCCCCCCC [CC] = AAAAAAAAAAAA[AA] SSSSSSSSSSSSSS [ss] MMMMMMMM [mm]: KKKKKKKKKKKKKKKK [kkkk] EEEEEEEEEEEEEEEE PPPPPPPPPPPPPPPPPP (VV): VVVVVVVV [VV] = KKKKKKKKKKKKKKKK [KKKK] MMMMMMMMMM [mm ] AAAAAAAAAAAA [AA] SSSSSSSSSSSSSS 3 [ss 3 ] MMMMMMMMMMMMMMMM FFiieeeeee (BB): TTTTTTTTTT [TT] = KKKKKKKKKKKKKKKK [kkkk] AAAAAAAAAAAA [AA] SSSSSSSSSSSSSS [ss ] RRRRRRRRRRRR oooo CCCCCCCCCCCCCCCCCC (RR): MMMMMMMMMM [mm] Show that Equation 6 is dimensionally correct. Be very clear with your substitutions and cancellations so that it evident that the units on both sides of the equation are equal. 3) Assume that the power supply has no limit to the current it can output to increase the magnetic field strength of the coil. As such, it can theoretically give the coil the ability for an ever-increasing, no-limit magnetic field strength. Think about what happens to the electron spot on the screen as the magnetic field increases as a result of increasing the current. If the magnetic field continues to increase there IS actually a limit to the strength of the magnetic field we can test with this equipment; and as a result the current. a) Solve Equation 6 for B Use V = 500 V Use the true value of e/m from Page 1 b) You'll have to calculate R using Equation 9. See the paragraph at the top of Page 6 for one of the necessary values. c) Finally, solve Equation 8 for I and substitute in the value for B you just calculated. Is this value of I reasonable? Why/why not?? The e/m Ratio of the Electron - Page 10