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Charge to Mass Ratio of the Electron

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					Charge to Mass Ratio of the Electron Daniel Okienko Department of Physics, Case Western Reserve University Cleveland, OH 44106 Abstract: The goal of this laboratory investigation was to determine a value of the ratio of charge to mass for an electron, e/m. By varying the current in Helmholtz coils that surround a Uchida chassis and by varying the voltage that accelerates electrons inside the chassis, we were able to determine an experimental value for e/m. The varying current alters the induced magnetic field which in turn affects the trajectory of an electron beam. Likewise, varying the voltage alters the velocity (and energy) of the electrons, and, therefore, affects the trajectory of the electron beam. We experimentally determined the ratio to be 1.49×1011±1.19×1010 C/kg. When compared to the CODATA recommended value of (1.75882017  0.00000007) 1011 C/kg, we see that there is a discrepancy. Our value very inaccurate compared to the recommended value when given our uncertainty. Despite this difference, our value is still relatively close to the accepted value. Introduction and Theory: Electrons can be freed in a vacuum tube from a hot filament. Electrons, once accelerated across a voltage difference due to an electric field will gain kinetic energy by the following relationship 1 2 (1) mv  eV 2 where m is the electron mass, v, is the electron’s speed, and e is the magnitude of the charge associated with an electron. Then, using magnetic fields, these free electrons can be focused. As an electron travels through the magnetic field B with velocity v it feels a force given by the equation (2) F  e( v  B) This magnetic force, because it is a cross product of two vectors (v and B), will be perpendicular to both the velocity and the magnetic field. As a result, an electron that moves with a constant speed, v, perpendicular to the magnetic field will travel in a circular trajectory that has a radius of R. See Figure 1 for an illustration.

Figure 1

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Because the magnitude of the magnetic force is evB , this force can then be applied to Newton’s Second Law as follows mv 2 evB  (3) R Rearranging Equations (1) and (3), we can obtain an expression for e/m e 2V  (4) m ( BR) 2 e Since the goal of this investigation is to experimentally determine a value of , we can m then compare our result to the CODATA recommended value which is e C  (1.75882017  0.00000007) 1011 m kg Experimental Procedure:

Figure 2

Figure 2 shows the experimental setup for this investigation. The vacuum tube, called the Uchida chassis, contains an electron gun used to create a beam of electrons. Current from the Elenco power supply will travel through the Helmholtz coils and will control the magnetic field that acts on the electron beam. The Pasco power supply provides the voltages that the electron gun requires to operate. Electrons are “boiled” off the filament in the electron gun. By accelerating the electrons through an electric potential of a few hundred volts, the energy (and velocity) of the electrons will increase, which will also result in circular trajectories that large enough radii that can be measured. Ultimately, the electrons’ circular trajectory can be seen as a faint glowing ring inside the vacuum tube. The experimental setup ensures that the magnetic field provided by the Helmholtz coil is perpendicular to the electron beam. The magnetic field at the center of the Helmholtz coil can by determined with 8 NI B 0 c (5) 5r 5 with μ0 is 4 107 T  m/A , Ic is the current of the coil, and N is the number of turns in the coil. For our setup, N=130 and r=0.150±0.005 m. The radius of the electrons’ path can be manipulated by changing the voltage that accelerates the electrons or current in the Helmholtz coil that controls the magnetic field. When measuring the radii of the electron beam, all three of us took our measurements separately. Once we each measured our values for the radius, we then took the average of all three measurements to use for further calculations. We decided to

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take this route because we would then each be working with the same numbers so that collaboration would be easier. First, we set the beam voltage to 300±1 V, and this voltage remained constant. We varied the current to find the upper and lower limits of the current that would allow us to clearly see the electron beam. It was very important to not go higher than a current of 2.5 A because the coils could be damaged. At each value of the current, each of us measured radius of the beam. Finally, we took radii measurements at three values of the coil current at current values between the upper and lower limits. The data we gathered, including our averaged values, can be found in Table 1 in the Results section. Next, the beam voltage was set to 300±1 V. We then adjusted the current going to the Helmholtz coil until the electron beam almost filled the entire vacuum tube. This current was set to 0.976 ±0.001 A. All three of us took radius measurements for these values of voltage and current. Then we kept the current value the same and varied the voltage. We found the maximum and minimum values for the voltage at which we could still clearly see the electron beam. Once we found the maximum and minimum values and their corresponding radii, we took readings at five additional voltages between the maximum and minimum voltages. This data can be found in Table 2 located in the Results section. Results and Analysis: Table 1. Varying Current with Radii Measurements
I = 0.851 A (Lower Limit) I = 2.425 A (Upper Limit) Name R Left (cm) R Right (cm) R Left (cm) R Right (cm) Fred 6 6 2.5 2.5 Ray 6.3 6.3 2.5 2.5 Dan 6.1 6.1 2.5 2.5 Average 6.13 6.13 2.5 2.5 I = 2.027 A I = 1.517 A Name R Left (cm) R Right (cm) R Left (cm) R Right (cm) Fred 2.9 2.9 3.6 3.6 Ray 3 3 3.8 3.8 Dan 2.8 2.8 3.8 3.8 Average 2.9 2.9 3.73 3.73 I = 0.997 A Name R Left (cm) R Right (cm) Fred 5.9 5.9 Ray 6.1 6.1 Dan 5.9 5.9 Average 5.96 5.96 Uncertainty = ±0.2 cm

Table 2. Varying Voltage with Radii Measurements
Name V=300 V; I=0.976 A V=276 V (Lower Limit) R Left (cm) R Right (cm) R Left (cm) R Right (cm)

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Fred Ray Dan Average

6.1 6 6 6

6.1 6 6 6

5.4 5.5 5.4 5.4

5.4 5.5 5.4 5.4

V=364 V (Upper Limit) V=350 V Name R Left (cm) R Right (cm) R Left (cm) R Right (cm) Fred 7.3 7.3 7.3 7.3 Ray 7.3 7.3 6.9 6.9 Dan 7.3 7.3 7 7 Average 7.3 7.3 7.1 7.1 V=333 V V=314 V Name R Left (cm) R Right (cm) R Left (cm) R Right (cm) Fred 6.8 6.8 6.5 6.5 Ray 6.8 6.8 6.3 6.3 Dan 6.8 6.8 6.4 6.4 Average 6.8 6.8 6.4 6.4 V=293V V=280V Name R Left (cm) R Right (cm) R Left (cm) R Right (cm) Fred 6.1 6.1 5.7 5.7 Ray 6 6 5.6 5.6 Dan 6 6 5.8 5.8 Average 6.03 6.03 5.7 5.7 Uncertainty=±0.2 cm

Using Equation (5), we are able to eliminate B from Equation (4) to give e 2V  (6) m (80 N / 5r 5) 2 I c 2 R 2 Equation (6) shows how the beam radius, R, is dependent on the coil current or the beam voltage. Furthermore, the dependence on Ic can be given as 1 (7)   Ic R α is a proportionality constant and can be expressed as 8 N e / m (8)  0 5r 10(V ) By adding an intercept, A, to Equation (7) which allows for the effect due to other fields parallel to the magnetic field of the coils yields 1   Ic  A (9) R Equation (9) is a plot of a straight line. Using our data from when the voltage was kept constant at 300±1 V and the current was varied, we plotted 1/R versus Ic; this plot is Figure 3. We fit a straight line to this plot and obtained the slope and intercept of the

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plot. Our fitted line returned a value of α to be 15.4345±0.0001 (m·A)-1 and A to be 2.72185±0.0001 m-1. Equation (8) can be rearranged to solve for e/m as follows e 125 2 r 2V (10)  m 320 2 N 2
e (125)(15.4345) 2 (0.150) 2 (300) C   2.35  1011 7 2 2 m (32)(4 10 ) (130) kg 10  e / m  1.57 10 Please see Appendix I for the full calculations determining δe/m. This data fit fairly well to straight line. Of course, it was not perfect, but it fit relatively well. Only one data point in particular stands out as not fitting very well. The theoretical value of A is given as 1 A    Ic R But, because the raw data is not completely linear, the value for A is not constant. However, by plugging in the values for R and Ic from Table 1 and the value for α into this equation, it is possible to see that the theoretical values for A are within ±1 the intercept A from the plot. The dependence on beam voltage can be expressed through the equation R  V 1/2 (11) β is a proportionality constant that can be calculated by 5r 10 (12)  80 NI c e / m Once again, an intercept term, B, can be added to Equation (11) to yield R  V 1/2  B (13) We plotted our data of R versus the square root of the beam voltage from the data we collected when the current was kept constant and the voltage was varied. We fitted a linear line to this data to determine values for β and B. This plot and its corresponding fitted line can be found in Figure 4. This plot returns a value of β to be 0.00738±0.0004 and a value of B to be 0.00738±0.0004. Equation (12) can then be rearranged to solve for e/m to give e 125r 2 (14)  m 32 2 02 N 2 I c 2
e (125)(0.150) 2 C   6.34 1010 2 7 2 2 2 m (32)(0.00738) (4 10 ) (130) (0.976) kg 9  e / m  8.09 10 Please see Appendix II for the full calculations in determining δe/m. A straight line does fit the data for this experiment. Once again, it is not a perfect fit, but nonetheless, the plotted data shows a linear trend. The theoretical value of B can be found with B  R  V 1/2

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Once again, since the data is not a perfect linear line, the data from Table 2 will not yield a constant value for B. However, the values for the theoretical B are close to the intercept B that was determined by the linear fit. Conclusions: From the coil current experiment, where voltage was held constant, we determined a value of e/m to be 2.35×1011 ±1.57×1010 C/kg. From the second experiment, in which current was held constant and voltage was changed, we determined e/m to be 6.34×1010±8.09×109 C/kg. There is a significant discrepancy between these two values for e/m. Taking the average of these two values yields 1.49×1011, which is much closer to the expected value of e/m as given by CODATA. Furthermore, a comparison of the uncertainties shows another discrepancy. The average of the two uncertainties is ±1.19×1010. Therefore, the single best value for e/m based on our data is going to be the average of our two values of e/m, which is 1.49×1011±1.19×1010 C/kg. When the uncertainty is added to e/m, our value is just shy of CODATA’s value. The fact that our e/m is smaller than the accepted value and has a greater error is no surprise considering that this laboratory investigation was not given a great amount in which to be completed. Also, our method for measuring the radius of the electron beam’s trajectory may have been flawed, which would ultimately affect our certainty in the measured values. By “eyeing” the radius based on a mirrored scale behind the vacuum tube, we increased our uncertainty dramatically. Eyeing up these measurements is very imprecise in that each person has different visual acuity and may have used a different method to line up the electron beam to the scale. Furthermore, light reflection may have also affected how accurately we were able to measure the radius. These flaws in the measurement of the radius have huge impacts on not only the calculations for e/m, but also our ability to determine a theoretical value for the intercepts of the plots. As a result, the scatter plots of the data will not be completely linear, thus making it difficult to determine a theoretical value of the intercepts. Our measurements of current and voltage can be considered to be fairly accurate because the meters used to measure these values have digital displays that show values to several significant figures. Acknowledgements: I would like to thank Raymond Zackowski and Fred Hatfull for their help in obtaining experimental data, preparing figures, and carrying out calculations. References: Buxton, Gavin and Diana Driscoll. Physics 122 Lab Manual: Electricity and Magnetism. CWRU Bookstore, Fall 2008.

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Appendix I. Error Propagation for e/m for Varying Current

 e / m  

 e/ m   250 r 2V     2 2     320 N

  (250)(15.4345)(0.150) 2 (300)       (0.0001) (32)(4 107 ) 2 (130) 2    6 e/ m  3.05 10   (250)(15.4345)2 (0.150)(300)  r     (0.005) (32)(4 107 )2 (130)2    10 e/ mr  1.57 10

 e/ m  
r

 250 2 rV  e/ m  r    2 2  r   320 N

 e/ m  
V

 e/ m   125 2 r 2   (125)(15.4345)2 (0.150)2  V   V   (1)  2 2  7 2 2   (32)(4 10 ) (130)   V   320 N  e/ mV  7.84 108
r V

 e / m  ( e / m )2  ( e / m )2  ( e / m )2  (3.05 106 )2  (1.57 1010 ) 2  (7.84 108 ) 2
 e / m  1.57 1010
Appendix II. Error Propagation for e/m for Varying Voltage

 e/ m  
r

   e/ m    250r (250)(0.150)   (0.005) r   2 2 2 2  r 2 7 2 2 2   (32)(0.00738) (4 10 ) (130) (0.976)   r   32 0 N I c   e/ mr  4.23109

 e / m

 e/ m      250r 2 (250)(0.150) 2      (0.0004)      32 3  2 N 2 I 2    (32)(0.00738)3 (4 107 ) 2 (130) 2 (0.976) 2     0 c       e/ m  6.89 109

 e/ m

I

   e/ m    250r 2 (250)(0.150) 2    (0.001)  I   2 2 2 3  Ic 2 7 2 2 3   (32)(0.00738) (4 10 ) (130) (0.976)   I   32 0 N I c   e / mI  1.27 108
c

 e / m  ( e / m )  ( e / m )  ( e / m )  (4.23 109 ) 2  (6.89 109 ) 2  (1.27 108 ) 2
2 2 2
r Ic

 e / m  8.09 109

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