User:Garrett E. McMath/Notebook/Junior Lab/2008/09/15
 E/M Ratio Lab
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Lab 2: The Ratio e/m for ElectronsNotes By: Garrett McMath and Paul Klimov SJK 00:22, 4 October 2008 (EDT) Your raw data notebook is very good, reflecting the very good work you did in class. As I mention below, make sure to take a look at Paul's notebook for more feedback. Purpose and ObjectivesThe purpose of this laboratory is to measure the e/m ratio of electrons. This will be done by accelerating electrons into a strong magnetic field, which will cause the electrons to rotate in a circle on a plane perpendicular to the field. TheoryThe Lorentz force, in the absence of an Electric field gives us: [math]\displaystyle{ \vec{F}=e(\vec{v} \times \vec{B}) = m \frac{\vec{v}^{2}}{R} }[/math] this implies: [math]\displaystyle{ \frac{e}{m}=\frac{|\vec{v}|}{R|\vec{B}|} }[/math] The electrons are accelerated through a potential V, implying: [math]\displaystyle{ \frac{1}{2}mv^{2}=eV }[/math] [math]\displaystyle{ v=\sqrt{\frac{2eV}{m}} }[/math] Now, we use this velocity in the above equation for the ratio e/m: [math]\displaystyle{ \frac{e}{m}=\sqrt{\frac{e}{m}}\frac{\sqrt{2V}}{RB} }[/math] This, then, boils down to give e/m ratio in terms of V, r, and B, all of which are variables that we can find: [math]\displaystyle{ \frac{e}{m}=\frac{2V}{(RB)^{2}} }[/math] From here, we must find the strength of the magnetic field which is produced by the Helmholtz coils: We use the Biot-Savart Law, to find the field due to one coil, and then add the field due to two coils. A single coil, with N loops, of radius R, stands with its opening aligned with the y-axis. The point of interest will be at y=a. Due to symmetry, we know that there will only be a field in the positive y direction (where current flows counter-clockwise, as seen looking from positive y). Theta will rotate in the xz plane, which will allow us to define a small current element. Biot-Savart Law: [math]\displaystyle{ d\vec{B}=\frac{\mu_{0}i}{4\pi}\frac{ \vec{dl}\times\vec{r}}{r^{3}} }[/math] we know that dl is perpendicular to r always, and so we can turn this into a scalar equation for the y coordinate. [math]\displaystyle{ dl = R d\theta }[/math] [math]\displaystyle{ B_{y}=\frac{N\mu_{o}i}{4\pi}\int_{0\leq\theta\leq2\pi}\frac{Rd\theta}{(a^{2}+R^{2})}\frac{R}{\sqrt{a^{2}+R^{2}}} }[/math] [math]\displaystyle{ B_{y}=\frac{N\mu_{o}i}{4\pi}\frac{R^{2}}{(a^{2}+R^{2})^{\frac{3}{2}}}\int_{0\leq\theta\leq2\pi}d\theta }[/math] [math]\displaystyle{ B_{y}=\frac{N\mu_{o}i}{2}\frac{R^{2}}{(a^{2}+R^{2})^{\frac{3}{2}}} }[/math] Now, because we have 2 coils, which make up the Helmholtz coils, we simply multiply the above expression to find the field as a function of the current: [math]\displaystyle{ B_{y}(i)=\frac{N\mu_{o}iR^{2}}{(a^{2}+R^{2})^{\frac{3}{2}}} }[/math] with this information, we should be able to complete the lab. Experimental Setup
Heater connected to SOAR PS-3630. Accelerating Electrodes connected in series with AMPROBE multimeter and then to the Gilman power supply. Voltmeter jacks connected to BK Precision multimeter Helmholtz coils connected in series with FLUKE multimeter to SOAR 7403 DC Power Supply.
Connected in series with an ammeter/voltmeter and connected to the heater.
Connected in series with an ammeter/voltmeter and connected to the Helmholtz coils.
Connected directly to the voltmeter jacks on the e/m experimental apparatus.
Connected in series with an ammeter/voltmeter and connected to the electron gun.
Connected in series to the SOAR 7403 and e/m experimental apparatus.
Connected in series between e/m apparatus and Gelman power supply.
Procedure1.) Turn on SOAR PS-3630 PSU for heater to:
Nothing connected to ground. This starts the heating of the filament. It starts radiating as it heats up. 2.) Turn on Gilman power supply for electron gun
3.) Settings on e/m experimental apparatus:
Experimental Procedure
DataSource of error: Our voltmeters accuracy doesn't go into the decimals. Each data point has an uncertainty of at least ±.1 cm because this is roughly the width of the beam Constant Current
1.)200V, 4.1±.1cm 2.)210V, 4.35±.1cm 3.)220V, 4.45±.1cm 4.)230V, 4.5±.1cm 5.)240V, 4.55±.1cm 6.)250V, 4.60±.1cm 7.)260V, 4.65±.1cm 8.)270V, 4.70±.1cm 9.)280V, 4.75±.2cm 10.) 290V, 4.80±.3cm 11.) 300V, 4.90±.3cm At the end, the beam seemed to get a bit thicker. This introduced more uncertainty into our measurements. Constant VoltageOne source of experimental error: The beam seems like it is somewhat elliptical. The radius on the right is slightly smaller than on the left. We have decided to use only the left radius. This will definitely introduce uncertainty into our calculations. However, after rotating the bulb, we have a spiral forming. With the spiral, we cannot take any measurements because there is simply too much going on. V=280V 1.)i=1.000A, 4.75±.2cm 2.)i=1.050A, 4.75±.2cm 3.)i=1.100A, 4.60±.1cm 4.)i=1.151A, 4.55±.2cm 5.)i=1.202A, 4.47±.1cm 6.)i=1.253A, 4.35±.1cm 7.)i=1.297A, 4.25±.2cm 8.)i=1.351A, 4.15±.2cm 9.)i=1.399A, 4.05±.2cm 10.)i=1.449A, 4.00±.2cm ANOTHER ROUND OF DATA FOR CONSTANT VOLTAGES:
V=299.8 1.)i=1.245A, R=3.90±.1cm 2.)i=0.989A, R=4.25±.2cm
V=249.6V 1.)i=.997A, R=3.95±.1cm 2.)i=1.252A, R=3.70±.2cm LAB SUMMARYSJK 00:34, 4 October 2008 (EDT)I do want your lab summary to be on a separate page. And also, actually a bit more concise. You will want to have this much analysis written down in your lab notebook, but then you want to sumamarize it on a separate page. Check out PK's summary for an example. Lab Manual Questions1. Why do we see the electron beam at all? The visual beam that we see is merely the ionization of the helium gas caused by the electrons losing energy to the helium causing the helium to emit photons of such wavelength in the visible light spectrum. 2. We ignored the Earth’s magnetic field in our procedure. How much error does this introduce into this experiment? Using the website http://www.ngdc.noaa.gov/geomagmodels/IGRFWMM.jsp, you can get the magnectic field and its components of your location. The total magnetic field in Albuquerque is 50,292.7 nT, with components of 23,027.7 nT North, 3858.3 nT East, 23,348.7 nT Horizontal, and 44,544.3 nT Upward. Our apperatus was more or less aligned with the east component of the earths B-field so plugging this value in using 300 V will give a reasonable approximation of the error the earths field could have caused. [math]\displaystyle{ R = \sqrt{\frac{2Vm}{e}}\frac{1}{B} = 15.15m }[/math] This value is actually pretty large using MatLab paul analyzed this result which you can see in the data analysis section.
Using the same formula as in question 2: [math]\displaystyle{ \frac{R_{p}}{R_{e}}=42.81 }[/math] So other than the difference in polarity the major difference would be a much stronger magnetic field would be needed to bend the beam. 4. Show that if the magnetic field is held constant, the time t required for an electron to make a complete circle in your e/m tube and return to the anode is independent of the accelerating voltage by deriving an expression for this time. [math]\displaystyle{ T=\frac{2\pi R}{v} }[/math] [math]\displaystyle{ R = \sqrt{\frac{2Vm}{e}}\frac{1}{B} }[/math] [math]\displaystyle{ v = \sqrt{\frac{2eV}{m}} }[/math] [math]\displaystyle{ T = \frac{2\pi m}{e B} }[/math] Data AnalysisSJK 00:06, 4 October 2008 (EDT)First of all, it is really great how you cite Paul for creating the MatLab codes and tell the reader where to find those codes. That is really great scientific practice. However, I'm going to clarify next week in lecture this point: For the lab summaries, your goal should be to do the data analysis and discussion "yourself." Meaning, it's OK to discuss and work with other people, but you want to learn how to do the calculations and analysis in the absence of the other person. For example, in this case, you'd want to know how to write your own matlab code. Or, if you don't have access or know how to use matlab, then you'd need to use another software application (like Excel). From what I read in your book, I don't think you learned as much as you could have, though I do know you were discussing these things with Paul. An example of what I'm saying is you put down a significant amount of results without explaining what they mean or where they came from. Like I said, we'll talk about this on Monday and please ask me for clarification as well. In your next lab, here's an example of what you could say, "During lab session, Paul and I analyzed our data using Matlab software that he wrote. We generated a bunch of figures and results that are here (link). I wrote my own Excel sheet to replicate these results, and the sheet is here (link)..." Then go on to discuss the methods of analysis and what you think it means.
      ProgramsUsing data analysis programs written in MATLAB by Paul klimov we calculated least squares regressions for both sets of data, constant voltage and constant current. The program then calculated an e/m ratio for both sets of data using the formula that is shown in theory section. Margin of ErrorAs will be discussed later in sources of error we had uncertainty in all of our measurements. The most obvious reason being the width of the beam caused a minimum uncertainty of one millimeter, though we often had larger uncertainty either due to thicker beams caused by higher voltage, or simply a large discrepency between Paul's and my measurements. Constant CurrentUsing the MatLab programs, the following information was obtained:
Constant Voltage
Slope Modification: Figure 2b
Final StatsSJK 23:46, 3 October 2008 (EDT)There is missing any description of what "overall" and "best" mean. There are also the last two figures without any discussion of why or what they are.
Sources of ErrorSJK 00:01, 4 October 2008 (EDT)This is a good discussion of error sources. This lab we were told has one of the worst percentage errors of any of the labs we will be doing this semester. With that in mind we watched very closely for sources of this while doing the lab.
Improvements
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