User:Boleszek/Notebook/Physics 307l, Junior Lab, Boleszek/2008/11/17

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= e/m lab = Darrell and Boleszek
 * Instrumentation
 * 1) Gelman Instrument Company - Delux Regulated Power Supply  (applied at electrode)
 * 2) Soar DC Power Supply 7403 (used for heater)
 * 3) Soar DC Power Supply (used for coils)
 * 4) Uchida e/m Experimental Apparatus Model TG-13
 * 5) Two voltmeters to monitor applied voltage to electrode and current through coils.

Equipment and Setup
The e/m apparatus set up and use is clearly stated in the Prof. Gould's manual. The procedure for setting up and using the equipment was followed with some care by us here. DMM's were used to measure the voltage on both the heating filament and the accelerating voltage. Current to the magnet coils was read from the output of the power supply.

We did experiment with higher ranges for the filament and settled on 10v. Adjusting the "focus" brings few results, but it does seem to produce a somewhat easier to see electron beam when twisted full clockwise.

The reading of the radius of the electron beam is very inaccurate. We feel that radius readings have an uncertainty of up to 0.5cm. We decide that we will both make separate estimates of the radius in order to reduce systematic error (of each of us having a slightly different way of reading) and also to produce more data points in hopes that their average will be more accurate. Thus there are four radius measurements for each configuration. Other measurement errors seem likely to pale in comparison. However, the DMM's on the voltage are likely accurate to at least .01 volts for the filament and 0.1Volts for the accelerating voltage. The current on the magnet supply reads to 0.1, and can readily be interpolated to 0.025 amp resolution. However, the accuracy of these front panel vernier displays is often questionable on older supplies.

Prior to beginning we notice that there is a 20mA current still flowing in the accelerating voltage power supply when the filament voltage is set to 0. This goes away when the power supply is unplugged from the e/m device. We are unsure where this small current leakage occurs. It seems likely that it is natural to the device. Accelerating voltage was set to 445 when this was measured. During normal operation it drew 40mA.

Day 1 Data
Filament Voltage: 10V

Accelerating Voltage   Magnet Current    Diameter 405                    1.35              4.3 5.6 5.5 5.5 445                     1.4               4.2 4.7 4.0 4.5 425                     1.4               4.5 4.0 3.9 4.3 425                     1.7               3.5 3.3 3.4 3.6 400                     1.6               3.5 3.6 3.5 3.8

339.8                  1.5               3.4 3.0 2.7 2.9 350                     1.5               3.0 3.5 3.0 3.3 359.8                   1.5               3.0 3.5 3.4 4.4 369.7                   1.5               3.7 3.2 3.7 3.2 440                     1.5               4.8 3.2 3.5 4.8

375                    1.0               3.7 3.5 3.8 3.8 375                     1.2               3.8 4.8 4.0 4.5 375                     1.3               3.8 4.2 3.9 4.2 375                     1.4               3.8 4.6 3.9 4.4 375                     1.5               3.8 4.0 3.5 4.1 375                     1.6               3.4 3.2 3.2 3.5

day 2
Used DMM's to read voltage/current on all three power supplies. This significantly increases our precision for the current reading. However, we do note that it tracks well with the front panel display which seems to be pretty accurate.

Again we find that adjusting the focus full clockwise gives best results. We also decide that lowering the filament voltage improves our ability to see where the beam is.

Though the manual asked us to make measurements of different combinations of I and V, it did not really make sense for us to do so because a calculation of e/m for this data would have to be done for each measurement instead of interpolated from the behavior of a graph. We choose to only make measurements with constant V and constant I today.

Constant Voltage data
Accelerating Voltage: 360

Magnet I (amps)  Radius (cm ± 1mm) 1.22             3.4 5.2 3.8 5.2 1.32              3.4 5.1 3.6 5.0 1.42              3.0 4.7 3.3 4.7 1.52              2.9 4.5 3.5 4.4 1.62              2.9 3.9 3.0 4.0 1.72              2.4 3.5 3.0 3.7 1.27              3.5 5.2 3.5 5.3 1.37              3.4 5.0 3.4 4.8 1.47              3.2 4.8 3.5 4.5 1.57              3.0 4.3 3.3 4.4

Constant Current Data
Magnet I: 1.35

Accel V          Radius (cm ± 1mm) 441.2            4.0 6.0 4.0 5.5 430.5             4.0 5.5 4.0 4.5 420.0             3.9 5.4 3.9 5.3 409.8             3.9 5.3 3.8 5.3 399.9             3.8 5.2 3.8 5.1 390.1             3.7 5.1 3.7 5.0 380.0             3.5 5.0 3.6 4.8 369.8             3.4 4.9 3.5 4.8 360.0             3.3 4.8 3.5 4.8 350.0             3.3 4.8 3.4 4.5

Constant Voltage Analysis
In the section entitled "Mathematical Background" I show how one can extrapolate the e/m ratio from a graph of $$\frac{(7.8*10^-4*R)^2}{2V} vs. \frac{1}{I^2}$$. .The graph displays A^-2 vs. T*m^2/A*V. The reason why the axes are scaled in this way (instead of just A vs m) is that this isolates the m/e ratio as the slope of the graph. These procedures were carried out in Matlab and are documented in the MATLAB code document. The reason why this value is not 1mm is because it accounts for the discrepancy between left and right radius measurements and disagreement between mine and Darrell's measurements. All this contributes to a much greater uncertainty in the "actual" radius (though and actual radius does not exist because the trajectory is not really circular"). I had a lot of trouble getting the error propagation to work. Ultimately I always got my error to be at least a full order of magnitude larger than my value, so I couldn't confidently report that. I believe I made a mistake in that I tried to calculate e/m all at once instead of calculating it for each value of I or V. Because I did not do this, the formula that I present below for the error of a slope does not work because I had ignored all the x's for each point and just used averages to do my calculations. When it came to summing over all the x's, I had only one radius per measurement, so there was no sum to be done! I regret not thinking about this earlier, but now that I have already done so much along the path I chose I will not have the time to redo all of it more tediously. Therefore I lay content with reporting the fractional uncertainty I had with the accepted value of 1.756*10^11 C/kg. I believe one of the reasons my answer is so bad is that the linear fit I performed with MATLAB did not pass through the origin and was therefore skewed. Since my y-intercept appears to be positive of the graph, moving it to the origin would probably increase my slope (m/e) which is desirable because then my e/m ratio would be diminished.
 * mean standard deviation of radii measurements:.008m
 * e/m ratio:4.3171*10^11C/kg with a whopping 146% error!!!!!!

Constant Current Analysis
The graph displays V vs. T*m^2. Curiously enough the standard deviation of the radii measurements is almost the same as those in the constant V data, which implies that our methods of measurement are at least consistent. Though the standard deviation of the radii for constant I is greater, on average, that that of constant V it is obvious that this value is much closer to the accepted value. I am actually surprised by this result, and had I more time I would redo the calculations in order to make sure I didn't copy a number wrong and to calculate the e/m ratio 4 times (using 1st, 2nd, 3rd, and 4th values of radius from each I and V) so that I could calculate a real standard deviation for my final result.
 * mean standard deviation of radii measurements:.008m
 * e/m ratio:2.4038*10^11C/kg with a not so bad 36.7% error

Final Result
Though I was very happy with my result from the constant I data, I mustn't ignore the failure looming just above it. I average these two values and obtain $$ \frac{e}{m}=3.3605*10^011 C/kg $$ with a deviation from the accepted value of $$ %error=.914% $$

Qualitative Observations
We observe that when the glass bulb is turned the emitted electrons follow a spiraling path that hits the wall of the bulb and spins back toward the source. This behavior perfectly represents the vectorization of trajectories. Since the Lorentz force is a cross product between B-field and velocity it follows that were the velocity perpendicular to the B-field then the force would be at all times tangential to the velocity, but if the velocity has a component along the B-field that component is untouched, so to speak, by the force and so the particle travels both in the direction parallel and perpendicular to the B-field, which is a spiral. It is possible to turn the apparatus so much that the velocity is entirely parallel to the B-field in which case no deflection occurs. We noticed that at small radii the top portion of the trajectory, whether circular or spiral, would appear weak violet while the rest of it remained pale blue. Since the light we see is the result of exited electrons of helium atoms dropping back down from higher energy levels this observation implies to me that this difference in color is the result of two different transitions. Violet is higher in energy than blue, so if I saw correctly the helium atoms are being exited to higher energy levels at the top than on the upwards and downwards legs of the trajectory. It could be that I saw wrong and that the light at the top was of lower frequency than the up and down legs, and this would make more sense to me is gravity had a role here, but the masses are so small that gravitational forces are much more negligible than the Earth's magnetic field. I really don't know why this happens.
 * Spiraling Trajectory
 * Color Change

Error Analysis
Error bars for each point of the raw data curves are presented as the standard deviation of the 4 measurements per point. I really am not sure if the errors in our measurements of radius are really random and not biased since we might have each tended to measure the radii differently (we did not formally agree to measure the left side of the beam, the middle, etc.. which I realize was something we should have done). Therefore the idea of a 68% confidence interval for these errors is imprecise because, for all I know, the parent distribution for radius measurements may have two humps (one for Darrell and I) instead of one bell curve. Were I to make sure I would have to take a lot more data at each point and make a bar graph of these readings. Since we only have four readings per point a bar graph here may not be very illuminating. We can see from the raw data curve for constant V that the errors tend to get smaller as radius decreases, which makes sense because the smaller radii were more concentrated and thus provided less room for measurement discrepancy between the two of us. But for constant I this does not appear to be the case. We made measurements at larger radii because we thought that a larger radius trajectory looked more circular and thus better fit the mathematical model we were going to apply to it. Our assumption may have paid off because the constant I result was much closer to the accepted value, though I can't be sure this would always be the case if we only made one run of constant I measurements. The fact that r decreases with increased I and increases with increased V makes sense because higher I increases the B-filed strength, which deflects the electrons into a tighter path, and higher V corresponds to faster emission velocities which travel a longer distance (larger loop) in the same amount of time.
 * Observational Error of Radii

I decide to only use the data from Day2 for my analysis. From this data I calculate the mean error of radii measurements. The National Geophysical Data Center (NGDC) reports the total magnetic field in Albuquerque to be only 50,292.7 nT (1T=1web/m^2), which, when compared to the $$7.8*10^{-4} web*m^{-2}$$(for current I=1A) B field in the coils is roughly 4 orders of magnitude smaller. Undoubtedly this has an effect, but I choose to make my life easier and ignore this complication. Therefore I only use the mean errors of the constant V and constant I measurements to find two errors for e/m. These are then averaged (which I realized much later to be a bad idea because it hindered actual error propagation) to find a single value for the standard deviation of the measured e/m ratio. The mathematics used in the error propagation is discussed below in the "Mathematical Background section".
 * Error Propagation for e/m

e/m ratio
In Prof. Gould's manual we are shown that from the Biot-Savart Law
 * $$ B=\frac{\mu R^2 N I}{(R^2+x^2)^{3/2}}$$

and the fact that for our setup x = R/2, $$ \mu=4\pi*10^{-7} web*A^{-1}*m^{-1}$$, N = 130, R = .15m the magnitude of the magnetic field through the Helmholtz coils is
 * $$ B=7.8*10^{-4} web*A^{-1}*m^{-2} * I$$

I was at first unsure how to get an expression for the e/m ratio from the magnetic field expression given in the manual so I did what all other scientists do when an answer is close by...asked another scientist. In particular I investigated Paul's Lab Notes. There I found a concise procedure for the mathematical calculations which I summarize here: If we assume that the electrons are deflected into an essentially circular path then the Lorentz force can be equated to the centrifugal force: $$\vec{F}=e(\vec{v} \times \vec{B}) = m \frac{\vec{v}^{2}}{R}$$

this implies:

$$\frac{e}{m}=\frac{|\vec{v}|}{R|\vec{B}|}$$

The electrons are accelerated through a potential V, implying:

$$\frac{1}{2}mv^{2}=eV$$

$$v=\sqrt{\frac{2eV}{m}}$$

Now, we use this velocity in the above equation for the ratio e/m:

$$\frac{e}{m}=\sqrt{\frac{e}{m}}\frac{\sqrt{2V}}{RB}$$

This, then, boils down to give e/m ratio in terms of V, r, and B (which depends on I), all of which are variables that we can find:

$$\frac{e}{m}=\frac{2V}{(RB)^{2}} $$

Then we can use this formula to calculate e/m for both constant v and constant I by solving R in terms of I and V. By simple algebraic rearranging we find
 * $$R^2=\frac{m}{e} \frac{2V}{B^2}$$

I recall that $$ B=7.8*10^{-4} web*A^{-1}*m^{-2} * I$$. I can rearrange the expression for R such that it is a "linear" graph with dependent variable I^-2 and slope m/e.
 * Constant V
 * $$\frac{(7.8*10^{-4}*R)^2}{2V}=\frac{m}{e} \frac{1}{I^2}$$

If I plot the right hand side of this expression versus I^-2 the slope should be m/e, which I can easy convert to e/m. for these measurements I=1.35A so $$B=7.8*10^{-4} web*A^{-1}*m^{-2} * 1.35A=1.053*10^{-3}web*m^{-2}$$. Then slope m/e is found from the expression
 * Constant I
 * $$\frac{(BR)^2}{2}=\frac{m}{e}V$$

Errors
Taylor explains that, given an uncertainty in y for the linear relation y = A + Bx, one can calculate the uncertainties of A and B by the familiar summation of errors in quadrature since both A and B (least square fits) are well-defined functions of the measured quantities y. The result of this propagation of error is
 * $$ \sigma_A = \sigma_y\sqrt{\frac{\sum_{i=1}^N x_i^2}{\Delta}}$$
 * $$ \sigma_B = \sigma_y\sqrt{\frac{N}{\Delta}}$$

where $$\Delta = N\sum_{i=1}^N x_i^2 - \left(\sum_{i=1}^N x_i\right)^2 $$ But when A=0, as it does in my case, the least squares fit expression for B changes and therefore so does its propagated error. In this case the error for B is
 * $$ \sigma_B = \frac{\sigma_y}{\sqrt{\sum_{i=1}^N x_i^2}}$$

This is the formula I use and it is provided on p.198 of Taylor's book. Since x and y are different for constant V and constant I measurements I must find B (turns out to be m/e) for both of them separately. It should be noted that I choose to isolate m/e as the slope so that my y-values are not just measured quantities but are actually the measured quantities square and scaled by constants such as V, 2, and 7.8*10^-4. Therefore the error $$\sigma_y$$ is not just the uncertainty in radius measurements, but is the propagated value of an uncertainty that is squared and multiplied by the scaling factors. All this is performed in the MATLAB code document.

Thoughts on the math
The mathematics applied in this analysis does not entirely correspond to the physical situation we observed because the "circular" path actually changes in radius. Of course, math is used to express an idealized model that closely resembles reality, but the strong asymmetry of the electron trajectory was quite apparent and measurable. Therefore the force acting on the particle was not just the centrifugal force $$mv^2/r = mr(d\phi/dt)^2$$ but included a radial component. Were I to seriously try to use this setup again to get the best results possible I would try to modify my mathematics to do so, or at least make sure to only make measurements of closely circular paths. Ultimately Thomson's method of balancing the trajectory of a B-field deflected electron with an E-field seems to be the safest way to avoid systematic error observation, but the accuracy with which ones instruments display voltage readings would still have to be considered as a possible source of error.