Roberto Sebastian Rosales/Notebook/Physics 307L/2010/12/01

Steve Koch 22:04, 21 December 2010 (EST):Again, very good notebook. Throughout the entire semester, you kept up an excellent notebook and did some very nice work. Thanks! =Equipment=
 * SOAR DC Power Supply Model PS-3630 (Electron Gun Heater)
 * e/m Experimental Apparatus -composed of a tube of dilute helium gas, an electron heater, focusing magnets, and a Helmholtz coil used to provide a constant magnetic field (this description is from Brian Josey's lab notbook - a link can be found in the references section of my primary notebook)
 * Gelman Instrument Company: Deluxe Regulated Power Supply
 * Hewlett Packard 6384A DC Power Supply (Helmholtz Coils)
 * Multiple BNC cables
 * 3 voltmeters (one of which was used as an ammeter)

=Safety= There were no major safety concerns other than the dangers associated with a high voltage source. Prof. Koch did mention that someone had been shocked with the high voltage source before, so we were extra careful with that piece of equipment.

=Setup=
 * We followed Prof. Gold's lab manual, which can be found here. For some reason, all the connections had been removed from the e/m setup. Fortunately, the e/m apparatus had labels for each set of connections. We connected the SOAR DC Power Supply to the heater connections on the apparatus, the Hewlett Packard Power Supply to the Helmholtz Coil connections, and the Gelman Power Supply to the accelerating voltage connections. We also connected a voltmeter in parallel with both the SOAR and Gelman Power Supplies in order to get a more accurate voltage reading that what the analog displays provided. Finally, we attempted to connect an ammeter in series with with the HP Power Supply and the Helmholtz Coil connections on the apparatus. As mentioned in the lab summary, Matt initially connected the series correctly, but I thought he was wrong so I had him switch it. After getting an electron beam that would not bend as we changed the current in the coils, Prof. Koch helped us connect the ammeter in series correctly. After this minor rearranging of wires, everything was up and running without any problems.

=Procedure= Our procedure for this lab was not very complicated at all. The measurements that we recorded were the the radius (both right and left) of the electron beam, the accelerating voltage, and the current in the coils. As suggested in the lab manual, we took 11 radius measurements at constant accelerating voltage and different coil currents, and 11 more at constant coil current and different accelerating voltage. The voltage and current were easy to adjust, but taking the measurement for the radius was a little tricky to master (I am still not sure that we did master it since our data in not very close to the accepted value). We had to try as best we could to line up the right or left hand side of the circular beam with its reflection in the mirrored ruler. This was difficult because the ruler was a little high vertically with respect to the center of the circular beam. Also the beam seemed a bit faint and not very intense.

=Data and Calculations=
 * Our raw data is as follows:


 * After taking only a few data points, we quickly realized the the circular beam was not centered with the 0cm mark. Because of this, I found the true radius to be:

$$True Radius = \frac{Radius Right + Radius Left}{2}$$


 * Our data analysis can be found below:


 * Magnetic Field (B Field) Calculation:
 * $$B = \frac {\mu R^2 N I}{(R^2+x^2)^{3/2}} = (7.8 \times 10^{-4}\times I)$$ $$ {T}\,\!$$ (as stated in the lab manual)
 * where:
 * $$\mu = 4\pi \times 10^{-7} \frac{N}{A^2}$$ which is the permeability of free space
 * $$N = 130 \,\!$$, the number of turns in the Helmholtz coils
 * $$I =\,\! $$ current in the Helmholtz coils
 * $$R = 0.15 m\,\! $$, radius of Helmholtz coils
 * $$x= \frac{R}{2}\,\! $$


 * e/m Calculation:
 * $$\frac{e}{m}=\frac{2V}{(rB)^2}$$


 * This is derived from the following:
 * $$eV=\frac{mv^2}{2}$$ and
 * $$ {F_{magnetic}} = {q}{v}{B}\,\!$$


 * Average $$\frac{e}{m}\,\!$$ and Standard Error of the Mean Calculation:
 * $$\bar x = \frac{\sum_{i=1} x_i}{n}$$
 * $$SEM\ = \frac{s}{\sqrt{n}}$$, where s is the standard deviation for the sample and "n" is the number of observations


 * I also plotted our data as called for in the lab manual. In the first method, I plotted $$r$$ vs $$\frac{1}{I}\,\!$$. From this I used the slope of the best fit line in the following equation to obtain the e/m ratio. I then used the "linest" function built into Excel to find the error in the slope, which then yielded a range for my e/m calculation.
 * $$\frac{e}{m} = \frac{2V}{r^2B^2}=\frac{2V}{s^2\frac{1}{I^2}(7.8\times10^{-4})^2I^2}=\frac{2V}{s^2\times(7.8\times10^{-4})^2}$$, where $$r=s\times\frac{1}{I}$$ and $$B=7.8\times10^{-4}\times I$$




 * In the second method, I plotted $$r^2$$ vs $$V$$. I then used the slope of the best fit line to calculate e/m in the following way:
 * $$\frac{e}{m} = \frac{2V}{r^2B^2} = \frac{2V}{(sVB^2)} = \frac{2}{s \times (7.8 \times 10^{-4} \times I)^2}$$ where $$r^2 = sV \,\!$$ and $$B=7.8\times10^{-4}\times I$$

\,\!

Final Results
I obtained four different values for $$\frac{e}{m}$$, one more than what is called for in the lab manual. My results are as follows: For Constant Accelerating Voltage: Numerical Calculation: $$\frac{e}{m} = ( 2.4 \pm 0.3 )\times 10^{11}\frac{C}{kg} $$ Graphical Calculation: $$2.4\times10^{11}\frac{C}{kg}\leq \frac{e}{m}\leq 2.5\times10^{11}\frac{C}{kg}\,\!$$

For Constant Coil Current: Numerical Calculation: $$\frac{e}{m} = ( 2.4 \pm 0.7 )\times 10^{11}\frac{C}{kg} $$ Graphical Calculation: $$2.3\times10^{11}\frac{C}{kg}\leq \frac{e}{m}\leq 2.4\times10^{11}\frac{C}{kg}\,\!$$

Accepted Value
The accepted value for the charge to mass ratio is: $$\frac{e}{m} = 1.76 \times 10^{11} \frac{C}{kg}$$.

Comparison
For Constant Accelerating Voltage:

Numerical Calculation: About 2 SEM's away from accepted value with a percent error of about 36%.

Graphical Calculation: About 36% percent error

For Constant Coil Current:

Numerical Calculation: About 1 SEM away from accepted value with a percent error of about 36% error.

Graphical Calculation: About 34% error


 * Using this formula, $$ % Error = \frac{|x_{calculated} - x_{actual}|}{x_{actual}}*100$$, we found the percent error in our data.

=References= During this Lab, Prof. Koch helped us out a great deal. I referenced Alexandra Andrego's Notebook and Brian Josey's "e/m Ratio" notebook for the lab write up.