Our setup using the booksThis was our measured reading on the micrometer
1. We first set the experiment up.
The step by step instructions are within the lab manual, rather then repeating the steps I'll simply show a photo before the setup and then after the setup.
2. After the setup, the optics need to be calibrated.
3. First thing is to measure the spacing between the plates, we measured 8.08mm SJK 15:04, 30 October 2009 (EDT)15:04, 30 October 2009 (EDT) This is a typo, should be mm not cm
4. Measure the thermistor resistance (our initial measurement is 1.90x10). After which we kept careful track of the resistance, should it decreases the temperature inside the well increases.
5. Next we plugged in the power supply to the apparatus, followed by the power supply for the lamp.
6. After all of the power is connected we focused the eyepiece and droplet focus, so that we could see the wired and grid.
7. Next, after everything is in focus, we switched the ionization source switch from off to spray droplet position.
8. As one of us sprayed the oil into the well, while the other person observed the oil droplets.
9. The oil and grid should be in focus through the eyepiece if that is the case, then measurements can begin.
10. The person looking through the eyepiece keeps track of an oil droplet measuring its rise and fall, while measuring the distance traveled against the grid, the other person keeps track of how long the droplet takes to travel using a stopwatch.
Data
In the data below, the Drop category is the drop that was measured, the rise time and fall time categories are the times it took the oil drop to rise and fall one major grid square, the temperature remained constant throughout the experiment, though only a few oil drops were measured, each drop was measured a minimum of ten times, wihle two drops 3i and 4i, were measured before and after ionizing the drop with thorium, however, I believe we might have lost track of 4i, and accidentally started measuring another droplet.
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Data Analysis
Since we are trying to analyze the charge of an electron using a fall time and a rise time we must first observe the relation given in the lab manual, [math]\displaystyle{ q=\left[400\pi d\left(\frac{1}{g\rho }\left[\frac{9\eta }{2} \right]^{3} \right)^{1/2} \right]x\left[\left(\frac{1}{1+\frac{b}{pa}} \right)^{3/2}\right]x\left[\frac{\nu _{f}+\nu _{r}\sqrt{\nu _{f}}}{V} \right]e.s.u. }[/math]
This formula is derived in the lab manual, the variables are defined as follows, where the numbers used are from our own measurements:
q=charge, in e.s.u
d=8.08mm; separation of the plates in the condenser.SJK 15:05, 30 October 2009 (EDT)15:05, 30 October 2009 (EDT) Same typo as above: it should be 0.808 cm (off by factor of 10)
[math]\displaystyle{ \rho }[/math]=0.886[math]\displaystyle{ gm/cm^{3} }[/math] density of oil. ([math]\displaystyle{ gm/cm^{3} }[/math])
g=981cm/s[math]\displaystyle{ ^{2} }[/math]; acceleration of gravity ([math]\displaystyle{ cm/s^{2} }[/math])
[math]\displaystyle{ \eta }[/math]=1.860[math]\displaystyle{ Nsm^{-2}x10^{-5} }[/math]; viscosity of air in poise (dyne [math]\displaystyle{ s/cm^{2} }[/math])
b=6.17x[math]\displaystyle{ 10^{-4} }[/math](cm of Hg)(cm)
p=75.7711cmHg; barometric pressure in cm of mercury.
a=radius of the drop in cm.
[math]\displaystyle{ \nu _{f} }[/math]=fall time. (in cm/s)
[math]\displaystyle{ \nu _{r} }[/math]=rise time. (in cm/s)
V=500V; potential difference across the plates in volts.
The accepted value for e is 4.083x10[math]\displaystyle{ ^{-10} }[/math] e.s.u., or 1.60x10[math]\displaystyle{ ^{-19} }[/math] coulombs.
The variable were all found in the following ways:
q-was calculated from the equation above.
d-was measured.
[math]\displaystyle{ \rho }[/math]-the density of oil is given in the manual.
[math]\displaystyle{ \eta }[/math]=[math]\displaystyle{ 1.860Nsm^{-2}x10^{-5} }[/math]this is found in appendix a from the measured temperature. The temperature was measured using the resistance read from the multimeter and referring to the table given on the apparatus.
b=6.17x[math]\displaystyle{ 10^{-4} }[/math](cm of Hg)(cm), this constant was given in the manual.
p-measured barometric pressure through the governments weather forecasting website in mb. Measured pressure was 1010.2 mb(750.06torr)= 7577.11mmHg= 75.711cmHg.link to the government weather webpage.
a-is dependent on fall times, therefore the varying a values are shown below.
[math]\displaystyle{ \nu _{f} }[/math]=measured through observation.
[math]\displaystyle{ \nu _{r} }[/math]=measured through observation.
V-measured and controlled through power supply.
Calculated values of a
We find a by using the equation in the manual,[math]\displaystyle{ a=\sqrt{\frac{9\eta \nu _{f}}{9g\rho }} }[/math]
Where [math]\displaystyle{ \eta }[/math]=1.860 and [math]\displaystyle{ \nu _{f} }[/math]=our fall times. g=981[math]\displaystyle{ cm/s^{2} }[/math], and [math]\displaystyle{ \rho =0.886gm/cm^{3} }[/math].
Calculating the above equation with the average of our recorded fall times we get the following results for a:
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Unfortunately this is not the correct radius for the drop of oil, since at .01cm radii and less the oil's fall time becomes affected by collisions with air molecules, and the original [math]\displaystyle{ \eta }[/math] value must be corrected. Once corrected a new equation is yielded, from this we can finally calculate the true radius of the oil drop.
[math]\displaystyle{ a=\sqrt{\left(\frac{b}{2p} \right)^{2}+\frac{9\eta \nu _{f}}{2g\left(\rho \right)}}-\frac{b}{2p} }[/math]
The new values for a are as follows:
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Calculated Values of q
Once the radii of the oil drops are known then it's simple a matter of plugging in the variables into the equation for q to discover what the value of q actually is:
Since the first part of the equation is not dependent on the oil drops, this can be solved for first then treated as a constant: [math]\displaystyle{ q=\left[400\pi 0.808\left(\frac{1}{981*0.886}\left[\frac{9*1.860x10^{-4}}{2} \right]^{3} \right)^{1/2} \right] }[/math]=8.31x10[math]\displaystyle{ ^{-4} }[/math]
The second part of the equation is only dependent on two variables, the barometric pressure and the radius of the oil drop, since the barometric pressure can be considered the same throughout the experiment, we only have to worry about the varying a values. [math]\displaystyle{ q=\left(\frac{1}{1+\frac{6.17e-4}{75.7711a}} \right)^{3/2} }[/math]
While the third part of the equation is only dependent on the rise and fall times, the voltage throughout the experiment remains the same:
[math]\displaystyle{ q=\left[\frac{\nu _{f}+\nu _{r}\sqrt{\nu _{f}}}{V} \right] }[/math]
The computed values of q for the second and third part of the equation are as follows:
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Using the techniques suggested in the lab manual I solved for q using a different set of variables in order to try to compute q in coulombs, from this I was able to recompute my data and came up with the following:
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The equation used was the same as above, but I used pascals instead of mmHG, and I kept the oil's density at 886kg/m^3. I also avoided calculating two sets of radii and computed only the accurate radius, this helped in avoiding confusion while I tossed numbers around back and forth.
Complications
We ran into several setbacks as we attempted this experiment, this is a list of some of those complications:
On day one much of our time was used trying to find a way to make a camera or video camera work through the eyepiece, this would've resulted in much greater accuracy in our experiment, unfortunately, due to time constraints we were unable to use video.
On our first attempt to spray the oil in we didn't realize that the ionization filter was in the off position, therefore blocking the oil from being seen.
The plastic cap was also placed upside-down at first, that too blocked the oil from being observed.
The grid was also at an angle at first, which required Dr. Koch and Pranav to go in and re-align the grid for us using the set screw on the side of the optical piece.
due to some of the previous circumstances the eyepiece was taken out of focus and required us to refocus once again.
I was unable to get the pie character to come up in excel and google docs so my calculations are not as accurate as I would like them to be, I simply typed in 3.1416 as an approximation. (Steve Koch 15:21, 30 October 2009 (EDT):!? Hopefully you just typed in 3.1416 or similar?)
Millikan Oil Drop
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