Oscilloscope LabSJK 03:27, 17 September 2008 (EDT)
The raw data from the lab is here.
In summary, I measured the peak-to-peak voltage and period of multiple sine waves of three different amplitudes the peak-to-peak voltage and period of a sine wave with a large DC offset, as well as the fall time of a DC Voltage read through AC coupling, measured by placing cursors at the peak of the function and at the 90% decay point of the function, as well as automatic calculation in the oscilloscope. I measured everything else by visual inspection, cursor inspection, and automatic calculation in the oscilloscope.SJK 03:12, 17 September 2008 (EDT)
|Pk-Pk 200 Hz||500 mV||508 mV||504~508 mV||100 mV||16 mV||1 mV|
|Period 200 Hz||5 ms||5.2 ms||5.128~5.144 ms||5 ms||0.8 ms||0.001 ms|
|Pk-Pk Hi-Amp||2200 mV||2220 mV||2180~2200 mV||200 mV||80 mV||10 mV|
|Period Hi-Amp||5 ms||5.2 ms||5.110~5150 mV||5 ms||0.8 ms||0.001 ms|
|Pk-Pk Lo-Amp||60 mV||59.2 mV||60.0~63.2 mV||20 mV||3.2 mV||0.1 mV|
|Period Lo-Amp||5 ms||5.2 ms||5.110~5.170 ms||5 ms||0.8 ms||0.001 ms|
|Pk-Pk Offset||900 mV||880 mV||864~872 mV||100 mV||32 mV||1 mV|
|Period Offset||5 ms||5.2 ms||5.130~5.140 ms||5 ms||0.8 ms||0.001 ms|
|Fall Time||n/a||58 ms||49.40~56.00 ms||n/a||4 ms||0.01 ms|
|Time Constant||n/a||26.3 ms||24.0 ms||n/a||1.8 ms||1.5 ms|
I am satisfied with this result, since nothing seems to be too unexpected. I would attribute the fluctuations to random error. If I had more time, I might examine Fourier transform phenomena in the wave functions.I am still confused by the nature of capacitive coupling. SJK 03:10, 17 September 2008 (EDT)
Planck's Constant LabSJK 12:38, 20 October 2008 (EDT)
In summary, I measured the stopping potentials (taken to be the maximum potentials measured) in a photodiode of multiple wavelengths of light diffracted from a mercury lamp at multiple levels of intensity, as well as the time taken to achieve this stopping potential for some of these wavelengths.
For Photon Theory Part A,
|Green Wavelength||100% Intensity||80% Intensity||60% Intensity||40% Intensity||20% Intensity|
|Stopping Potential||0.886~0.887 V||0.886~0.887 V||0.875 V||0.873 V||0.871 V|
|Mean Time to Reset||48.20 s||58.64 s||38.00 s||46.89 s||91.08 s|
|Standard Deviation||5.94 s||3.82 s||1.90 s||0.34 s||8.50 s|
|Number of Measurements||5||5||3||3||3|
For Photon Theory Part B,
|Mean Stopping Potential||0.746 V||0.885 V||1.526 V||1.752 V||2.057 V|
|Standard Deviation||0.005 V||0.007 V||0.009 V||0.016 V||0.016 V|
|Number of Measurements||3||3||3||3||3|
Longer wavelengths of light have lower stopping potentials, while
higher shorter wavelengths have higher potentials, by extension, shorter wavelengths have higher energies.
This lab supports a photon-based model of light, since the wavelength (rather than the intensity) has the greatest effect on stopping potential by far.The slight drop in stopping potential is possibly due to the physical limitations of the photodiode used to capture wavelengths, which may not function properly for low intensities of light.SJK 12:07, 20 October 2008 (EDT)
For Determination of Planck's Constant,
|Mean Stopping Potential||0.731 V||1.296 V||1.544 V||1.755 V||2.091 V|
|Standard Deviation||0.003 V||0.013 V||0.004 V||0.001 V||0.004 V|
|Number of Measurements||3||3||3||3||3|
The slope of this linear fit is h/e (Planck's constant over elementary charge) and the y-intercept is W0/e (Work function over elementary charge).SJK 12:30, 20 October 2008 (EDT)
From these 6 data sets of V vs. ν, we obtain these values:
|Values||h||W0||σ (h)||σ (W0)|
|Divided by e||4.102e-15 J*s/C||1.283 J/C||3.2983-16 J*s/C||0.267 J/C|
|Raw Value||6.571e-34 J*s||0.205e-18 J||5.284e-35 J*s||4.285e-20 J|
Compared with the accepted value of 6.626e-34 J*s for h, this value obtained in the experiment is quite close, and is on exactly the right order of magnitude.
These results seem good, aside from the low-intensity anomalies in the first part of the experiment, through which error crept in. If I had more time, I might examine the intensity relationship for other wavelengths. I did take extra measurements in addition to what the experiment demanded, to get more accurate results.
Poisson Distribution LabSJK 02:18, 1 November 2008 (EDT)
In this lab, I measured counts of background radiation with a photomultiplier tube. I took multiple readings with multiple dwell times.SJK 02:08, 1 November 2008 (EDT)
|Dwell Time||10 ms||20 ms||40 ms||80 ms||100 ms||200 ms||400 ms||800 ms||1 s||2 s||4 s||4 s Revised*||8 s||8 s Revised*|
|Mean # Counts||0.1729||0.3223||0.6523||1.3262||1.6914||3.2832||6.7139||13.5625||17.0566||35.4111||3.7427e3||82.3503||331.6879||171.8591|
|σ # Counts||0.4057||0.5694||0.7479||1.0653||1.2175||1.4678||1.8282||2.3434||2.6681||3.5962||9.8137e3||12.3347||1.6070e3||44.8021|
|Square Root of Mean||0.4158||0.5677||0.8077||1.1516||1.3005||1.8120||2.5911||3.6827||4.1300||5.9507||61.1776||90.7471||1.8212e3||13.1095|
- The revised data have all anomalous readings removed. I have included analysis for original data for comparison.
The standard deviation is very close to the mean for small dwell times, but the quantities diverge for very large dwell times.
I am not entirely satisfied with these results: the standard deviations seem inaccurate for large dwell times.
If I had more time, I would compare these results with real Poisson distributions to check for consistency, and perform X2-analysis to find the probability distribution of the background radiation. I took far more sets of data than the original experiment required (12 as opposed to 3).
Charge to Mass Lab
In this lab, we activated a Helmholtz coil apparatus in order to measure the charge-to-mass ratio of an electron based on the deflection of an electron beam by the magnetic field generated from the Helmholtz coils, as well as the voltage used to accelerate the electron beam and the current used to generate that magnetic field.
We applied various combinations of voltages and currents until paths of electrons could be observed (left). Some of these paths glowed in the UV spectrum (center). Then the current and voltage were adjusted upward to obtain large circular paths (right), which were suitable to measure.
Using the formula (e/m) = 2*V/(B*r)^2 where
- V = accelerating voltage
- B = magnetic field strength = 7.8 * 10^4 Wb/(A m^2) * I
- I = current
- r = deflection radius,
we can calculate the following out of 12 sets of data:
|Mean Value||Standard Error of the Mean|
Therefore, we can say with 68% confidence that (e/m) = 3.829 ± 0.335 * 10^11.SJK 00:54, 3 November 2008 (EST)
We also qualitatively measured the effects of activating deflection plates in the Helmholtz coil. Applying a current to the deflection plates in the indicated direction, the electron beam is deflected upward; applying current in the opposite direction, the beam is deflected downward.
Lab Questions:1. We see the electron beam due to the bremsstrahlung effect, which exists due to the helium in the glass tube.SJK 02:00, 3 November 2008 (EST)
2. According to the NGDC, there is an effect of roughly 23 μT. Given the magnitude of our experimental field, we can ignore this.
3. If protons were emitted, the beam would always be deflected in the opposite direction.
4. According to Wikipedia frequency f = 1/t = B * (e/m), therefore t = 1/(B * (e/m)). For constant B, time t has no dependence on V.
5. According to the equation v = rB(e/m), velocity v = 0.05021±0.00190 *c. (Mean ± Standard Error of the Mean) For such speeds v << c, relativistic corrections would not matter much.
I acknowledge this website from which I got some equations.SJK 03:12, 3 November 2008 (EST)
If I had more time I would attempt to find the source of this error. I would also attempt to physically rotate the tube to determine the effects of deflecting the electron beam, and to further explore the effects of the Earth's magnetic field. I would also try to check my data against data taken against constant current, as well as data taken against constant voltage.
Further Analysis and NotesSJK 00:49, 3 November 2008 (EST)
The mean and standard error of the mean (SEM) values for voltage, current, magnetic field strength, and beam radius are included not for any intrinsic value, but because we originally believed they might have aided in the calculation of the SEM value for the charge-to-mass ratio through the principle of propagated error.
They were useful for computing the mean of the ratio. However, they were useless for the SEM since the values of voltage, current, and beam radius are not independent. As such, traditional formulas for error propagation for independent variables do not apply, and since we have all values in the computer, we can compute the SEM directly (which we ultimately did).
The mean of e/m ratios was calculated from inputting the means of the variables into the equation for e/m. The SEM of e/m was calculated from the (automatic) standard deviation function applied over each individual e/m ratio as calculated from corresponding sets of voltage, current, and beam radius, divided by the square root of the number of ratios. (All other SEM values were calculated in a similar way; the standard deviation of the set of all values, divided by the square root of the size of the data set).
The MATLAB M-file used to compute all quantities is here:
function RatioLab format long;clc V = [268.4 244.7 290.4 264.2 289.4 283.6 231.7 292.2 255.1 281.2]; Vmean = mean(V) I = [1.283 1.208 1.180 1.409 2.099 1.330 1.002 1.418 1.380 1.265]; Imean = mean(I) r = [4.0 3.5 3.8 4.2 4.0 3.6 3.9 4.1 4.6 3.6 4.5 4.7 3.7 3.5 3.0 3.3 2.0 1.7 1.8 2.0 4.1 3.5 3.9 4.1 3.9 3.6 4.1 4.4 3.6 3.4 3.3 3.5 2.7 2.5 2.6 2.9 4.3 3.8 4.0 4.2]; Bmean = 7.8e-4 * Imean rmean = mean(r)/100 ratiomean = 2 * Vmean/((Bmean * rmean)^2) % mean VSEM = std(V)/sqrt(length(V)) ISEM = std(I)/sqrt(length(I)) rSEM = std(r)/sqrt(length(r)) BSEM = 7.8e-4 * std(I)/sqrt(length(I)) r = [4.0 3.5 3.8 4.2;4.0 3.6 3.9 4.1;4.6 3.6 4.5 4.7 ;3.7 3.5 3.0 3.3 ;2.0 1.7 1.8 2.0 ;4.1 3.5 3.9 4.1; 3.9 3.6 4.1 4.4; 3.6 3.4 3.3 3.5 ;2.7 2.5 2.6 2.9; 4.3 3.8 4.0 4.2]; ratio = 2 .* V./(((7.8e-4 .* I) .* mean(r'/100)).^2) ratioSEM = std(ratio)/sqrt(length(ratio)) % standard error of the mean vel = mean(r'/100) .* (7.8e-4 .* I) .* ratio velmean = mean(vel) velC = velmean/3.0e8 velSEM = std(vel)/sqrt(length(vel)) velCSEM = velSEM/3.0e8
Balmer Series Lab
In this lab, we observed the wavelengths of light emitted from various elemental lamps in order to determine the Rydberg constant.
First, we had to calibrate the spectrometer by shining a mercury lamp at it, and comparing the measured wavelengths with accepted values. Then we measured the visible second-order emmission spectra from regular hydrogen and deuterium lamps (also known as the "Balmer series") with the spectrometer in order to determine the Rydberg constant, as well as a krypton lamp to determine how well our equipment could resolve.
Our spectrometer seemed to be able to visibly resolve wavelengths within ~1 nm of each other and no closer. We determined this by observing the violet end of the krypton spectrum.
The value for the Rydberg constant for hydrogen was measured as 10964446.07 ± 7548.89 1/m (mean ± SEM). Compared with the official value of 10967758 1/m, we find an astonishingly low 0.0302% error (thus we note that the mean value is within one standard deviation the accepted value).
The value for the Rydberg constant for deuterium was calculated as 11055173 1/m, which has a 0.7970% difference from the Rydberg constant for hydrogen. Qualitatively, however, there is no clear difference between the spectra for regular hydrogen and deuterium.
I am satisfied with the results; I obtained a value very close to the canonical value for the Rydberg constant with multiple measurements, with small error that appears random.
Although I did not have time to calculate the precise canonical differences in second-order wavelengths from hydrogen to deuterium, I was able to take 3 sets of measurements each.
Astonishingly, all sets of measurements were extremely consistent; I obtained the same values for wavelengths on each of 3 measurements of both the hydrogen and deuterium spectra. I do not know why that is the case.