User:Garrett E. McMath/Notebook/Junior Lab/2008/11/17: Difference between revisions
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<!--{{SJK Comment|l=02:47, 8 December 2008 (EST)|c=Many of the parts you have here are very good. However, as noted below there are significant things missing (for example, figures & references!). Please make sure to read all these comments and ask me for any clarifications.}}--> | <!--{{SJK Comment|l=02:47, 8 December 2008 (EST)|c=Many of the parts you have here are very good. However, as noted below there are significant things missing (for example, figures & references!). Please make sure to read all these comments and ask me for any clarifications.}}--> | ||
<!--{{SJK Comment|l=17:32, 15 December 2008 (EST)|c=Your improvements to your final draft are very good -- I really like your final version! Due to the end-of-semester time crunch, my feedback is complete, but I do have a few comments below. Nice work!}}--> | |||
<br style="clear:both;"/> | <br style="clear:both;"/> | ||
=Estimating Planck's Constant via the Photoelectric Effect= | =Estimating Planck's Constant via the Photoelectric Effect= | ||
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==Equations== | ==Equations== | ||
<!--{{SJK Comment|l=17:05, 15 December 2008 (EST)|c=I like your revised abstract above. An "equations" section is not typical ... they would normally be worked into the methods or possibly a "theory" section if the equations were novel.}}--> | |||
<math> Vo=\frac{h}{e}f - \frac{W_{o}}{e} </math> | <math> Vo=\frac{h}{e}f - \frac{W_{o}}{e} </math> | ||
*(where Vo is the stopping potential, e is the fundamental electron charge, f is the frequency of the photon, h is Planck's constant, and Wo is an intrinsic property of the material called the work function) | *(where Vo is the stopping potential, e is the fundamental electron charge, f is the frequency of the photon, h is Planck's constant, and Wo is an intrinsic property of the material called the work function) | ||
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==Introduction== | ==Introduction== | ||
<!--{{SJK Comment|l=02:29, 8 December 2008 (EST)|c=Also for this section, you will need citations to original references. The last sentence is ambiguous with the term "voltage difference" (you probably mean energy difference?). Combined with the above, you have a really good background part of the introduction. There are a couple things to add to make this a very solid introduction: (1) information about how Planck's constant is precisely measured to produce today's accepted value (NIST CODATA is a place to go for this info). (2) a concluding part of the introduction where you mention what you will be reporting in this paper as a way of leading in to the rest of the paper.}}--> | <!--{{SJK Comment|l=02:29, 8 December 2008 (EST)|c=Also for this section, you will need citations to original references. The last sentence is ambiguous with the term "voltage difference" (you probably mean energy difference?). Combined with the above, you have a really good background part of the introduction. There are a couple things to add to make this a very solid introduction: (1) information about how Planck's constant is precisely measured to produce today's accepted value (NIST CODATA is a place to go for this info). (2) a concluding part of the introduction where you mention what you will be reporting in this paper as a way of leading in to the rest of the paper.}}--> | ||
<!--{{SJK Comment|l=17:16, 15 December 2008 (EST)|c=I like your improvements to the introduction...it introduces your research very well.}}--> | |||
This experiment relies heavily on the physics of the photoelectric effect. The photoelectric effect, discovered by accident by Heinrich Hertz(1857-1894) in 1887, was explained by Einstein in his Annalen der Physik'''[2]'''. Einstein stated that the energy quantization used by Planck in solving the Rayleigh-Jeans problem (known as the ultraviolet catastrophe) defined light energy as discrete quanta each with energy hf.'''[3]''' These quanta, known as photons, can be completely absorbed by electrons. With enough energy, a photon can raise the energy of the electron that absorbs it so much that it is ejected from the metal with some kinetic energy. The minimum amount of energy needed to eject an electron is known as the work function and is specific to the metal the electron is in. Therefore the maximum kinetic energy of an ejected electron is the energy of the photon it absorbed(hf) minus the work function of the metal. The main equation that allows us to measure Planck's constant is obtained by setting this maximum kinetic energy equal to a potential multiplied with the fundamental charge of an electron. When the two sides of the equation are equal that potential is what is called the stopping potential because in the experiment what is measured is the voltage difference caused by the ejected electrons. The electrons are ejected into a negative potential difference which doesn't allow any electrons to reach the anode unless they have a kinetic energy greater than the potential. Thus if the energy difference is less than the stopping potential no electrons will reach the anode. Modern determinations of Planck's constant are extremely accurate and are far more advanced than the scope of this lab. The modern accepted value as recorded by the National Institute of Standards and Technology (NIST) is obtained primarily from a method known as the Watt Balance. A Watt Balance is an apparatus that measures two powers one in watts and the other in standard electrical units which produces the measure of their product. Using a value known as the von Klitzing constant (used in quantum Hall effects) this produces a direct measure of Planck's Constant from the equation'''[4]''', where ''R''<sub>K</sub> = ''h''/''e''<sup>2</sup>, | This experiment relies heavily on the physics of the photoelectric effect. The photoelectric effect, discovered by accident by Heinrich Hertz(1857-1894) in 1887, was explained by Einstein in his Annalen der Physik'''[2]'''. Einstein stated that the energy quantization used by Planck in solving the Rayleigh-Jeans problem (known as the ultraviolet catastrophe) defined light energy as discrete quanta each with energy hf.'''[3]''' These quanta, known as photons, can be completely absorbed by electrons. With enough energy, a photon can raise the energy of the electron that absorbs it so much that it is ejected from the metal with some kinetic energy. The minimum amount of energy needed to eject an electron is known as the work function and is specific to the metal the electron is in. Therefore the maximum kinetic energy of an ejected electron is the energy of the photon it absorbed(hf) minus the work function of the metal. The main equation that allows us to measure Planck's constant is obtained by setting this maximum kinetic energy equal to a potential multiplied with the fundamental charge of an electron. When the two sides of the equation are equal that potential is what is called the stopping potential because in the experiment what is measured is the voltage difference caused by the ejected electrons. The electrons are ejected into a negative potential difference which doesn't allow any electrons to reach the anode unless they have a kinetic energy greater than the potential. Thus if the energy difference is less than the stopping potential no electrons will reach the anode. Modern determinations of Planck's constant are extremely accurate and are far more advanced than the scope of this lab. The modern accepted value as recorded by the National Institute of Standards and Technology (NIST) is obtained primarily from a method known as the Watt Balance. A Watt Balance is an apparatus that measures two powers one in watts and the other in standard electrical units which produces the measure of their product. Using a value known as the von Klitzing constant (used in quantum Hall effects) this produces a direct measure of Planck's Constant from the equation'''[4]''', where ''R''<sub>K</sub> = ''h''/''e''<sup>2</sup>, | ||
:<math>h = \frac{4}{K_{\rm J}^2 R_{\rm K}}</math> | :<math>h = \frac{4}{K_{\rm J}^2 R_{\rm K}}</math> | ||
While our experiment had no chance of duplicating the accuracy of a the Watt Balance, We measured Planck's constant using the much of the same physics as were used in its original determination. | It is worth noting however that before all the new age physics, Planck managed to calculate the value to within 1.2% of today's accepted value using nothing more than blackbody radiation data and some statistical mechanics'''[7]'''. While our experiment had no chance of duplicating the accuracy of a the Watt Balance, We measured Planck's constant using the much of the same physics as were used in its original determination. | ||
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<!--{{SJK Comment|l=02:32, 8 December 2008 (EST)|c=As I have been mentioning on other students' reports, lists of equipment are uncommon in research publications...probably due to history of limited page space. You should merge this section with the next by mentioning the equipment as you describe the procedure. Since many things come from PASCO, you can say, "All equipment from PASCO (CITYNAME) unless otherwise specified." And as an example, you can say, "we followed a procedure as described in reference ___ (cite the PASCO manual in your reference list). There were three separate methods: blah. Also, you will want to briefly describe the quantitative analysis methods in the methods section. For example, "Data were fit using the LINEST function in Excel (Microsoft, Redmond WA).}}--> | <!--{{SJK Comment|l=02:32, 8 December 2008 (EST)|c=As I have been mentioning on other students' reports, lists of equipment are uncommon in research publications...probably due to history of limited page space. You should merge this section with the next by mentioning the equipment as you describe the procedure. Since many things come from PASCO, you can say, "All equipment from PASCO (CITYNAME) unless otherwise specified." And as an example, you can say, "we followed a procedure as described in reference ___ (cite the PASCO manual in your reference list). There were three separate methods: blah. Also, you will want to briefly describe the quantitative analysis methods in the methods section. For example, "Data were fit using the LINEST function in Excel (Microsoft, Redmond WA).}}--> | ||
[[Image:Apparatus.jpg |thumb|left|'''Figure 1''': PASCO h/e Apparatus Setup'''[ | [[Image:Apparatus.jpg |thumb|left|'''Figure 1''': PASCO h/e Apparatus Setup'''[1]''']] | ||
[[Image:Schematic.jpg |thumb|left|'''Figure 1''': Schematic diagram of PASCO h/e Apparatus'''[1]''']] | |||
<!--{{SJK Comment|l=17:23, 15 December 2008 (EST)|c=You would want to have analysis as part of your methods--i.e. describing the use of Excel for your fitting (and specific algorithms such as LINEST)...this would be in place of the "excel is a microsoft application" that you have in your figure legends<br><br>Oh, I now see that you describe this below. That is good, but it would be better in the methods section, not mixed in with the results (though that is OK, and probably commonly done).}}--> | |||
The overall goal of this lab was to establish the quantum nature of light and to accurately measure Planck's Constant. This was accomplished through the following experiments. | The overall goal of this lab was to establish the quantum nature of light and to accurately measure Planck's Constant. This was accomplished through the following experiments. | ||
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#Measurement of stopping potential with variable order and frequency | #Measurement of stopping potential with variable order and frequency | ||
All the equipment used for the experiments were PASCO apparatuses, except for a Wavetek 85XT RMS DVM multimeter, and can be seen in the | All the equipment used for the experiments were PASCO apparatuses, except for a Wavetek 85XT RMS DVM multimeter, and can be seen in the PASCO Manual for Apparatus'''[1]''' along with the procedure followed for each experiment. Each experiment proved different aspects of the quantum nature of light while the measurement of Planck's Constant was found through data analysis of the data from the 2nd and 3rd experiments. | ||
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<!--{{SJK Comment|l=02:39, 8 December 2008 (EST)|c=this is good discussion of the data. The link to the raw data would be good to include as an appendix. (In most of today's publications any raw data would be linked as "supplemental online material.") So, what's missing here is a nice table or graph of the data -- something to make it easy for the reader to see your data. You will want to number the table or figure and provide a detailed caption so the reader can understand the table or caption without having to search through the text. Then in the text you will also want to refer to the table or figure by number and describe what it is.}}--> | <!--{{SJK Comment|l=02:39, 8 December 2008 (EST)|c=this is good discussion of the data. The link to the raw data would be good to include as an appendix. (In most of today's publications any raw data would be linked as "supplemental online material.") So, what's missing here is a nice table or graph of the data -- something to make it easy for the reader to see your data. You will want to number the table or figure and provide a detailed caption so the reader can understand the table or caption without having to search through the text. Then in the text you will also want to refer to the table or figure by number and describe what it is.}}--> | ||
[[Image:ICT1.jpg |thumb|left|'''Figure 1a''': Intensity of yellow light vs. mean charge time. Three different trend lines and their R-squared values have been added: linear, exponential, and power. Intensity was was controlled by a filter with five sections corresponding to 20%, 40%, 60%, 80%, and 100%. The filter used computer generated dots to block out a certain amount of light. A filter was also used to allow only the yellow spectrum of light through]] | [[Image:ICT1.jpg |thumb|left|'''Figure 1a''': Intensity of yellow light vs. mean charge time. Three different trend lines and their R-squared values have been added: linear, exponential, and power. Intensity was was controlled by a filter with five sections corresponding to 20%, 40%, 60%, 80%, and 100%. The filter used computer generated dots to block out a certain amount of light. A filter was also used to allow only the yellow spectrum of light through. (Excel is a Microsoft® based application)]] | ||
[[Image:ICT2.jpg |thumb|left|'''Figure 1b''': Intensity of green light vs. mean charge time. Three different trend lines and their R-squared values have been added: linear, exponential, and power. Intensity was was controlled by a filter with five sections corresponding to 20%, 40%, 60%, 80%, and 100%. The filter used computer generated dots to block out a certain amount of light. A filter was also used to allow only the green spectrum of light through]] | [[Image:ICT2.jpg |thumb|left|'''Figure 1b''': Intensity of green light vs. mean charge time. Three different trend lines and their R-squared values have been added: linear, exponential, and power. Intensity was was controlled by a filter with five sections corresponding to 20%, 40%, 60%, 80%, and 100%. The filter used computer generated dots to block out a certain amount of light. A filter was also used to allow only the green spectrum of light through. (Excel is a Microsoft® based application)]] | ||
====Experiment 1-Qualitative==== | ====Experiment 1-Qualitative==== | ||
<!--{{SJK Comment|l=17:21, 15 December 2008 (EST)|c=Actually the effect on charging ''initial charging '''rate''''' should be proportional to intensity (not time). You're not really measuring initial rate...but if you were, you'd expect the power to be the best fit (inverse of x).}}--> | |||
Experiment 1 proved the non classical result that the stopping potential i.e. the maximum kinetic energy of the electrons is not related to intensity. At each of the five intensities the stopping potential was unaffected other than the small charge leak due to the non infinite impedance in the apparatus. The experiment also proved that the charge time is affected by intensity. We know from physics that the affect on charge time should be linear however as seen in the figures 1a and 1b ours was more exponential we believe this is due to the charge leak, and we could not find any reasonable information on the apparatus to allow us to calculate this loss and therefore was not adjusted. The overall conclusion was the intensity of light affects how many electrons are ejected from metal but not the speed they are ejected at. | Experiment 1 proved the non classical result that the stopping potential i.e. the maximum kinetic energy of the electrons is not related to intensity. At each of the five intensities the stopping potential was unaffected other than the small charge leak due to the non infinite impedance in the apparatus. The experiment also proved that the charge time is affected by intensity. We know from physics that the affect on charge time should be linear however as seen in the figures 1a and 1b ours was more exponential we believe this is due to the charge leak, and we could not find any reasonable information on the apparatus to allow us to calculate this loss and therefore was not adjusted. The overall conclusion was the intensity of light affects how many electrons are ejected from metal but not the speed they are ejected at. | ||
====Experiment 2 and 3-Quantitative==== | ====Experiment 2 and 3-Quantitative==== | ||
Experiments 2 and 3 involved the measuring of the stopping potential of the different spectra of mercury in both the first and second order spectra. Data analysis provided the Planck's constant and work function. Comparing the first and second order provided evidence that the green spectra in the second order was being overlapped/corrupted with another band of light. The PASCO manual confirmed this suspicion stating that the frequency of the second order green band is interfered by an ultraviolet band of the third order. | |||
[[Image:SPF1.jpg |thumb|left|'''Figure 2a''': Mean stopping potentials for each interference fringe are shown, with the SEM of each point represented by error bars. A least squares line is fit to the data without constraints. In addition, a slightly modified line is also shown, which was generated by moving the least squares line within, or in close proximity, of the error bars. (Excel is a Microsoft® based application)]] | |||
==Data Analysis== | ==Data Analysis== | ||
<!--{{SJK Comment|l=17:27, 15 December 2008 (EST)|c=Just a quick note that you have too many digits on your measured values, which makes them difficult to read.}}--> | |||
*'''Work function:''' 1.36(08)eV'''[6]''' | *'''Work function:''' 1.36(08)eV'''[6]''' | ||
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|} | |} | ||
* (Wavelengths'''[1]''', frequencies calculated using c/λ (c=speed of light, λ=wavelength)) | * (Wavelengths'''[1]''', frequencies calculated using c/λ (c=speed of light, λ=wavelength)) | ||
[[Image:SDF.jpg |thumb|left|'''Figure 3''': Standard Deviation vs Frequency, A plot of the standard deviation of each point plotted against the frequency corresponding to that point. A very rough trend showing that the data got less precise as the frequency got higher. (Excel is a Microsoft® based application)]] | |||
The analysis of the data was done in Excel | The analysis of the data was done in Excel (Excel is a Microsoft® based application). The analysis involved taking averages of our stopping potential data, since we performed the experiment several times for each frequency, and making a scatter plot of the average stopping potential vs the corresponding frequency of light. A best fit linear line was was put on the graph and its R-squared value to show the linearity of this relationship. The actual Planck's constant and uncertainty values were calculated using Excel's LINEST function on the average stopping potential and frequencies. The results are as follows: | ||
{|border="1" | {|border="1" | ||
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*(Accepted value for Planck's Constant'''[5]''', Accepted value for work function'''[6]''') | *(Accepted value for Planck's Constant'''[5]''', Accepted value for work function'''[6]''') | ||
Clearly we did not achieve the optimal results. Our data did not encompass the accepted values | Clearly we did not achieve the optimal results. Our data did not encompass the accepted values even at the most extreme ranges of both uncertainties. However, given the systematic error we could not account for (i.e. the charge leak and multimeter errors) the data was within our expectations. Figure 3 show the standard error of the data sets vs the frequency they were taken at. It shows a very rough trend that at higher frequency we got larger standard deviations meaning our data was getting less precise. Obviously the experiment must be done differently or with a correction for the charge loss in order for the data to within any reasonable confidence interval. | ||
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<!--{{SJK Comment|l=02:44, 8 December 2008 (EST)|c=I don't really understand this first sentence. Also, any comment about accuracy is difficult when your comparison to the accepted value is missing (as mentioned above).}}--> | <!--{{SJK Comment|l=02:44, 8 December 2008 (EST)|c=I don't really understand this first sentence. Also, any comment about accuracy is difficult when your comparison to the accepted value is missing (as mentioned above).}}--> | ||
This lab was simple in nature and because of that it was very possible to eliminate human sources of error. However the price you pay is its nearly impossible to correct for the systematic error inherent in the apparatus. The only errors that make a significant difference in this lab are the charge leak from the op-amp in the apparatus and the always present multimeter correction for non infinite impedance. I attempted a side experiment in an attempt to find a way to correct for the charge leak. The idea behind the experiment was that given longer times of charging, the leak would become more and more a factor. Therefore it seemed that by doing the stopping potential experiment at different intensities and graphing the results would produce a line whose slope would be used as a correction factor for the leak. | This lab was simple in nature and because of that it was very possible to eliminate human sources of error. However the price you pay is its nearly impossible to correct for the systematic error inherent in the apparatus. The only errors that make a significant difference in this lab are the charge leak from the op-amp in the apparatus and the always present multimeter correction for non infinite impedance. I attempted a side experiment in an attempt to find a way to correct for the charge leak. The idea behind the experiment was that given longer times of charging, the leak would become more and more a factor. Therefore it seemed that by doing the stopping potential experiment at different intensities and graphing the results would produce a line whose slope would be used as a correction factor for the leak. The experiment failed to give such a result. I did not have sufficient enough time to explain the results, but they were basically the exact opposite of what we had hoped would happen. The percent error in the calculation of Planck's constant actually improved when less intensity was used in the data acquisition, instead of worsening as hoped. I could find no plausible physics reason for this, it seems to be part of the systematic error of the lab.<!--{{SJK Comment|l=17:30, 15 December 2008 (EST)|c=I am really glad you carried out this experiment and reported the unexpected result. I am still very curious and your experiment may lead a future student to discover some unknown way of eliminating the systematic error.}}--> The conclusion reached is that there is unequivocal evidence for the quantum nature of light and that given an accurate measure of Planck's Constant, we can describe the quantization of the energy in that light. These conclusions though somewhat elementary in this day and age are still some of the most profound and influential to modern physics as we know it. | ||
<br style="clear:both;"/> | <br style="clear:both;"/> | ||
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<!--{{SJK Comment|l=02:04, 8 December 2008 (EST)|c=Good job on this section.}}--> | <!--{{SJK Comment|l=02:04, 8 December 2008 (EST)|c=Good job on this section.}}--> | ||
My thanks to Paul Klimov for his help in | My thanks to Paul Klimov for his help in performing these experiments and in the valuable information gained from our discussions of the results. Also much appreciation must be given to Dr. Steven Koch and Aram Gragossian for their help in understanding the safety aspects of the lab as well as invaluable information gained from talking about the inner workings of the PASCO apparatus. | ||
==References== | ==References== | ||
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# Ayar, Eric. Griffith, Dave (editor). h/e apparatus and h/e apparatus accessory kit. ''Instruction Manual and Experiment guide for the PASCO scientific Mode AP9368 and AP 9369''. Roseville, CA: PASCO Scientific. 1989. | # Ayar, Eric. Griffith, Dave (editor). h/e apparatus and h/e apparatus accessory kit. ''Instruction Manual and Experiment guide for the PASCO scientific Mode AP9368 and AP 9369''. Roseville, CA: PASCO Scientific. 1989. | ||
# Einstein, Albert. On a Heuristic Viewpoint Concerning the Production and Transformation of light. ''Annalen Der Physik'' 17: 1905. | # Einstein, Albert. On a Heuristic Viewpoint Concerning the Production and Transformation of light. ''Annalen Der Physik'' 17: 1905.<!--{{SJK Comment|l=17:13, 15 December 2008 (EST)|c=Presuming you didn't read the original, it would be helpful to link to the english translation}}--> | ||
#Llewellyn, Ralph A;Tipler, Paul A. ''Modern Physics''. 5th ed New York, NY: W.H. Freeman and Company. pp119-132. 2008. | #Llewellyn, Ralph A;Tipler, Paul A. ''Modern Physics''. 5th ed New York, NY: W.H. Freeman and Company. pp119-132. 2008. | ||
#Mohr, Peter J; Taylor, Barry N; Newell, David B. ''CODATA Recommended Values of the Fundamental Physical Constants''. Reviews of Modern Physics 8:633-730. 2006 | #Mohr, Peter J; Taylor, Barry N; Newell, David B. ''CODATA Recommended Values of the Fundamental Physical Constants''. Reviews of Modern Physics 8:633-730. 2006 | ||
#''Physics Laboratory''. Jun.1994/Apr. 2008 National Institute of Standards and Technology. 14 Dec. 2008 <http://physics.nist.gov> | #''Physics Laboratory''. Jun.1994/Apr. 2008 National Institute of Standards and Technology. 14 Dec. 2008 <<nowiki>http://physics.nist.gov</nowiki>> | ||
#''Tech Note 303 Detail''. 5 Dec. 2001/10 Oct. 2003. PASCO 10 Dec. 2008 <http://www.pasco.com/support/technical-support/technote/techidlookup.cfm?technoteid=303> | #''Tech Note 303 Detail''. 5 Dec. 2001/10 Oct. 2003. PASCO 10 Dec. 2008 <<nowiki>http://www.pasco.com/support/technical-support/technote/techidlookup.cfm?technoteid=303</nowiki>> | ||
#Planck, Max. On the Law of | #Planck, Max. On the Law of Distribution of Energy in the Normal Spectrum. ''Annalen der Physik'' 4: 553. 1901 | ||
==Links== | |||
*[[User:Garrett E. McMath/Notebook/Junior Lab/2008/11/17/Data1|Experiment 1 Data]] | |||
*[[User:Garrett E. McMath/Notebook/Junior Lab/2008/11/17/Data2|Experiment 2 & 3 Data]] | |||
*[http://www.pasco.com/support/technical-support/technote/techidlookup.cfm?technoteid=303 PASCO material and work function information] | |||
*[http://physics.nist.gov National Institute of Standards and Technology] | |||
*[ftp://ftp.pasco.com/Support/Documents/English/AP/AP-9368/012-04049j.pdf PASCO Manual for Apparatus] | |||
*[http://physics.nist.gov/cuu/Constants/codata.pdf CODATA Recommended Values of the Fundamental Physical Constants] | |||
*[http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html Max Planck: On the Law of Distribution of Energy in the Normal Spectrum] | |||
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Revision as of 13:24, 22 December 2008
 Planck's Constant and the Photoelectric Effect
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Estimating Planck's Constant via the Photoelectric Effect
AbstractPlanck's Constant is one of the most fundamental quantities in all of physics, and essential for quantum mechanics. In this experiment we set up a way to measure this fundamental value with reasonable accuracy. We used a commercial apparatus to estimate Planck's constant via the photoelectric effect from varying frequencies of light. In so doing we were able to achieve results that correlated with the accepted value with only 9% error. Given the rudimentary nature of the lab setup we felt this was a very good estimate of Planck's constant given the equipment we had to work with.
Equations[math]\displaystyle{ Vo=\frac{h}{e}f - \frac{W_{o}}{e} }[/math]
to determine Planck's constant h. This is done by the fact that when Vo and f are graphed in Cartesian coordinates they will produce a straight line with a slope equal to Planck's constant and an intercept equal to the work function of the material.
HistoryMax Planck, the originator of Planck's Constant, was a German physicist who lived from 1858 to 1947. In the year 1900 he announced that by making some then strange assumptions he had found a solution to the problem of the Rayleigh-Jeans equation at short wavelengths. The Rayleigh-Jeans equation was used at the time to describe the energy density distribution function of light. This equation was very accurate at long wavelengths but experimentally fell apart at shorter wavelengths. This was due to the fact that the equation had been derived using classical mechanics, that is that the average energy per mode of oscillation is same as a harmonic oscillator in one dimension. The problem is that this predicts that as the wavelength approaches zero the energy density distribution goes to infinity while experiments had shown it actually approaches zero. Planck remedied this problem by assuming that the average energy of the oscillating charges and conversely the radiation they produce was a discrete variable meaning they had to take values of 0,ε,2ε,3ε,...nε where n=0,1,2,3... This meant that Planck could now write the energy as [math]\displaystyle{ E_n=n{ε}=nhf }[/math]. This idea that energy was quantized spurred quantum mechanics/modern physics as we know it. The actual idea was actually brought to full fruition by Einstein in 1905 when he used Planck's ideas to both explain the photoelectric effect as well as postulate that this quantization was a fundamental characteristic of light and not some mysterious property of oscillators in cavity walls.[3] IntroductionThis experiment relies heavily on the physics of the photoelectric effect. The photoelectric effect, discovered by accident by Heinrich Hertz(1857-1894) in 1887, was explained by Einstein in his Annalen der Physik[2]. Einstein stated that the energy quantization used by Planck in solving the Rayleigh-Jeans problem (known as the ultraviolet catastrophe) defined light energy as discrete quanta each with energy hf.[3] These quanta, known as photons, can be completely absorbed by electrons. With enough energy, a photon can raise the energy of the electron that absorbs it so much that it is ejected from the metal with some kinetic energy. The minimum amount of energy needed to eject an electron is known as the work function and is specific to the metal the electron is in. Therefore the maximum kinetic energy of an ejected electron is the energy of the photon it absorbed(hf) minus the work function of the metal. The main equation that allows us to measure Planck's constant is obtained by setting this maximum kinetic energy equal to a potential multiplied with the fundamental charge of an electron. When the two sides of the equation are equal that potential is what is called the stopping potential because in the experiment what is measured is the voltage difference caused by the ejected electrons. The electrons are ejected into a negative potential difference which doesn't allow any electrons to reach the anode unless they have a kinetic energy greater than the potential. Thus if the energy difference is less than the stopping potential no electrons will reach the anode. Modern determinations of Planck's constant are extremely accurate and are far more advanced than the scope of this lab. The modern accepted value as recorded by the National Institute of Standards and Technology (NIST) is obtained primarily from a method known as the Watt Balance. A Watt Balance is an apparatus that measures two powers one in watts and the other in standard electrical units which produces the measure of their product. Using a value known as the von Klitzing constant (used in quantum Hall effects) this produces a direct measure of Planck's Constant from the equation[4], where RK = h/e2,
It is worth noting however that before all the new age physics, Planck managed to calculate the value to within 1.2% of today's accepted value using nothing more than blackbody radiation data and some statistical mechanics[7]. While our experiment had no chance of duplicating the accuracy of a the Watt Balance, We measured Planck's constant using the much of the same physics as were used in its original determination.
Procedure/Setup  The overall goal of this lab was to establish the quantum nature of light and to accurately measure Planck's Constant. This was accomplished through the following experiments.
All the equipment used for the experiments were PASCO apparatuses, except for a Wavetek 85XT RMS DVM multimeter, and can be seen in the PASCO Manual for Apparatus[1] along with the procedure followed for each experiment. Each experiment proved different aspects of the quantum nature of light while the measurement of Planck's Constant was found through data analysis of the data from the 2nd and 3rd experiments.
Experiments  Experiment 1-QualitativeExperiment 1 proved the non classical result that the stopping potential i.e. the maximum kinetic energy of the electrons is not related to intensity. At each of the five intensities the stopping potential was unaffected other than the small charge leak due to the non infinite impedance in the apparatus. The experiment also proved that the charge time is affected by intensity. We know from physics that the affect on charge time should be linear however as seen in the figures 1a and 1b ours was more exponential we believe this is due to the charge leak, and we could not find any reasonable information on the apparatus to allow us to calculate this loss and therefore was not adjusted. The overall conclusion was the intensity of light affects how many electrons are ejected from metal but not the speed they are ejected at. Experiment 2 and 3-QuantitativeExperiments 2 and 3 involved the measuring of the stopping potential of the different spectra of mercury in both the first and second order spectra. Data analysis provided the Planck's constant and work function. Comparing the first and second order provided evidence that the green spectra in the second order was being overlapped/corrupted with another band of light. The PASCO manual confirmed this suspicion stating that the frequency of the second order green band is interfered by an ultraviolet band of the third order.  Data Analysis
Accepted values of Mercury spectra
The analysis of the data was done in Excel (Excel is a Microsoft® based application). The analysis involved taking averages of our stopping potential data, since we performed the experiment several times for each frequency, and making a scatter plot of the average stopping potential vs the corresponding frequency of light. A best fit linear line was was put on the graph and its R-squared value to show the linearity of this relationship. The actual Planck's constant and uncertainty values were calculated using Excel's LINEST function on the average stopping potential and frequencies. The results are as follows:
Clearly we did not achieve the optimal results. Our data did not encompass the accepted values even at the most extreme ranges of both uncertainties. However, given the systematic error we could not account for (i.e. the charge leak and multimeter errors) the data was within our expectations. Figure 3 show the standard error of the data sets vs the frequency they were taken at. It shows a very rough trend that at higher frequency we got larger standard deviations meaning our data was getting less precise. Obviously the experiment must be done differently or with a correction for the charge loss in order for the data to within any reasonable confidence interval.
ConclusionsThis lab was simple in nature and because of that it was very possible to eliminate human sources of error. However the price you pay is its nearly impossible to correct for the systematic error inherent in the apparatus. The only errors that make a significant difference in this lab are the charge leak from the op-amp in the apparatus and the always present multimeter correction for non infinite impedance. I attempted a side experiment in an attempt to find a way to correct for the charge leak. The idea behind the experiment was that given longer times of charging, the leak would become more and more a factor. Therefore it seemed that by doing the stopping potential experiment at different intensities and graphing the results would produce a line whose slope would be used as a correction factor for the leak. The experiment failed to give such a result. I did not have sufficient enough time to explain the results, but they were basically the exact opposite of what we had hoped would happen. The percent error in the calculation of Planck's constant actually improved when less intensity was used in the data acquisition, instead of worsening as hoped. I could find no plausible physics reason for this, it seems to be part of the systematic error of the lab. The conclusion reached is that there is unequivocal evidence for the quantum nature of light and that given an accurate measure of Planck's Constant, we can describe the quantization of the energy in that light. These conclusions though somewhat elementary in this day and age are still some of the most profound and influential to modern physics as we know it.
AcknowledgmentsMy thanks to Paul Klimov for his help in performing these experiments and in the valuable information gained from our discussions of the results. Also much appreciation must be given to Dr. Steven Koch and Aram Gragossian for their help in understanding the safety aspects of the lab as well as invaluable information gained from talking about the inner workings of the PASCO apparatus. References
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