Physics307L:People/Ierides/

Planck's Constant

In this lab we were to achieve a measured value for Planck's Constant [math]\displaystyle{ h\,\! }[/math] using an h/e apparatus combined with a mercury light source shining beams of different colored light through the filters as seen in my lab report. From our observations we were to confirm the independence of the energy stored in light from the light's intensity and the dependence of the light's energy on it's frequency. The main overview and summary of the procedure and lab is provided by this link:

Prof. Gold's Lab Manual

The following link leads to my Planck's Constant Lab entry that includes the data, procedure, and observations that were made:

Planck's Constant Lab Notebook Entry

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Data Summary

• Derivations & Calculations (from lab notebook): [1]
  

The total maximum energy of the electrons leaving the cathode is:

[math]\displaystyle{ E =h \nu= KE_{max} + W_0 \,\! }[/math]

[math]\displaystyle{ KE_{max}=\frac{1}{2}m_ev^2 }[/math]

where [math]\displaystyle{ E=h\nu\,\! }[/math] is the initial energy of the photon and [math]\displaystyle{ E=KE_{max}+W_0\,\! }[/math] is the resulting energy containing the final kinetic energy of the electron plus the energy loss due to the electron overcoming the work function; :[math]\displaystyle{ m_e\,\! }[/math] is the rest mass of the electron and [math]\displaystyle{ v\,\! }[/math] is its final velocity.

The negative potential, [math]\displaystyle{ V_s\,\! }[/math], needed to stop the flow of electrons is derived by equating the potential barrier, :[math]\displaystyle{ eV_s\,\! }[/math], to the electron's kinetic energy where [math]\displaystyle{ e\,\! }[/math] is the charge of an electron and:

[math]\displaystyle{ eV_s=KE_{max}\,\! }[/math]

So

[math]\displaystyle{ E=eV_s+W_0=h\nu\,\! }[/math]

[math]\displaystyle{ eV_s=h\nu-W_0\,\! }[/math]

[math]\displaystyle{ V_s=\frac{h\nu-W_0}{e}\,\! }[/math]

From this equation we can see that there is a linear relation between the stopping potential [math]\displaystyle{ V_s\,\! }[/math] and the frequency :[math]\displaystyle{ \nu\,\! }[/math] with slope [math]\displaystyle{ \frac{h}{e}\,\! }[/math].

Using the slope from our best-fit line and the electron's charge, [math]\displaystyle{ e\,\! }[/math], we can approximate the value of Planck's constant:

[math]\displaystyle{ e=1.602\times {10^{-19}} C\,\! }[/math]

[math]\displaystyle{ h=me\,\! }[/math]

where [math]\displaystyle{ m\,\! }[/math] is the slope of our line. So,

[math]\displaystyle{ m_{first order}=4 \pm 0.001\times 10^{-15} Vs\,\! }[/math]

[math]\displaystyle{ h_{measured, first order}=me=(4\pm 0.001\times 10^{-15} Vs)(1.602\times {10^{-19}} C)\,\! }[/math]

[math]\displaystyle{ \simeq 6.408\pm 0.0016\times 10^{-34} Js\,\! }[/math]

[math]\displaystyle{ m_{second order}=3\pm 0.001\times 10^{-15} Vs\,\! }[/math]

[math]\displaystyle{ h_{measured, second order}=me=(3\pm 0.001\times 10^{-15} Vs)(1.602\times {10^{-19}} C)\,\! }[/math]

[math]\displaystyle{ \simeq 4.806\pm 0.0016\times 10^{-34} Js\,\! }[/math]

Also, using the y-intercept from our graph we can find the work function, [math]\displaystyle{ W_0\,\! }[/math]:

[math]\displaystyle{ y=mx+b\,\! }[/math]

[math]\displaystyle{ y_{intercept}=\frac{W_0}{e}\,\! }[/math]

[math]\displaystyle{ W_0=ey_{intercept}\,\! }[/math]

[math]\displaystyle{ y_{first order}=(4\pm 0.001\times 10^{-15})x-1.5483\pm 0.001\,\! }[/math]

[math]\displaystyle{ y_{intercept, first order}=-1.5483\pm 0.001\,\! }[/math]

[math]\displaystyle{ W_{0measured, first order}=(-1.5483\pm 0.001 V)(1.602\times {10^{-19}} C)\,\! }[/math]

[math]\displaystyle{ \simeq -2.48\pm 0.0016\times 10^{-19} J\,\! }[/math]

[math]\displaystyle{ y_{second order}=(3\pm 0.001\times 10^{-15})x-0.94\pm 0.001\,\! }[/math]

[math]\displaystyle{ y_{intercept, second order}=-0.94\pm 0.001\,\! }[/math]

[math]\displaystyle{ W_{0measured, second order}=(-0.94\pm 0.001 V)(1.602\times {10^{-19}} C)\,\! }[/math]

[math]\displaystyle{ \simeq -1.506\pm 0.0016\times 10^{-19} J\,\! }[/math]

• Error Claculations:

Percent error from accepted value

[math]\displaystyle{ \% error=\frac{h_{accepted}-h_{measured}}{h_{accepted}} }[/math]

[math]\displaystyle{ h_{accepted}=6.62606896(33)\times10^{-34} Js }[/math]

[math]\displaystyle{ \% error_{first order}=\frac{6.62606896(33)\times10^{-34} Js-6.408\times 10^{-34} Js}{6.62606896(33)\times10^{-34} Js} }[/math]

[math]\displaystyle{ \simeq 3.29 \%\,\! }[/math]

[math]\displaystyle{ \% error_{second order}=\frac{6.62606896(33)\times10^{-34} Js-4.806\times 10^{-34} Js}{6.62606896(33)\times10^{-34} Js} }[/math]

[math]\displaystyle{ \simeq 27.47 \%\,\! }[/math]

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Conclusion

From our results we find that our values for our second order maxima may have been wrong. The error estimated for our measured Planck's constant was very large. I'm not particularly sure for why that is. From our table, our values for the stopping potential was significantly different for the green spectra between first and second order. We were not able to figure out why that is either. For our first order measurement of Planck's Constant we found that we managed to get quite close to the current accepted value. Although the transmission narrowed the intensity per part by 20 %, our measured stopping potential only varied by a small amount for each time measured. We found that the energy stored in light is independent of intensity but depends on the incident frequency. Also, due to our error estimation, I find that the measurement for the first order stopping potential should be more correct and hence see that it only takes about 1.5 eV for the electrons to leave the cathode inside the h/e apparatus. Overall this lab was informative and exciting in the sense that we were able to understand one way in which constants--Planck's to be exact--are determined. Of course it takes more trials to conclude the actual value, but we had some problems with our equipment (check our uncertainty notes in lab notebook). [2]