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# Final Lab Report 2 (MH and LL): Determination of the Boltzmann Constant and Elementary Electron Charge through Noise Measurements

Abstract

Two sources of noise, Johnson noise and shot noise, are investigated in this experiment.Grammar errors: The Johnson noise, which is the voltage fluctuations across a resistor that arose from the random motion of electrons, is measured using the Not defined Noise Fundamentals box. The noise was measured across different resistances and at different bandwidths at room temperature, resulting in a calculation of the Boltzmann constant of $$1.4600 \pm0.0054 \cdot 10^{-23}\textrm{ m}^2 \textrm{ kg} \textrm{ s}^{-2} \textrm{ K}^{-1}$$ and $$1.4600 \pm0.0052 \cdot 10^{-23} \textrm{ m}^2 \textrm{ kg} \textrm{ s}^{-2} \textrm{ K}^{-1}$$. The shot noise occurs due to the quantization of charge, and was measured by varying current in the system, with which we calculated the electron charge of $$1.649 \pm 0.007 \cdot 10^{-19} \textrm{ Coulombs}$$. They agree quite well with the accepted values of $$1.38064852 \cdot 10^{-23} \textrm{ m}^2 \textrm{ kg} \textrm{ s}^{-2} \textrm{ k}^{-1}$$, and $$1.64 \cdot 10^{-19} \textrm{C}$$ for the Boltzmann constant and electron charge respectively. Errors are discussed.

Editor’s Notes:

1. a reader won’t know what a “Noise Fundamentals” box is. More helpful – and more relevant – would be “a specially designed low noise preamplifier, a combination of high and low pass filters and amplifiers, and a voltage multiplier.” But it isn’t clear you need this phrase at all in a sentence explaining what Johnson noise is, since the next sentence explains what you measured: the dependence of Johnson noise on the resistance and bandwidth.

2. the “quite well” statement doesn’t match your quoted uncertainty for your measurement of $$k_B$$. The percent deviation of your best estimate of the value of kB from the accepted value is $$5.8\%$$, but if you are using 1 standard deviation σ as the uncertainty in your measurement of kB , then your value differs from the accepted value by 16 $$\sigma$$. If we assume that your estimate of uncertainty is correct, then the two values don’t agree at all, much less quite well. A difference of 16 $$\sigma$$ simply isn’t possible if you are really measuring the same thing. If, on the other hand, we assume that $$6\%$$ really is close agreement, then you have substantially underestimated your uncertainty or the accuracy of your measurement is substantially worse than the precision (meaning there is an unaccounted for source of systematic error leading to an excessively high value).

3. in this case, you appear to have confused the accuracy with which a computer can fit a straight line through your $$k_B$$ data with the precision and accuracy with which you acquired that data. Disappointingly, it also appears you didn’t collect the data you would need to do so and so don’t have the uncertainties needed to calculate a weighted fit nor calculated the chi-square probability between that fit and your data. These determinations and calculations also appear to be missing for the Shot noise determination of $$e$$. A determination of uncertainty — and ‘error bars’ for plots — and a measure of the goodness of fit are essential elements of an advanced experiment.

4. I suspect you have made at least one miscalculation somewhere along the line for $$k_B$$, and that may have lead to the deviation of your result from what would be expected. One reason I suspect this is that you have data points that have negative bandwidths. Looking at your data on plotly, however, it is not possible for me to figure out how that occurred. I’ve tried, but there’s simply no way to trace your calculations or reproduce your results. This is because not all the needed information is there. Consider, for example, the $$10 \textrm{ k}\Omega$$ data table. The ‘dark current’ offset isn’t listed, so I don’t know if that was taken into account or not. the gains aren’t listed, and even if I use the ones from your writeup, I don’t know what ‘output voltage’ in the Plotly table means? What does it correspond to in terms of the equations and variables in your paper? And what does $$V^2$$ mean in that table? It can’t be the same as the $$V^2$$ values in your tables, as the plotly table has values on the order of 1.425e-13, and the data table has units of 1.0e+2. Why such a careless and incomplete approach to the data collection and analysis?