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Lucy Liang added subsection_Analysis_Error_analysis_for__1.tex
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\subsection{Analysis}
Error analysis for Johnson Noise
The discrepancy between our result ($1.46 \pm0.0054 \cdot 10^{-23}\textrm{ m}^2 \textrm{ kg} \textrm{ s}^{-2} \textrm{ K}^{-1}$ and $1.46 \pm0.0052 \cdot 10^{-23} \textrm{ m}^2 \textrm{ kg} \textrm{ s}^{-2} \textrm{ K}^{-1}$) and the accepted Boltzmann constant ($1.38 \cdot 10^{-23} \textrm{ m}^2 \textrm{ kg} \textrm{ s}^{-2} \textrm{ K}^{-1}$)is approximately $5.8\%$ . Both sets of data we obtained (with a $1\textrm{k}\Omega$ resistor and a $10\textrm{k}\Omega$) gave a consistent value for the Boltzmann constant which is a matter of accuracy rather than precision, so this error is most likely a systematic error.
Assuming that our thermometer is accurate (for room temperature), the temperature within the instrument, where the resistor is, may be higher. As we can see in equation \ref{eq:Equatoin}, with a higher $T$, we would have a lower $k_b$ value which is opposite to what our value is, so this is unlikely what is causing the error.
The measured values(Table \ref{table:True_Resistance}) for Rin is about $0.3\%$ different from the claimed value. Rin is also not the main source of error.
Unfortunately, we only took one error value for all of the Johnson data. We found that the average error for the $1 \Omega$ resistor was $0.003 \textrm{Volts}^2$. Because we did not take any other error data, it is impossible to find the total error in our measurements.
