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\section{Experiment on Johnson Noise}  \subsection{Method for Johnson Noise}  In order to get the desired data, we decided to vary $\Delta f$ in order to vary the bandwidth. To do this, we kept the resistance ($R$) constant, but did three different trials where each trial had a different $R$ value. The values for $R$ are as follows: $10 \textrm{ k}\Omega$, $1 \textrm{ k}\Omega$, and $1 \Omega$. As with any resistor, there is error in the measurement and Table \ref{table:True_Resistance}, shows the true, measured value of the resistance. You can perform the Johnson Noise experiment by varying resistance for a given bandwidth ($\Delta F$), f$),  but we chose to vary $\Delta f$ and keep resistance constant. We chose to do this because if our apparatus and measurements are correct, all of the different $R_{in}$ should generate the same value for the Johnson Noise, $k_B$. We chose the three different resistances because the $10 \textrm{ k}\Omega$ and $1 \textrm{ k}\Omega$ were used to calculate the $k_B$ while the $1 \Omega$ measurement was used as an error source that needed to be subtracted. As with all electronics, there is some innate error associated with the device, in order to subtract this, we took data with the $1 \Omega$ resistor. We want to find a ratio in which both the numerator and denominator get bigger as you increase $R_in$ (the resistor that is the source of the voltage fluctuations), so the slope $k_B$ remains the same.