In order to find \(k_B\), we plotted \(V^2/4TR\) versus \(\Delta f\) to find the slope which is proportional to \(k_B\). Or, if you prefer, you can create a plot where the slope is equal to \(k_B T \Delta f\) -(plotting \(V^2\) versus \(4R\)), than solve for \(k_B\). In order to find the bandwidth, we had to use values that came from the Noise Fundamentals Test Data (TeachSpin Manual 7.3) (\ref{table:Johnson2}), which gave us the actual measured values from the different filters and amplifiers. You can see the values we used in \ref{table:Johnson2}, and as you can see, the measured values (from TeachSpin Manual 7.3) are different from the nominal values (values given on the apparatus). After using the measured values from the Noise Fundamentals test data, we had to calculate the Equivolent Noise Bandwidth (ENBW) in order to find \(\Delta f\). To do this, we used an equation:

\[\label{eq:ENBW} ENBW = \left(\frac{\pi}{2} \ Q\right) f_c\]

Where \(f_c\) is the High Frequency filter values (\(f_2\)) and Low Frequency filter values (\(f_1\)) subtracted shown in \ref{table:True_Bandwidth1} and \ref{table:True_Bandwidth2}. The equation for finding \(f_c\) is: \(f_c = f_2 - f_1\)