# Measuring Boltzmann constant using thermal noise

For my physics lab I have to come up with a way to measure Boltzmann constant using thermal noise.

The circuit that I 'm going to use is something like the one in the figure below:

Than I'm going to attach an arduino UNO board to the output and measure the $\Delta V$, using a sampling rate of $50$KHz, and by computing the st.deviation, get a value for $k_b$.

1. Is the LF 355 a good op-amp candidate for my experiment? Considering that the thermal noise input will be very weak and thus very close to the GND value is it better to use a RRIO op-amp?
2. Is the sampling rate of Arduino enough to see a consistent variation of the $\Delta V$? Does the fact that the board can measure only positive difference of potential will alter the calculation of the st.deviation?

You'll need more gain than that to see 5 uVrms -- the arduino probably has a resolution of 10 bits (?), equivalent to about 5 mV. To see the signal, you'll need about 50 mV, so a gain of 10,000 is needed. The opamp's DC offset might be an issue, so just use a low value of DC gain, and bypass for AC -- that also provides your LPF.

Now check if the opamp has enough gain*BW to do this at 50 kHz.

If this actually works, you can then arrange to short your 100k R and do a 2nd measurement -- this would measure the opamp's noise, and you can subtract the measurements (actually sqrt(difference in squares)) to get the R's noise.

You need to avoid splitting the signal in half - just use 2x 200k resistors; one to ground and one to Vcc - now your signal will be central but, you'll also need to do the same with the 100 ohm resistors.

However, the op-amp's 20nV per root Hz internal noise might cause a problem. Across a 50 kHz bandwidth, this is a net noise of $\sqrt{50,000}\times 20nV$ = 4.47 uV.

A 100k resistor at ambient produces a noise of 9 uV across a 50 kHz bandwidth so you have problems using this op-amp because it's own voltage noise source is half that of the 100k resistor. The internal noises of the bias current of the op-amp chosen are very small and won't significantly affect things.

However, if you look at figure 32 in the data sheet for the TI LF355 you'll see a graph of noise voltage: -

This is indicating that there is considerable noise at the low-end of the spectrum (as per most op-amps) and this could produce a few more micro volts of noise. To avoid this I would high pass filter the output so the LF stuff was pretty much avoided. You should be able to do this in software.

• Thank you! I forgot to write about passing the data through an high-pass filter so I'll take your observation as further confirm. I didn't get the part where I have to attach the 220k resistor to the Vcc (positive supply voltage of the op-amp). How I'm going to supply the op amp then? (this can be a newbie question but we didn't studied op-amp yet) – damides Feb 6 '16 at 15:57