The prediction of Nyquist's theory is that thermal fluctuations yield = 4 KTRAf, where k is Boltzmann's constant, T the absolute temperature (in Kelvin), and A f is the effective frequency bandwidth over which the voltages are averaged. After amplification, these fluctuations yield measured voltages of =4 KT R Af G2/10, where G is the product of all amplifier gains. Note that mathematically, the total noise diverges if there is no limitation on the bandwidth; but physically A f is always limited. The high-frequency response is usually limited by amplifier technology, or more fundamentally by the maximum speed that electrons can move during thermal fluctuations in the resistor. In any case, for accurate measurements it is important to limit the response to a well-defined bandwidth where the amplifiers are well-behaved, which is best achieved by electronic filters. In the experiment, the high-level electronics (wooden box) has low-pass and high-pass filters for accurately defining A f. A table of the effective noise bandwidths for each setting on the low-pass (column) and high-pass (row) filters is pasted below. Table 1.5 Effective noise bandwidths, f, given in Hertz, computed for model filter responses f2 = 0.33 kHz 1 KHz 3.3 kHz 10 kHz 33 kHz 100 kHz fi = 10 Hz 355 1,100 3,654 11,096 36,643 111,061 30 Hz 333 1,077 3,632 11,074 36,620 111,039 100 Hz 258 1,000 3,554 10,996 36,543 110,961 300 Hz 105 784 3,332 10,774 36,321 110,739 1000 Hz 9 278 2,576 9,997 35,543 109,961 3000 Hz 0.4 28 1,051 7,839 33,324 107,740 These computed values are all subject to uncertainties of order 4%; (see Section 5.2 for details on how any of them can be more carefully calibrated). They are all computed (by the methods of Section 2.2) for ideal filter responses, ignoring systematic effects. Inclusion of those effects may raise values in the rightmost column by (3+1)%, and may raise values in the next-to-rightmost column by (111)%. There areTwo sets of measurements of the mean-squared voltage amplitude yield the data listed below, with the resistors, filter settings, and gain factors given for each set. The data were taken at room temperature, 7=295 K. Recall that the measured mean-squared voltage is related to Boltzmann's constant via: =4 KT R Af G2/10. For each set of data, calculate the ratio of the experimental value of k, divided by the known value of k. Specifically, obtain the average experimental values of / (4TRA f G2 / 10) for each set of data, then divide by the accepted value of k = 1.3806 x 10" 23 J/K. Your answer will then be a dimensionless ratio, quantifying the discrepancy between the measurement and the accepted value. For the best set of measurement (the set with the least discrepancy), this dimensionless ratio is k(exp)/1.381x10-23 = 100kOhm; 0.1-1 1MOhm; KHZ; 0.1-1 KHz; G=2.4x10*6 G=6.0x10^5 0.99 0.559 0.987 0.567 0.985 0.582 1.021 0.6 0.97 0.566 0.98 0.59 0.964 0.601 1.004 0.577 0.995 0.574 0.999 0.573 1.025 0.581 0.986 0.597 0.993 0.587D Question 2 2 pts Three sets of measurements of mean-squared voltage from three different resistors (R) are listed below. Also given are the gain factors from the low-level and high-level amplifiers. All data were taken at room temperature (7=295 K), with the high-pass filter at 0.1 kHz, and low-pass filter at 10 kHz. This net mean-squared voltage includes a contribution from the amplifiers, so that when normalized by the gain: 10/G2 = + . Here, Nyquist's prediction is =4KTRA f, while the amplifier noise () is dominated by the first pre-amp in the low-level electronics (metal box), so its dependence on the high-level gain (wooden box) is negligible. From these data, the best experimental value for the dimensionless ratio of experimental value divided by accepted value is k(exp)/1.381x10-23 = Hint: depends on R, but does not. R(Ohms) low- high- level level 1000 0.7495 600 5000 10000 0.8032 600 3000 100000 0.6723 600 1000