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Note: Equating the integrands of two definite integrals cannot be justified mathematically. A strict derivation would then appeal to the balance of emitted and absorbed power over an arbitrarily small frequency interval, rather than simply over the full range; this is the argument Nyquist constructs in Fig. 10.9. There, he considers the following circuit thought experiment : Imagine that over all frequencies, the exchange of power between the two resistors (at equal temperature) is equal, but over a small range of frequencies, the resistor on the left radiates more power than it absorbs from that on the right. Then, introduce a non-dissipative circuit element between the two that interferes more with the transfer of energy in the frequency range of interest than in any other range, so that now more total power flows from the right resistor to the left than in the other direction. But since the resistors are initially at the same temperature, to let the system evolve would be to raise the temperature of a hotter thermal body using the heat of a colder body, by simply introducing a passive circuit element. So, there must be a detailed balance of power over each tiny range of frequency, in accordance with the second law of thermodynamics.) Then, in the classical limit, h kBT , S( ) SV ( ) = 4RkBT . (10.23)

Equation (10.23) shows the Nyquist relation between the voltage power spectrum and temperature. Accordingly, the current power spectrum is closely related to the following equation, SI ( ) = 4k T 1 S ( ) = B R R2 V (10.24)

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Repeat Problem 8.43, displaying the list of calculated values within a combo box, as in Example 8.10. Write a Visual Basic program that will generate a table of sin x, cos x, tan x, loge x and ex. Generate 101 entries for uniformly spaced values of x between 0 and (i.e., let x = 0, /100, 2 /100, . . . , 99 /100, ). Place the values within a multidimensional array. Include a feature that will display the value of all five functions for any tabulated x-value selected from a text box. Write a Visual Basic program that will sum the elements in each row and each column of a table of numbers. Use a multidimensional array to represent the table. Display the sum of each row and the sum of each column. Then calculate the sum of the row sums and the sum of the column sums (they should be equal). Label the sums appropriately. Test the program using the following table of numbers. 6 8 11 13 27 0 15 3 2 3 12 5 1 13 0 4 5 17 9 14 17 18 12 24 8 21 0 7 4 10

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Since the actual resistor will be situated in a circuit with an amplifier designed to enhance the minuscule voltage fluctuations, and since the transmission line will exhibit capacitive effects, a realistic configuration of the circuit diagram would therefore, appear as in Fig. 10.9. The parallel combination of current source i and resistor R is the Norton equivalent of R and a small fluctuating voltage source,

illustrated in Fig. 10.9. The net input voltage is then denoted by Vin, and the additional current and voltage associated with the operation of the amplifier itself are labeled Iamp and Vamp. The input voltage power spectrum takes the simple form SV = SI [ (R| C)]2 + SI, amp [ (R| C)]2 | |

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