GENERATION, CONTROL, AND MEASUREMENT 241 



noise voltage as a result of thermal agitation of the electrons. This noise 

 voltage was experimentally evaluated by Johnson (1928) and theoretically 

 treated by Nyquist (1928) and others (Aloullin, 1938). The magnitude 

 of the Johnson noise is given by the formula 



E^ = {4.kTR A/)'•^ (3-28) 



where k is the Boltzmann constant, T the absolute temperature, R the 

 resistance of the circuit element, and A/ the frequency limits or band 

 pass of the measuring system. The formula reduces to 



En = lA X \0-^{TR A/)^^ 



or, for a room temperature of 300°K, to 



Em ^ 1.3 X lO-^i^ A/)^, (3-29) 



where R is in ohms and A/ in cycles per second. For a 40-ohm thermo- 

 couple and an amplifier that passes all frequencies between 1 and 11 cps, 

 A/ = 10, and the Johnson noise is En = 0.026 /zv. For a 10-megohm 

 photocell load resistor, it is 1.3 ^v. 



In the ideal thermal detector, temperature fluctuations in the receiving 

 element determine the ultimate limit of sensitivity. However, in practi- 

 cal detectors there is relativ'ely weak coupling between the radiation field 

 and the electrical signal; i.e., only a small proportion of the radiant 

 power absorbed appears in the signal. As a result of this weak coupling, 

 the sensitivity of practical thermocouples and bolometers is always 

 limited by Johnson noise. The magnitude of this noise signal can be 

 calculated from the resistance of the detector. From the noise voltage 

 En and the sensitivity in volts per watt *So, one can calculate the mini- 

 mum detectable power in watts of radiant flux. If the thermocouple 

 has a sensitivity of 6.5 v w~\ the minimum detectable power would be 

 4 X 10~^ w. If a slow meter with a period of about 10 sec is used in 

 the output of the amplifier, the effective band pass for the system is 

 reduced by about 100, and the minimum detectable power is reduced 

 by a factor of 10. 



The selective detector, such as the photomultiplier, is inherently much 

 more sensitive in the spectral region of its useful range than the thermal 

 detector, because most of the thermal radiation is outside the region of 

 spectral sensitivity, and the coupling between the radiation field and the 

 electrical signal is much stronger. Maximum coupling is attained when 

 the quantum efficiency is 1 and individual electrons can be counted. 



The cooling of both thermal and selective detectors to dry-ice or liquid- 

 air temperatures greatly reduces the magnitude of the stray radiation 

 field, and, in case of thermal detectors, greatly decreases the thermal- 

 resistance noise, as predicted by the Nyquist formula, Eq. (3-29). 



