SENSITIVITY OF A DETECTOR, AND THE WEBER-FECHNER LAW 55 



not a simple proportionality, but rather a logarithmic one. Thus, the sensa- 

 tion, or loudness, L, is given by 



L oc log///° 



where 1° is background intensity, and / is the intensity, over background, of 

 the signal to be detected. This is the basic form of the Weber-Fechner law. 

 It has many manifestations. For instance, if there are two signals equally 

 strong, with different backgrounds, the resolution of (difference in loudness), 

 L 2 - L, , is related to the ratio of the intensities of the two backgrounds, 

 1° and I 2 °, as follows: 



L 2 — L { oc log I°/I° 



This is a law which has rather wide application, not only in the psycho- 

 logical sensations but in detection of electromagnetic waves of many fre- 

 quency ranges, from the radio to the infrared. Therefore its implications 

 should be very thoroughly contemplated. 



Because of this logarithmic law, it is convenient to express power ratios 

 by a logarithmic unit, so that sensation becomes approximately linearly pro- 

 portional to this unit. The unit is called the ^bel," (b) and is equal to the 

 logarithm of the ratio of two sound intensities if they are in a ratio of 10 : 1. 

 The number of bels then is given by 



b = log 1/1° 



For sound, the value 1° is arbitrarily chosen to be the lowest one which a 

 human ear can detect (10 -16 w/cm 2 ; or, in pressure units, 0.0002 dynes/cm 2 , 

 since the same conversion factor applies to numerator and denominator). 

 The bel unit is too large for convenience, and the decibel, one tenth of a bel, 

 has received wider use. Therefore, the number of decibels is: 



db = io log i/r 



Another form of the Weber-Fechner law, then, is 



L « db 



It holds true for all sensory receptors. 



Some minimum discernible relative changes,** (/, - 7°)//° (where I, is 

 threshold intensity), which man can detect are: 



Brightness of light: 1 per cent 



Lengths of lines: 2 per cent 



Feeling of weight: 10 per cent 



Loudness of sound: 30 per cent 



** Remember relative error, defined in Chapter 1 ? 



