THE CARBON MICROPHONE 191 



For the case of uniform hills distributed on the surface of a sphere 

 it may be shown that equal numbers of hills will be added for equal 

 increments in x, in which case Nx = constJ From this it follows 

 that n = and N" = 2.5 in agreement with the measured value, 

 curve 11.^ For N" = 3.2 as obtained with the hemispheres of random 

 height on a plane, curve ///, n would have the value 0.7. The 

 corresponding distribution function N^ would approximate to that of 

 the portion of an ordinary error curve near its maximum. A rough 

 determination of the distribution of heights amongst the small rubber 

 hemispheres showed in fact that they approximated closely to an 

 error curve and that the displacement range covered that portion of 

 the curve near the most probable height. 



It would appear from this analysis that the elastic behavior of our 

 carbon contacts under conditions of relatively large strain is adequately 

 explained on the very simple assumption that the hills which we 

 observe under the microscope have a random distribution of heights 

 and behave like smooth spherical surfaces. We have, however, still 

 to account for the hysteresis and the large values of N" corresponding 

 to small values of AD as well as the values of N (Fig. 22). 



It is unlikely that the hills which we observe under the microscope 

 are submicroscopically smooth, in which case we would expect a 

 small plastic movement in these secondary hills arising from overstrain. 

 We have direct evidence for this in the fact that contacts once estab- 

 lished — even without the passage of current — require relatively small 

 but finite forces to break them. Such junctions within the contact 

 region could well account for hysteresis and a stiffening up of the 

 contact in the region of small strains. Furthermore it is to be expected 

 that they might affect the resistance behavior to a much greater 

 extent than the elastic, and over a wider range of strain, since the 

 junctions — ^though too weak to affect appreciably the contact stiffness 

 — ^might well carry a relatively large proportion of current; in which 

 case the value of N would be smaller than that calculated on the 

 assumption of smooth spherical surfaces. 



We will now derive an expression relating resistance and force for 

 the type of contact considered in the derivation of equation (5), 

 assuming smooth hills. 



Classical theory ^ gives the following formula for the conductance 



' This argument rests on the fact of geometry that if A is the area of contact 

 between a sphere of radius r and a plane, dA/dx = 2irr. 



* This agreement between theory and experiment shows that the compression 

 of some of the hills by an amount in excess of 1 per cent, of their radii has not aflfected 

 the applicability of equation (3) to our problem. 



" Riemann Weber. 



It is here assumed that the mechanical and electrical areas of contact are coinci- 

 dent, which according to the ideas of wave mechanics may not be the case. 



