FLUCTUA TIONS IN MICROPHONES AND OTHER RESISTA NCES 221 



where D is the total displacement from the first coincidence. This 

 number can also be expressed in terms of the contact resistance R by 

 using Goucher's ^ derived equation: IjR = Const. Z)''^''/^. Thus the 

 above expression can be written 



Nc. = Const. 7?-(2»+2)/(2n+3)_ 



{'» 



The area of secondary electrical conduction surrounding each area of 

 coincidence is precisely that area which would be added were the 

 contact compressed by an increment of compression A.r. From the 

 theory of Hertz ^^ one can show that for smooth spherical hills in con- 

 tact, A^/Ax is independent of the total hill compression and hence 

 that the total area of secondary electrical conduction in a contact 

 Ac is proportional to Nc, giving 



Ac = Const. Nc = Const. i?-(2n+2)/(2n+3)^ 



(9a) 



For purposes of analysis let us think of the secondary conduction 

 area surrounding each primary area of contact as divided up into small 

 elements of like nature, and further that the secondary conduction 

 through each of these elements of area is independent of that in every 

 other element. If such a contact is connected in a circuit as shown in 

 Fig. 12 A then the equivalent a.-c. circuit can be thought of as that 

 shown in Fig. 15, where R is the mean resistance of the entire contact, 



Fig. 15 — The equivalent a.-c. circuit of a contact with v elements of secondary 

 conduction area, through each of which there is an independently fluctuating current, 

 when such a contact is connected in a circuit as shown in Fig. 12^. 



*i, • ■ ' 12 are the instantaneous deviations of the current from the 

 mean current in each of the v elements of secondary conduction area, 

 and e is the instantaneous value of the fluctuation voltage across the 

 measuring device with impedance Z. Taking into account the random 

 nature of the fluctuation currents through each element of secondary 



'"A. E. H. Love, "Mathematical Theory of Elasticity," 2nd ed., p. 192. This 

 has been experimentally confirmed by J. P. Andrews, Phys. Soc. Proc. 43, 1 (1931). 



