NOISE IN RESISTANCES 169 



Ordinarily, the magnitude of F„, is very small compared with F© ; /o is 

 very small compared with I, and Xm is very small compared with x. 



Suppose V„, were held constant, say, by putting a conducting plane of 

 potential Vm at x^ . Then, the electrons which pass this plane are quite 

 independent of the low energy electrons which are turned back, and hence 

 in the current passmg ocm there will be pure shot noise. 



t^ = 2eIoB. (39) 



Now suppose we change Vm . The change in /o will be, from (38), 



dio = dVjRm (40) 



Rm = {eU/kT)-K (41) 



If "we use a constant current instead of a constant voltage d-c supply, then 

 Vm must fluctuate in such a way as to cause a current equal and opposite to 

 (39), or, there must be a fluctuating voltage vL such that 



^ = lel^Rl 



(42) 

 = {\/2)AkTRmB. 



Suppose we consider the noise fluctuation of the anode voltage of a space 

 charge limited diode supplied from a constant-current source. If there 

 were no fluctuations in the voltage drop between the potential minimum 

 at Xm and the anode at x, (42) would give the noise voltage fluctuation of 

 such an "open circuited" diode. Actually, much larger fluctuation voltages 

 are observed, and we must conclude that they arise in the space between the 

 potential mmimum and the anode. As the current is constant in this region 

 (by definition — we have assumed a constant-current supply) we are forced 

 to conclude that such fluctuations are due to a variation of mean electron 

 speed in this region. The field at .t„» is necessarily zero. If, with a constant 

 current, electrons travel more rapidly between Xm and the anode, there is 

 less electronic charge ever>^vhere in this region, the rate of change of field 

 with distance, and hence, the field, are everywhere smaller, and the voltage 

 between x„, and the anode at x will be smaller. 



It is somewhat involved to treat the problem of multi-velocity flow 

 exactly; this has been done by Rack^ and others^-^°^^; however. Rack has 

 shown that an approximate treatment yields very nearly the correct result 

 over a fairly wide range of conditions. In this approximation, the stream 

 of electrons with many velocities and a fluctuating mean velocity is replaced 

 by a stream in which all electrons have the same velocity, and this has a 

 mean square fluctuation equal to that of the multi-velocity stream. 



Let us now measure x from the potential minimum. Suppose we con- 

 sider an electron which passed the potential minimum (:c = 0) at / = 0. 



