NOISE IN RESISTANCES 171 



From (51) and (52) 



dVo= ^2'''(^^~'^\l''dxo. (53) 



Now, if (dxo)^ is the mean square fluctuation in velocity, the mean square 

 fluctuation in voltage will be 



v^ = 2{e/m)-' Vo(d±o)''- (54) 



The assumptions leading to (54) are those leading to Child's law, and thus 

 we can use (37) in connection with (54), giving 



V' = 3{e/m) 'loR {dx^y. (55) 



It now remains to evaluate {dx^-, the mean square fluctuation in the 

 velocity of the electrons passing the potential minimum; to do this, we 

 return to (25). Suppose N is the number of input electrons per second. 

 The output current can then be written 



h = nNe (56) 



and we can call the fluctuation in it 



t' = {bnNef. (57) 



Equation (25) applies for no fluctuation in /o and hence for no fluctuation 

 in iV; e is a constant, and thus we may write (25) as 



m' = ^-§in''-n% (58) 



We may generalize this to say that each electron has a probability p of 

 producing some effect of magnitude n and the fluctuation in the magnitude 

 of the effect is (5^)^. Before, we said that an electron had a probability 

 p of producing n secondaries. Now we will say instead that an electron 

 has an uncorrelated probability p of having a velocity u, and obtain for the 

 mean fluctuation in the velocity, {dx^"^ 



(d^2 = ^^(u^-u'). (59) 



In a MaxweUian distribution, the number of electrons passing a plane 

 perpendicular to the direction of motion per second having velocities lying 

 in the range du at u is 



dn = Aue-^"'''"'''''^^ du. (60) 



