NOISE IN RESISTANCES 161 



line from both resistances over a period L/c. If p is the power per unit 

 bandwidth from one resistance, then 



2p(L/c) = w= kT/{c/2L) 

 p = kT. 



(7) 



Or, we may say that the power flow from a resistance into a matched load 

 (the available power) is, for a bandwidth B 



P = kTB. (8) 



Sometimes it may be desired to know the mean squared fluctuation voltage 

 integrated over all frequencies. Carrying out such an integration for the 

 voltage between a pair of terminals connected by a comphcated network 

 would seem to be a difficult procedure. However, if the pair of terminals is 

 shunted by a capacitance, the integrated fluctuation voltage can be ob- 

 tained by direct application of the principles of statistical mechanics. 



In a lumped network composed of capacitive, inductive and resistive 

 elements* each capacitance and each inductance constitutes a degree of 

 freedom; that is, the electrical state of the network can be specified com- 

 pletely by specifying the voltage across each capacitance and the current 

 in each inductance**. According to statistical mechanics, the average 

 stored energy per degree of freedom is kT/2. The stored energy in a 

 capacitance is Cv-/2. Thus, the mean squared noise voltage of all frequen- 

 cies across a capacitance C must be 



^2 = kT/C. (9) 



Similarly, the mean squared noise current of all frequencies flowing in an 

 inductance L is 



i"2 = kT/L. (10) 



We have conveniently thought of Johnson noise as generated in the 

 resistances in a network. We need not change this concept and say that 

 the voltage and current of (9) and (10) are generated in the capacitance or 

 inductance any more than we would say that the thermal velocities of 

 molecules are generated by the molecules' mass. Relations (9) and (10) 

 merely represent necessary consequences of the laws of statistical mechanics 

 as, indeed, does (1). 



It is of some interest to illustrate the use of (9) and its connection with (1) 



* Strictly, such a lumped network is an unrealizable ideal. There are no pure capaci- 

 tances, inductances, or resistances. The conditions under which actual condensers, coils 

 and resistors can be represented satisfactorily by these idealizations must be judged b> 

 measurement or calculation or by past experience or intuition. 



** In enumerating the degrees of freedom, capacitances in series or shunt are lumped 

 together as one element; the same holds true for inductances. 



