ELECTRON STREAMS IN A DIODE 835 



the total current density / flowing in the diode is a function of time alone; 

 it has the same value at all planes along the ic-axis, and is given by 



/=-pi;-,g (3) 



at 



The first term is the conduction current density, the second term is the 

 displacement current density, and I is measured according to circuit con- 

 ventions in the direction opposite to the motion of the electrons. The charge 

 density p is 



P=.g (4) 



and its substitution in (3) gives 



u'^ + '4=-l (5) 



dx dt e 



which is the differential equation for electric intensity. 



Before passing, it should be noted that the conduction current density Qy 

 measured according to circuit conventions, is 



Q= -pU = -.U^^ (6) 



dx 



The two differential equations for U and E are now repeated as a group 



I,|^ + f=_,£ (2) 



dx dt 



U^ + ^=-l (5) 



dx dt € 



and in this group the total current density / may be regarded as any known 

 or arbitrarily assigned function of time. These are the basic equations whose 

 solution is sought in the present theory of electron streams. They are a de- 

 scription of the whole diode space, and they tell how U and E occur and 

 vary with time throughout that whole space. They are first order equations, 

 linear in their derivatives, and it is known from the theory of differential 

 equations that their general solution is the complete solution, and that it 

 will contain two arbitrary functions. So if we find a solution containing two 

 arbitrary functions, we may be quite sure that it is the complete solution. 

 The equations can be solved by the Lagrange method, as outlined in Ap- 

 pendix I. But that is a rather abstract operation, and the solution is here 

 obtained by another method that has more physical meaning and is really 

 equivalent to the Lagrange method. 



