590 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



generalized in some of those cases to take into account the existence of 

 groups of electrons with different "effective masses" 7n. 



3. THE SOLUTION OF MAXWELL's EQUATIONS 



Maxwell's ecjuations will now be solved in a cylindrical waveguide 

 filled with a hypothetical medium which contains the ferrite and the 

 plasma as a special case. It will be supposed therefore that both its per- 

 meability and dielectric constant are tensors of the form previously 

 considered. 



3.1 Field components 



The following notation will be found convenient. The projection of a 

 vector A upon the plane normal to the ^-axis will be written At . If the 

 components of A, are a, ^ then an associated vector having components 

 (/3, —a) is devoted by A*- A similar notation is used for differential 

 operators. Thus, if V denotes {d/dx, d/dy), V* denotes (d/dy, — d/dx)'\. 

 Denoting scalar products by a dot, the following identities are evident 



At*-At* = ArAt ; (At*)* = -At ; ArAt* = 0; 

 AfBt* = -At*-Bt) 

 and 



At-Bt* = ^-component of [A X B]. 



Also if k is a unit vector along the positive ^-axis, k X A = —At*. 

 Similar relations hold for differential operators. If one denotes the star- 

 ring operation by the symbol P then clearly 



where o is a number. 



Maxwell's equations may noAV be written, for that case in which the 

 dependence of any component upon t and z is of the form e^'''^'^~^^\ in the 

 form: 



V*//. + mt* = jo^eEt + c^vEt*, 

 \/-Ht* = jo:e.E., 



\/*E, + j^Et* = -jo^iiHt - o^KHt*, and 



y-Et*= -j^n^H,, 



where use is made of equations (3) and (6). 



t The operator V* is called "flux" by SchelkunofT. Strictly, one should write 

 V« and V«*, rather than V and V*, but this is needlessly cumbersome. 



(9) 



