CIRCULAR CYLINDER CAVITY RESONATOR 33 



as the current formerly on the surface of the removed metal crowds over 

 onto the adjacent metal, but this is a second-order effect. 



To determine the best location of such cuts, therefore, it is necessary to 

 know the vector distributions of the wall currents for the various modes. 

 This current vector, /, is proportional to and perpendicular to the mag- 

 netic vector, //, of the field at the surface. Expressions for the components 

 of the //-vector at the surfaces of the end plates and side walls are given in 

 Fig. 1. 



End Plate: Contour Lines 



At the end plates, the magnitude of the //-vector at any point is given by: 



IP = H,' + lie'. (1) 



Xow substitute values of Hp and He from Fig. 1 into (1); drop any constant 

 factors common to Hp and He as these can be swallowed in a final propor- 

 tionality constant; introduce the new variable x: 



X = kip = r ^. (2) 



where R = D/2 = cavity radius. Thus is obtained; 



//' = [J fix) cos (df + 



- J fix) sin (6 



X 



(3) 



Now Jf and Jf, are expressed in terms of Jf^i and Jf^i and a further re- 

 duction leads to. 



//"' = (//_ cos (d)' + iJf+ sin Cey (4) 



where 



Jf. = Jf.,ix) - Jf^.ix) (5) 



and 



Jf+ = Jf.r(x) + Jf,:ix) (6) 



The formulas (4) to (6) apply to both TE and TM modes. The values 

 obtained depend on r, which is different for each mode. 



When ^ = 0, / is proportional to Jf. and when 6 — ir/lf, I is proportional 

 to Jf+ . Relative values of / are thus easily calculated for these cases, 

 once tables of // are available. Such tables have been prepared and are 

 attached. For TE modes, when d = 0, He — 0, and the currents are all 

 in the 6 direction. For TM modes, when 6 = 0, Hp = 0, and the currents 

 are all in the p-direction. When d = tt/K, the converse holds. 



Figures 3 to 18 are a set of curves showing the relative magnitude of H 

 (or /) for several of the lower order TE and TM modes. The abscissae 



