The Electrostatic Field in Vacuum Tubes With Arbitrarily 

 Spaced Elements 



By W. R. BENNETT and L. C. PETERSON 



VACUUM tubes with close spacing between electrodes have become of 

 increasing importance in recent years. The higher transconductances 

 and lower electron transit times thus obtained combine with other features to 

 raise both the frequency and band width at which the tube may operate 

 satisfactorily as an amplifier. Specific designs have been discussed in papers 

 by E. D. McArthur and E. F. Peterson^ and by Kremlin, Hall and Shatford^. 

 The important contributions to structural technique made by E. V. Neher 

 have been described in the Radiation Laboratory Series^. Further im- 

 portant advances in the art have been recently announced by J. A. Morton 

 and R. M. Ryder of the Bell Laboratories at the recent I.R.E. Electronics 

 Conference held at Cornell University in June, 1948. The material of the 

 present paper represents work done by the authors over a decade ago, and 

 naturally there has been considerable publication on related topics in the 

 intervening years. It has been suggested by our colleagues, however, that 

 some of the results are not available in the technical literature and are of 

 sufiicient utility to warrant a belated publication. These results have to do 

 with the variation of the electric intensity, amplification factor, and current 

 density which takes place along the cathode surface because of the nearby 

 grid wires. 



We shall deal mainly with the approximate solution which neglects the 

 efifect of space charge. The correction required to take account of space 

 charge is in general relatively small as shown by both qualitative argument 

 and experimental data in an early paper by R. W. King'^. More recent 

 theoretical work^^ extending into the high frequency realm has confirmed 

 the minor nature of the modification needed. The problem is thereby re- 

 duced to one of finding solutions of Laplace's equation which reduce to con- 

 stant values on the cathode, grid, and anode surfaces. The original work 

 on this problem was done by Maxwell^ who calculated the electrostatic 

 screening effect of a wire grating between conducting planes long before the 

 vacuum tube was invented. All subsequent work has followed the methods 

 outlined by Maxwell. In particular he suggested the replacement of the 

 conducting planes by an infinite series of images of the grid wires and 

 described an appropriate solution in series for the case of finite size wires. 

 The useful approximation obtained when the diameter of the grid wires is 



303 



