ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 561 



ductor is tanh 7r/2 = 0.92. When u = lir, the ratio reaches its 

 first maximum coth x = 1.004. At 1 megacycle the optimum thick- 

 ness of a copper conductor is about 0.1038 mm. 



By a method of successive approximations, H. B. Dwight has ob- 



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 



a 



Fig. 5 — The skin effect in cylindrical shells. The curve represents the ratio of the 

 a-c. resistance of a typical shell to its d-c. resistance. 



tained the impedance of a tubular conductor with an external coaxial 

 return. 2^ His final results appear as the ratio of two infinite power 

 series, which converge for all values of the variables involved, though 

 they can be used advantageously in numerical computations only 

 when the frequencies are fairly low and the convergence is rapid. 

 We shall merely indicate how Dwight's formula and other similar 

 formulae can be obtained directly from the exact equations (75). 



Let us replace the outer radius h of (75) by o + ^, where / is the 

 thickness of the wall, and replace the various Bessel functions by their 

 Taylor series in t\ 



hiab) = Ioi<Ja + at) = Z 



n=o n\ 



/o^-K^a), 



Ko(ab) = Ko(aa + at) 



n=o n\ 



(<rty 



(88) 



hicb) = lo'(ab) = E^^/o^-^'Ko-a), 



n=0 W! 



- K,{<rb) = Ko'iab) = t i^Ko^n+i)(^^a) 



2" "Skin Effect in Tubular and Flat Conductors," A. I. E. E. Journal, Vol. 37 

 (1918), p. 1379. 



