CHAP. XI] INDUCTANCE OF TRANSMISSION LINES 193 



Prob. 8. At what ratio of b to a in Fig. 46 is the magnetic energy 

 stored within the inner conductor equal to that stored between the 

 two conductors ? Ans. 1.28. 



Prob. 9. It is required to replace the solid inner conductor A in 

 Fig. 46 by an infinitely thin shell of such a radius a' that the total 

 inductance of the cable shall remain the same. What is the radius of 

 the shell? Hint: (/x/2;t)Ln(o/o') - p/8*. 



Ans. a'/a-e-- -0.779. 



Prob. 10. Prove that the part of the inductance due to the linkages 

 within the outer conductor in Fig. 46 is expressed by 



. . . (114) 



Hint: d(? p -ttdx/2KX', rip = l -*(x-&)/*(cV& 1 ). 



Prob. 11. Deduce eq. (112) from formula (114), assuming the 

 ratio t/b to be small as compared to unity. Hint: Put c = &(!+?/) 

 where y = t/b is a small fraction. Expand Ln(l -y) into an infinite 

 series, and omit in the numerator of eq. (114) all the terms above t/*; 

 expand (c 2 6 2 ) in the denominator in ascending powers of y, and divide 

 the numerator by this polynomial. 



60. The Magnetic Field Created by a Loop of Two Parallel 

 Wires. Let Fig. 47 represent the cross-section of a single-phase 

 or direct-current transmission line, the wires being denoted by 

 A and B. With the directions of the currents in the wires shown 

 by the dot and the cross, the magnetic field has the directions 

 shown by the arrow-heads, one-half of the flux linking with each 

 wire. Before calculating the inductance of the loop it is instruct- 

 ive to get a clear picture, quantitative as well as qualitative, of the 

 field itself. 



The field distribution is symmetrical with respect to the line 

 AB and the axis 00'. The whole flux passes in the space between 

 the wires, so as to be linked with the m.m.f. which produces it, and 

 then extends to infinity on all sides. The flux density is at its 

 maximum near the wires and gradually decreases toward OO' 

 and toward oo, as is shown by the curve pqsts'q'p'. The ordi- 

 natcs of this curve represent the flux densities at the various points 

 of the line passing through the centers of the wires. The reason 

 for which the flux density is larger near the wires is that the path 

 there is shorter, although the m.m.f. acting along all the paths 

 is the same. This m.m.f. is numerically equal to the current i in 

 the wires, UK numlx-i <>f turns being equal to one. 



It is prnvnl U-low that the magnetic paths outside theconduc- 



