Shielding of Concentric Spherical Shells, 1 23 



Write f = <l + e), 



where e is so small that terms in e may be neglected, unless 

 divided by a small quantity. 



Then equation (30) may be written 



(>+?){<4)'+*°(l-i)} 



-5(|-i)( i+ -)-i-»- 



The terms in e may therefore be neglected unless R/« 3 is 

 large. 



As an example, let R=10, i. e. let the volume of the metal 

 be one tenth of the volume of the enclosure. 



If then, as before, we take yit=501, a Q =l, 



/. f = 97(1 + 0-009) and e = 0*009. 

 If we neglect all the terms in e we get 



- = 2-17. 



Hence eR/a 3 = '1953, 



which cannot be neglected when compared with unity. 

 Taking this term into account, 



l/« 3 = 1-875. 



As the radii vary inversely as the cube roots of the u's, the 

 two values of the external radius are 1*233 and 1*294. As 

 the shielding varies slowly when near a minimum value, as 

 the outer shell is displaced outward, it is probable that no 

 very grave error would be introduced even in this case by the 

 use of the approximate equation. The error would, how- 

 ever, increase rapidly as the volume of material employed 

 diminished. 



Using the more correct value we get 



«!=«/! -g) =1-0*0533 = 0*9467, 



-i= 4---- + - = 0*1-1*875-1*056 + 1 = -1*831. 



«2 -ft a 3 «1 a 



Hence we have for the best arrangement, 



a 3 =0*5333, a 2 =0-5461, ^=0*9467, « =1, 

 or « 3 = 1*233, a 2 = 1*223, «i= 1*018, a =l. 



