Shielding of Concentric Spherical Shells. 121 



Now equation (30) may be thrown into the form 



_I^_1\ ± 

 R a 3 \a 3 U) R 2 



= 0; 



and putting 



111 



a 3 R «o' 



the left-hand side becomes 



<i + i)* {*-?-§}■ 



Hence the condition that there shall be an arrangement 

 which gives a minimum external field is that 



^>^- F J -, i.e. >e, i.e. > 



R I ' " ' ' " (2/. + lj(2 + ^)' 



If we confine ourselves to the case of a small magnet in 

 the centre of the shell : in the limiting case 



L = ^ = l + g=l + e, 



as against L = (1+v^) 2 , 



obtained when the internal and external radii were given. 

 If fi is nearly equal to unity, this reduces to 



i > -; 



R % 



i. e, there will be no " best arrangement *" unless the volume 

 of the shielding material exceeds the volume of the space 

 enclosed. If fju is very large, the expression becomes 



1 1_ 9_ 



R ' a 2fjb 



TV, 'f KM 1 ^ ' 009 



lhus, ir/i= 500, t> > • 



There will be no " best arrangement " unless the volume of 

 the shielding material is greater than (in round numbers) 

 one hundreth of the volume enclosed ; i. e. unless it is more 

 than sufficient to form a shell the thickness of which is 

 = 0*003 of the radius. 



In this case therefore, as in that in which the radius of the 



