30 Prof. W. H. Bragg on the " Elastic 



displacement IricyP 2 "" Typs) alon g PO, an( * this is equal to 



a cos 0. But this is also the displacement along P pro- 

 duced by moving the sphere a distance a to the left. 



The surrounding medium therefore will take up that strain 

 (in addition to its previous strain) due to charges — %irr z at 

 0, and 27rr 3 at 0'. 



The difference of pressures at P and P' produced by such 

 a strain is easily found. 



The original difference before the sphere was displaced was 

 2r cos (f> . E x b. In consequence of the new charges the pres- 

 sure at P is lessened by the amount 



Ej_ /2th' 3 __ 2tt/- 3 \ 



47T*VOP 0P7 



__ E x 27JT 3 . a cos (j) 



47r * r 2 



__ Ejm cos (j> 

 ~2~~ ; 



and the pressure at P' is increased by that amount 

 The difference therefore amounts now to 



2rcos$E 1 fr — Ex?' a cos <j>. 



The difference of pressures at P and P' due to the strain of 

 the medium within the sphere is 2(b + a)r cos<£.E 2 . Equate 

 these two, we have 



2.E 2 (& + a) = 2E 1 6-E 1 a 



= 2b. 



E x — E 2 



E! + 2E 2 * 



This value of a is independent of <j>; if then a have this value, 

 the pressures balance everywhere. 



So the total displacement at B, which is sometimes called 

 the density of the charge, 



= a + b 



3Ei 



=b 



Ei + 2E 2 * 



If Ei = E 2 , a=0, as of course it should. 

 If E 2 = 0, i.e. the sphere is a conductor, the density of the 

 charge at B = 36. 

 13. It is easy to find the loss of energy caused by the 

 presence of the sphere in the field. We can first of all find 

 the work that must be done to restore the external medium 



