112 STATIC ELECTRICITY 



conductor as if it were a doublet, the forces on the external charges 

 producing the field will be the same as those of the doublet. 



We have just seen that two almost overlapping spheres with 

 equal radii a and equal and opposite densities of charge, will change 

 the internal intensity from E to 0, i.e. by an amount E if they are 

 equivalent externally to a doublet of moment Ea 8 . Then in 

 order that the internal intensity may be changed from E to E', i.e. 

 by an amount E E', the pair of spheres must be equivalent 

 externally to a doublet (E E')a 3 . 



Let us assume that when a dielectric sphere, radius a, and with 

 dielectric constant K is substituted, the field is uniform and equal 

 to E' within, and without is E with the field due to the doublet 

 (E - E')a 8 superposed on it. In order that this may be the actual 

 arrangement of intensity it has to satisfy the two conditions: 

 (1) equality of intensity in the two media tang-nti:il to and close 

 to the surface of the sphere; and (2) equality of strain in the t\\o 

 media normal to and close to the surface. 



The tangential intensities at a point on the surf'.. -nt // 



in the field assumed are : just outside, E sin 9 (E I '.') >in = 

 E' sin ; just inside, E' sin 6. 



So that any uniform value of E' will sati-fx tlii> condition. 



The normal intensities at the same point are: just ouNidc, 

 2(E-E') cos 0+ E cos =(,'3K-^K') cos th. ju>t ins: 



Since the strains are respectively intensity /4-Tr in air and 

 K x intensity /4?r in the sphere, we" must have for equalitx 

 normal strain 



3E - 2E' = KE' 



whence E' 



The moment of the doublet equivalent for the outside is 

 (E - E') a 



and the values of the internal and external fields thus obtained 

 satisfy the conditions, and so constitute a solution. We shall assume 

 that it is the only solution. 



Since the field without is modified as if the dielectric \\i-re 

 replaced by the doublet, the action on any external charge will be 



TjT _ I 



the same as that of the doublet, or will be -^ - -^ times the action 



JV + * 



of the equal conducting sphere. Similarly, the reaction of the 



tr _ 1 



charge on the dielectric will be ^ - times the action on the 



equal conducting sphere. We shall see how Boltzmann used this 

 result to obtain K. 



