301 



Polarization of single sphere in external field. 



Before proceeding with the solution of the problem it will be 

 convenient to derive an expression for the stale of polarization of 

 a dielectric sphere placed in a known external tield. Charges of 

 polarization are induced. If the electric intensity due to these.charges 

 be Ei and if the impressed electric intensity be ^g, the total intensity 

 is E^Ee -\- Ei. Let us suppose that Ei may be derived from a 

 potential 



An "n (COS I J ) COS mff 



V— 2 — — — ...... (4) 



n,m 



y.n-\-\ 



outside the sphere. Then it must be derivable from 



V z= Z A'n r« P,"' {cos d) cos nuf (5) 



inside the sphere since the potential is continuous at the surface. 

 Denoting the components along the outward drawn normal by the 

 sufTix n and referring to the state just inside the sphere by n, and 

 to the state just outside by n^ we have the boundary condition 



«0 {Een + Ein^) == Egn + ^/«a 



or 



(g, 1) Een = Ein^ — €^ Ein^ 



' Using (4) and (5) 



{e,--\) Een = ^ (ne^ -\- n i- I) An Pn {cos &) cos mrp ■ • (6) 



n,7ii 



Thus if Een can be expanded in a series of surface harmonics 

 the coefficients ^'" may be determined fron) (6) and hence the state 



of polarization of the sphere may be obtained. 



Derivation of expansion for Eg»- 



In order to solve the problem it will be sufficient to express Een 

 in terms of A"^ and substitute the result in (6). 



A " 

 The average polarization of the medium being -y^- we have 



the first summation being extended over all the spheres inside a 

 large sphere having its centre at the origin, the dielectric sphere 

 situated at the origin being omitted from the summation as indicated 

 by the accent. Using the notation : 



