152 



DIELECTRIC CONSTANT. ABSORPTION AND SCATTERING 



will HOW lie expaiided, hi analogy witli llie incident 

 field (Ei, Hi). Thus, 



E. = i?oZ(-j)" 



(«„ mj,/ + jbn n„/) , 



nin + 1) 



(9) 



H. = - ^ Z i-j)- 4^ (6: m^:!> - jal n«>) , 

 ij'i „=i n{n+ 1) 



valid at distances r^a, i.e., outside the si^here in 

 medium 2. Clearly in equations (6) and (7), in the 

 expressions of m and n, ]i„ re23laces h according to 

 equations (5) and (6). 



Inside the sphere (complex wave numlier l\. in- 

 trinsic impedance i;i), the transmitted field is ex- 

 panded ill the following way: 



„_i n{n+l) 



H, 



E. 



U(-jy 



2/1 H- 1 



(10) 



nJ ^1 • / „(1)\ 



[o„ m,„ —ja„ n„„ ) . 



VI „ = i n{n+ 1) 



The final determination of the scattered and trans- 

 mitted fields is thus reduced to finding the coefficients 

 (or amplitudes) a^^, 5^^ and a^, b^. 



The jDreceding formulas permit one to write down 

 rapidly the polar conpionents of the different field 

 strengths (E,,-, Hi), (E„ H^), and (E„ H,). The 

 boundary conditions at the surface of the sphere de- 

 mand the continuity of the tangential components of 

 the total field outside the sphere and the transmitted 

 field. If we denote these tangential components by 

 subscripts 6 or (/>, the boundary conditions take on 

 the following form : 



E'o + El = E'o,m + H' = H'e,r= a. (11) 



These lead to the following systems of equations for the 

 determination of the coefficients (a', b],) and (aj,, &/ ) : 



aU,?' (A^p) - a;^, sf (p) = zHHp). (12) 



M2a„ 



-4- iNpz?^ (Np)] -Mi«:, |-[p~^f' (P)] 

 d{Np) dp 



= p,±[p^^^{p)-], 

 dp 



n,Nb'J^ {Np)-^,hl^? (p) = M.4" (P), 



[iVp2<,"(iVp)] 



d{Np) 



mi^[p^Hp)] = N^[pz}^{p)l 

 dp dp 



where N = l\/l\_, p = l\a and the z 'J' (.r) and 

 Zi'^' (.r) are the spherical Bessel functions defined in 



&!, = - 



equation (7). Elimination of a[^ from the first pair 

 and that of V^ from the second pair of these equa- 

 tions leads to 



s ^ _ Mi~-i" {Np) [p^f," (p)]' -p.^f," (p) [Np:^ jNp)-]' 

 " P^}^ {Np) [p^f (p)]' -p=.f (p) [Npz^^ {Np)]' ' 



(14) 

 pi^;," (p) [A^p^w {Np)]'-p,N'-z'^' {Np) [pzl!'> {p)Y 

 p,zl?>{p)[Npzir{Np)]'-p,NhlP{Np)[pzf\p)]'' 



The primes at the square brackets stand for differen- 

 tiation with respect to the argument of the Bessel 

 function inside the brackets. Similarly, eliminating 

 (/;' and 6',, respectively, one would get a[^ and &^| 

 aijpearing in the field strengths inside the sphere. 

 For the computation of either the scattered or ab- 

 sorlicd radiation, one needs to know the field strengths 

 at a large distance r from the center of the sphere, 

 i.e., for r ^ a, or Ji„r ^ h^a. It is important to notice in 

 this connection that the coefficients a„ and h,, become 

 small for n > k\a, and the summation over n may 

 then he limited to the integer ii' — Ji\a. At great dis- 

 tances r, l\r > n; in other words, the order n of the 

 terms of importance is less than the argument (k^r) 

 of the spherical Bessel functions. Under these condi- 

 tions the asymptotic expressions of these functions 

 can be used. These are given by^^^ 



zirun-) ^ 



cos 



(-^) 



zlp(kr) 



1 



-i{kr- 



(15) 



From these asymptotic expressions one sees that the 

 radial comjKuients of the scattered field strengths 

 can be practically neglected; they decrease with r as 

 I//'-, in contrast with the 6 and i^ components which 

 decrease as 1/r. This means that for large r, the field 

 is transverse to the direction of propagation (radia- 

 tion zone). Hence, 



El = III = 0, r » a, 

 and \\ilh o-„ — 



(16) 



Is ^ " lis '" " 1 , 



l««-^^ + &„-— )cos</., 

 \ sin d dd / 



(13) £■« = - yTiH% = 



(£) 



EoC 



2n + 1 (17) 



„ = 1 n{n + 1) 

 / „ dPl Pi \ . 



\ dd sm 6 / 



