DIELECTRIC CONSTANT, ABSORPTION AND SCATTERING 273 
equation (7). Elimination of a‘, from the first pair 
and that of b' from the second pair of these equa- 
tions leads to 
‘ at (Np) [oa (e)]'— rm (pe) [N oa (Np)]’’ 
(14) 
_ mr202(p) [N p20? (N p)’—u2N 22 (Np) Lozn” (e)] 
mre (0) EN pen? (N p)\'—n2N22n” (Np) [zn (e)]! 
The primes at the square brackets stand for differen- 
tiation with respect to the argument of the Bessel 
function inside the brackets. Similarly, eliminating 
a* and 6%, respectively, one would get a’, and b! 
= 
appearing in the field strengths inside the sphere. . 
For the computation of either the scattered or ab- 
sorbed 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 kr >> k,a. It is important to notice in 
this connection that the coefficients a, and b, become 
small for n > ka, and the summation over n may 
then be limited to the integer n ~k,a. At great dis- 
tances r, k,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*®* 
a 
2 (kr) & == 5 
. n+1 
2)(lir) ae 2 —itkr— 5 ™) (15) 
kr 
From these asymptotic expressions one sees that the 
radial components of the scattered field strengths 
can be practically neglected; they decrease with r as 
1/r?, in contrast with the 6 and ¢ 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, 
E; = H; = 0,r >a, (16) 
and with o, = 0 
; 2n+1 
= nll, =(2) trem nS) Sian 
2 nn tl 
wales an) 
(«: sin 0 ap User cose, 
tad 2 2n+1 (17) 
B= — nity —(f ») Ber : f= n(n +1) 
ee dP P} 7 p 
sing. 
” sin 6 
Since the resultant field at any point outside the 
sphere is obtained by superposition of the incident 
and scattered or reflected fields, one has 
E= E;+E,;H=H,;+H,. (18) 
In view of equation (16), the complex Poynting vector 
associated with this resultant field is radial, so that 
S. = 3 (He Hy* — Ey Ho*), (19) 
where an asterisk denotes the complex conjugate. 
Using equation (18) one gets 
S.= 3 (By Hy" — Ei, Hj") +4 (B4 H3* — Bi, Hy") 
+ 4(HGH,* + Ey H,* — Ey, Ho* — Ey, H5*). (20) 
The first term on the right-hand side is the rate of 
flow of energy in the incident wave and the second 
term is the rate of flow in the scattered wave. The 
total scattered power is then 
2x 
P,= pRe/ ih (Ey Hi,* — Ey H5*) r? sin odédd, 
0 0 (21) 
where Re denotes “Real part of . . .” and the integral 
is extended over the surface of a large sphere of radius 
r. In our case, using equations (16) and (17), one 
gets 
Pi= 5 -Ref ae (| #5 |2-+ | #$| 2) 7?sin ededg. 
(22) 
In the case of an absorbing sphere, the net flow of 
energy across a closed surface around the sphere is 
absorbed energy flow, and it is directed inward. One 
may thus write that this absorbed energy, which dis- 
appears in the form of heat, is 
2n f(r 
Pasi i _ |, (Sa rsin dado. (28) 
0 
Since the integral of the incident flow across a closed 
surface is zero, equation (23) in connection with 
equation (21) leads to the definition of the rate of 
flow of total energy or the power subtracted from 
the beam, i.e., (Pay + P,), as an integral over a closed 
surface of the third term on the right-hand side of the 
radial component S_ of the Poynting vector, equation 
(19). Thus 
2x cr 
Pox Past P= 3(— Re) [ i; 
(Ey H* + EB; Hi,*— E’, Hi* — E%, H5*) r? sin 6dédd. 
(24) 
Substituting equations (16) and (17) into equation 
(24) and remembering that the ¢ integration leads 
one finds, 
2 @o 
P, = 22S Qn +) la, +1640), @5) 
272 
n=1 
eee 
iL 0 (= F dé sin iad 
2 
= pac il [In(n+1)]?, 
to a factor 7 and that the integrais over products of 
the associated Legendre polynomials P}(z) are dif- 
ferent from zero only in the following combination of 
these products appearing in equations (22) and (24), 
