Particular solutions and boundary conditions

Now let the origin of the coordinate system be the particle center, and the direction of propagation of the incident radiation be along the z- (or 9 = 0) axis in the positive direction. Further, let the electric vibration of the incident wave be in the x—z plane ( /) =0°), and k and k2 be the propagation constants outside and inside the particle of radius a, respectively. 

It is useful to think of the total field (Eq, Hq) as being composed of three partial fields an incident and a scattered field outside the particle, as well as a field within the particle. Solutions for these fields can be expressed as expansions in u and V with undetermined coefficients, each term representing a particular integral. The coefficients can then be determined from the boundary conditions, which are that the four tangential components of the total field Eqs, Eq, Hoe, and Ho,p remain continuous across the spherical surface r = a even though the propagation constant k and magnetic permeability /x are discontinuous. The conditions that the radial components Eor and Hor are also continuous across the surface then follow automatically from Maxwell s equations. 

We recall from Eqs. (3.8.3) and (3.8.4) and the related discussion that Eq and Ho of the incident field are independent of time, and define the amplitude of the electric vector to be normalized to unity. Because of the coordinate system chosen, the Poynting vector is along the positive z-direction and Eq is in the x—y plane. Hence, from Eq. (3.8.3), the electric field for the incident wave is [see Eq. (1.3.22)] 

An analysis of Hor completely analogous to that just undertaken for Eor reveals that 

Equations (3.8.39) and (3.8.41) together suffice to describe the complete incident field. 

---