Scattering 1 Classical scattering of a charged particle (Rutherford ...

Now work out the Fourier components of the distribution.ρ k = ∑ A n cos(n[φ − ωt])The coefficients A n are found using othorgonality of the Fourier functions, so that;Then;ρ k =q δ(r − a)(n + 1)2π a 2 δ(θ − π/2) ∑ ncos(2πkn/N) cos(n[φ − ωt])ρ = N−1 ∑k=0ρ kThe sum over k provides the followingN−1 ∑k=0cos(2πnk/N) = cos(nπ[1 − 1/N])sin(nπ)sin(nπ/N)This is zero unless n = N or 0. Thus the radiation has multipolarity N. But if N → ∞there will be no radiation. In effect the radiation from each charge elecment contributescoherently, cancelling the radiated power.10 Index of refractionWe choose to look at a simple model that will demonstrate that the index of refraction isdue to an interference between the incident wave and the scattered wave represented byradiation from absorption of the incident wave on atomic electrons. Look at fig 7. Theatomic electrons in a ring of radius ρ in the (x, y) plane are caused to harmonically oscillatedue to a plane wave incident along the z axis. We assume the electric vector of this wave ispolarized in the x direction.The electrons are bound to atoms, we assume by a linear force, resulting in the non-relativisticharmonic force equation;m d2 xdt 2 + kx = qE e iωtThe 1 2t term is the intertial force, ma, the 2 nd term is the restoring force keeping the electronbound to the nucleus, and the 3 rd term is the driving force due to the EM field.The solution is ;x = x 0 e iωt = qE/mω 2 0 − ω 2 e iωt 14