A non-magnetic dielectric consists of N atoms per unit volume. Model the polarization of this system as the dipole moment per unit volume P = −Ner, where r(t) is the displacement of each electron from its nucleus. The dynamics of each electron is modeled as a damped harmonic oscillator:

(a) Do not assume any particular time dependence for E and establish that

where S = μ₀ ⁻¹  E × B, u_EM = 1 / 2 ϵ₀(E · E + c²B · B), and u_mech is the total mechanical (kinetic plus potential) energy density of the Lorentz oscillators. Physically interpret the last term on the left-hand side above.
(b) Now assume that a time-harmonic wave propagates through this medium. The index of refraction  n̂ (ω) = n̂₁(ω) + in̂₂(ω) is

where ω²_p = Ne² mϵ₀. Show that (S) = 1 / 2 ϵ₀ c_n̂1|E|² k̂ is the time-averaged Poynting vector if the real and imaginary parts of k are parallel.

(c) Show that the energy density (u) = (u_EM + u_mech) = 1/2ϵ₀ |E|² (n̂²₁ + 2ωτn̂₁n̂₂) when similarly averaged so that the “energy velocity”

(d) Prove that ν_E(ω)

Transcribed Image Text:

d²r 1 dr + dt² t dt + wor -E. m