I’m trying to derive the relationship between conductivity and permittivity:
ε(K,ω) = 1+ (iσ(K,ω))/(ε0 ω)
using the equations :
(1) D(K,ω) = ε0 ε(K,ω)E(K,ω)
(2) J(K, ω) = σ(K,ω)E(K,ω)
(3) D = ε0 E + P , and
(4) J = ∂P/∂t
D: Dielectric displacement
J: Current density
P: Polarization
ε: Dielectric function
σ: Conductivity
E: Electric field
t: Time
K: Wave Vector
ω: Frequency
"Recognizing that in the Fourier domain ∂/∂t → −iω"
(This is from the textbook, Plasmonics: Fundamentals and Applications, page 9 by Stephen Maier)
However, I’m not overly strong with the Fourier Transform, I only know the basics.
To derive the relationship between conductivity and permittivity, we can use the equations given:
Starting with equation (3), we can substitute equation (1) into it to get:
D(K,ω) = ε0 ε(K,ω)E(K,ω) = ε0 (1+ (iσ(K,ω))/(ε0 ω))E(K,ω)
Rearranging this equation, we get:
E(K,ω) = D(K,ω) / [ε0 (1+ (iσ(K,ω))/(ε0 ω))]
Well, to make things a little easier, you can assume the plane wave character of the electric field without any loss of generality. Combining eqs. (1) to (4) and taking a time derivative, you'll reach the relation:
dE/dt * (ε0ε) = ε0 * dE/dt + σE.
Now use the plane wave assumption: E = E0 * exp(K dot x - ωt). You'll end with with a relation that can be used to derive ε = 1 + iσ/ωε0.
Hope that helps.
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