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Polarizaition and susceptibility

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KFC
#1
Jun3-09, 12:11 PM
P: 369
In some unit, the relation of (linear) polarization and susceptibility can be written of

[tex]P(t) = \chi E(t)[/tex]

but I also read some expression in other text reads

[tex]P(\omega) = \chi(\omega) E(\omega)[/tex]

why change the time to frequency? Why polarization depends on frequency?
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Andy Resnick
#2
Jun3-09, 12:27 PM
Sci Advisor
P: 5,524
Your equations are not really written correctly. The first one, the time dependent one, should really be written as a convolution: The polarization of a linear isotropic medium with a local but noninstantaneous response (but still independent of time) is:

P(t)=[itex]\int \chi(t-\tau)E(\tau)d\tau[/itex]

And taking the Fourier transform of this equation provides your second expression.

If the material responds instantaneously and has no memory[[itex]\chi(t-\tau) = \chi\delta(t-\tau)[/itex]], then the convolution integral reduces to your first expression.

Having a frequency-dependent susceptibility is simply dispersion.
KFC
#3
Jun3-09, 12:54 PM
P: 369
Quote Quote by Andy Resnick View Post
Your equations are not really written correctly. The first one, the time dependent one, should really be written as a convolution: The polarization of a linear isotropic medium with a local but noninstantaneous response (but still independent of time) is:

P(t)=[itex]\int \chi(t-\tau)E(\tau)d\tau[/itex]

And taking the Fourier transform of this equation provides your second expression.

If the material responds instantaneously and has no memory[[itex]\chi(t-\tau) = \chi\delta(t-\tau)[/itex]], then the convolution integral reduces to your first expression.

Having a frequency-dependent susceptibility is simply dispersion.
Oh ... I just wonder why in textbook they don't say it is a convolution! So you mean in frequency domain susceptibility is the repsonse function?

BTW, can you tell me one text in which the author show clearly the convolution relation b/w polarization, susceptibility and field? I am writing a short report and need a reference


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