Drude Model Permittivity Formula - e^iwt or e^(-iwt)?

In summary, the conversation discusses a problem encountered while deriving the permittivity formula using the Drude model. It is revealed that the issue was caused by using the wrong expression for complex field vectors. The conversation also touches on the convention for using either eiwt or e-iwt and how it affects the results. The conversation concludes with a discussion on the different signs used in AC circuit analysis and the implications for the complex part of the susceptibility in the fundamental linear equation.
  • #1
IcedCoffee
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4
I've been having a sign problem while deriving the permittivity formula using Drude model,

and I found out that the problem came from the fact that complex field vectors are expressed with e-iwt, not eiwt, thus producing (-iwt) term when differentiated:

http://photonics101.com/light-matter-interactions/drude-model-metal-permittivity-conductivity (See "Show Solutions")

Now, I guess that when you pick either eiwt or e-iwt and derive complex parameters like permittivity, you have to stick with it from then on in order to obtain valid real-part values with correct phase,

but is e-iwt the conventional one? I've seen the same formula for permittivity in Wikipedia... Is this because i(kx-wt) is more natural for waves propagating to positive x direction?
 
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  • #2
:welcome:In the linear response theory in the time domain, the fundamental equation is ## V_{out}(t)=\int\limits_{-\infty}^{t} m(t-t') V_{in}(t') \, dt' ##. The Fourier transform ## \tilde{F}(\omega)=\int\limits_{-\infty}^{+\infty} F(t) e^{-i \omega t} dt ##. The convolution theorem gives ## \tilde{V}_{out}(\omega)=\tilde{m}(\omega) \tilde{V}_{in}(\omega) ##. Then, using the inverse transform operation, ## F(t)=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} \tilde{F}(\omega) e^{+i \omega t} \, d \omega ##. The mathematics works equally well if the F.T. is defined with ## e^{+i \omega t} ##, and the minus sign in ## e^{-i \omega t} ## is then used during the inverse F.T. operation. ## \\ ## Because of the sign convention, the physicist uses impedances ## Z_L=j \omega L ## and ##Z_C= -\frac{j}{\omega C} ##while the EE has the signs reversed in analyzing AC circuits.## \\ ## (I think the physicist uses ## V(t)=V_o e^{+i \omega t} ## for a sinusoidal voltage in an AC electrical circuit, but when considering traveling waves, they switch to ## E(x,t)=E_o e^{i (kx-\omega t)} ##. I will need to check this result with some calculations, but I believe that is the case). ## \\ ## In an AC circuit analysis, the physicist's phasor diagram (because of the sign on ## i \omega t ##), in a graph of the complex ## V ## as a function of time, rotates counterclockwise, while the EE has their phasor diagram rotating clockwise. ## \\ ## And to conclude: Your assessment is accurate. The complex part of the susceptibility ## \tilde{\chi}(\omega) ## will have a different sign dependent on whether ## E(t)=E_o e^{+ i \omega t} ## or ## E(t)=E_o e^{-i \omega t} ## is assumed. ## \\ ## The fundamental linear equation here is ## P(t)=\epsilon_o \int\limits_{-\infty}^{t} \chi(t-t') E(t') \, dt' ##, and ## \tilde{P}(\omega)=\epsilon_o \tilde{\chi}(\omega) \tilde{E}(\omega) ## for the expression involving Fourier transforms from the convolution theorem. The susceptibility ## \tilde{\chi}(\omega) ## is actually the Fourier transform of the linear response function ## \chi(t) ##, and it is often written simply as ## \chi(\omega) ##.
 
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  • #3
Charles Link said:
:welcome:In the linear response theory in the time domain, the fundamental equation is ## V_{out}(t)=\int\limits_{-\infty}^{t} m(t-t') V_{in}(t') \, dt' ##. The Fourier transform ## \tilde{F}(\omega)=\int\limits_{-\infty}^{+\infty} F(t) e^{-i \omega t} dt ##. The convolution theorem gives ## \tilde{V}_{out}(\omega)=\tilde{m}(\omega) \tilde{V}_{in}(\omega) ##. Then, using the inverse transform operation, ## F(t)=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} \tilde{F}(\omega) e^{+i \omega t} \, d \omega ##. The mathematics works equally well if the F.T. is defined with ## e^{+i \omega t} ##, and the minus sign in ## e^{-i \omega t} ## is then used during the inverse F.T. operation. ## \\ ## Because of the sign convention, the physicist uses impedances ## Z_L=j \omega L ## and ##Z_C= -\frac{j}{\omega C} ##while the EE has the signs reversed in analyzing AC circuits.## \\ ## (I think the physicist uses ## V(t)=V_o e^{+i \omega t} ## for a sinusoidal voltage in an AC electrical circuit, but when considering traveling waves, they switch to ## E(x,t)=E_o e^{i (kx-\omega t)} ##. I will need to check this result with some calculations, but I believe that is the case). ## \\ ## In an AC circuit analysis, the physicist's phasor diagram (because of the sign on ## i \omega t ##), in a graph of the complex ## V ## as a function of time, rotates counterclockwise, while the EE has their phasor diagram rotating clockwise. ## \\ ## And to conclude: Your assessment is accurate. The complex part of the susceptibility ## \tilde{\chi}(\omega) ## will have a different sign dependent on whether ## E(t)=E_o e^{+ i \omega t} ## or ## E(t)=E_o e^{-i \omega t} ## is assumed. ## \\ ## The fundamental linear equation here is ## P(t)=\int\limits_{-\infty}^{t} \chi(t-t') E(t') \, dt' ##, and ## \tilde{P}(\omega)=\tilde{\chi}(\omega) \tilde{E}(\omega) ## for the expression involving Fourier transforms from the convolution theorem. The susceptibility ## \tilde{\chi}(\omega) ## is actually the Fourier transform of the linear response function ## \chi(t) ##, and it is often written simply as ## \chi(\omega) ##.

Ahh, so it's FT convention. That slipped my mind just thinking about the differential equation. Thank you!
 
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  • #4
One additional minor correction to the above: I think in MKS units, they often define the susceptibility so that ## \tilde{P}(\omega)=\epsilon_o \tilde{\chi}(\omega) \tilde{E}(\omega) ## with an ## \epsilon_o ## in the equation. And yes, I see the "link" you posted uses it with the ## \epsilon_o ##. Let me make that correction above.
 
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  • #5
Yeah, it's convention, and this involves some schizophreny even within the physics community alone (if you switch between physics E&M and electrical-engineering E&M textbooks, which often treat other subjects than the physics textbooks, you get completely confused).

Usually the convention in full Maxwell theory is to choose the time dependence for harmonically evolving fields as ##\exp(-\mathrm{i} \omega t)##, while for circuit theory where you deal with integrated quantities like voltages and currents they choose ##\exp(+\mathrm{i} \omega t)##, which is of course nuts, because the integrated quantities are just the fields integrated over space after all. Why they choose a different convention, you must not ask me. I don't know; maybe they like to confuse students even more than the subject itself is confusing, and it's confusing enough for the beginner (particularly when the SI units are used, but that's another story).

Then there are the various conventions concerning Fourier transformations of functions. In field theory, at least in the HEP community, the convention usually is like
$$\psi(t,\vec{r}) = \int_{\mathbb{R}} \frac{\mathrm{d} \omega}{2 \pi} \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 k}{(2 \pi)^3} \tilde{\psi}(\omega,\vec{k}) \exp(-\mathrm{i} \omega t+\mathrm{i} \vec{k} \cdot \vec{r}).$$
The inverse transformation then follows to be
$$\tilde{\psi}(\omega,\vec{k})=\int_{\mathbb{R}} \mathrm{d} t \int_{\mathbb{R}^3} \mathrm{d}^3 r \psi(t,\vec{r}) \exp(+\mathrm{i} \omega t-\mathrm{i} \vec{k} \cdot \vec{r}).$$
It may well be that in other communities you have to convention used by Charles Link in #2. In the engineering literature it's also not uncommon to write ##\mathrm{j}## for the imaginary unit.
 
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1. What is the Drude Model Permittivity Formula?

The Drude Model Permittivity Formula is a mathematical equation used to describe the dielectric constant (or permittivity) of a material at a given frequency. It is based on the Drude Model, which explains the behavior of free electrons in a material.

2. How is the Drude Model Permittivity Formula represented mathematically?

The formula is represented as e^(iωt) or e^(-iωt), where e is the base of the natural logarithm, i is the imaginary unit, ω is the angular frequency, and t is time.

3. What does the e^(iωt) component of the formula represent?

The e^(iωt) component represents the phase of the material's permittivity, which is a measure of how much the material's polarization lags behind an applied electric field at a given frequency. It is a complex number, with a real part representing the amplitude of the permittivity and an imaginary part representing the phase angle.

4. How is the Drude Model Permittivity Formula used in scientific research?

The formula is used to calculate the permittivity of a material at a specific frequency, which is important in understanding its electrical properties. It is often used in studies of materials such as metals, where free electrons play a significant role in the material's behavior.

5. Are there any limitations to the Drude Model Permittivity Formula?

Yes, the Drude Model Permittivity Formula is based on assumptions and simplifications, so it may not accurately represent the behavior of all materials. It also does not take into account factors such as temperature and impurities, which can affect a material's permittivity.

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