How can I calculate dilectric constant from conductivity

[Aseel5]

Hello,
could you help me please about how can I calculate dielectric constant form electrical conductivity?

 

[Charles Link]

There is a connection between the linear response equations for a dielectric medium:
$P(x,t)=\int \chi(x-x',t-t') E(x',t') \, d^3 x' \, dt'$ and $J_p(x,t)=\int \sigma(x-x',t-t') E(x', t') \, d^3x' dt'$. Taking Fourier transforms these become: $\tilde{P}(k,\omega)=\tilde{\chi}(k,\omega) \tilde{E}(k,\omega)$ and $\tilde{J}_p(k,\omega)=\tilde{\sigma}(k,\omega) \tilde{E}(k,\omega)$. The equation $J_p=\dot{P}$ (for the polarization current=it follows also from the continuity equation) and its Fourier transform $\tilde{J}_p(k,\omega)=-i \omega \tilde{P}(k,\omega)$ tie these together, along with $D(x,t)=\int \epsilon(x-x',t-t') E(x',t') \, d^3 x' \, dt'$ and its Fourier transform, $\tilde{D}(k, \omega)=\tilde{\epsilon}(k,\omega) \tilde{E}(k,\omega)$ so that $\tilde{\epsilon}(k,\omega)=1+4 \pi \tilde{\chi}(k,\omega)$. $\tilde{\sigma}(k,\omega)$ is the conductivity, and $\tilde{\epsilon}(k,\omega)$ is the dielectric constant. I used cgs units so that $D=E+4 \pi P$, but conversion to any other units can be readily done. Hopefully this was helpful. (With a little algebra, you can solve for $\tilde{\sigma}(k,\omega)$ in terms of $\tilde{\epsilon}(k,\omega)$). For a reference, Ichimaru's Plasma Physics book has much of this in the first couple of chapters.

 

[Charles Link]

Note: The continuity equation is $\nabla \cdot J_p+\frac{\partial \rho_p}{\partial t}=0$. Since $\rho_p=-\nabla \cdot P$, this gives $\nabla \cdot (J_p-\dot{P})=0$ which gives the result that the polarization charge current density $J_p=\dot{P}$.

 

[Aseel5]

Thanks so much Charles,

 

[Phellippe Marques]

Why can't I see the equations, only what I presume to be the code for equations?

 

[NascentOxygen]

Why can't I see the equations, only what I presume to be the code for equations?

I think the app does not run whatever is needed to interpret MathJax. So users of the PF app will not see the rendered equations.

 

[f95toli]

Hello,
could you help me please about how can I calculate dielectric constant form electrical conductivity?

You have to be more specific? Do you mean in theory (see Charles response)? Or in practice (i.e. you have some data for the conductivity of a dielectric) ?
If it the latter the answer is that you can't, at least not in the general case. I can think of a few situations where it might work, but you would need data which -as far as I am aware- tends to be quite difficult to measure. It usually makes more sense to measure the dielectric constant directly by measuring the capacitance or -even better- putting the material in a resonator.