# How can I calculate dilectric constant from conductivity

1. Jan 31, 2017

### Aseel5

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

2. Jan 31, 2017

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.

3. Jan 31, 2017

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 $J_p=\dot{P}$.

4. Jan 31, 2017

### Aseel5

Thanks so much Charles,

5. Feb 1, 2017

### Phellippe Marques

Why cant I see the equations, only what I presume to be the code for equations?

6. Feb 1, 2017

### Staff: Mentor

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.

7. Feb 1, 2017

### f95toli

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.