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I am able to measure relative permittivity $\varepsilon_r$ of a fluid, and I want to calculate conductivity of same fluid. Can anyone suggest me how to do this?

I found one formula to calculate conductivity but i am not sure it is right or wrong. The formula is $$\varepsilon=\varepsilon_r\varepsilon_0 + j\frac\sigma\omega,$$ but i know that $$\varepsilon=\varepsilon_r\varepsilon_0 .$$

If this is equal then $\sigma$ should be zero according to this formula. I think the above one is wrong.

Can anyone tell me how to calculate a material's conductivity from its permittivity?

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2 Answers 2

In general, you do not have enough information to do this. A material's permittivity and its conductivity are fundamentally different quantities, and cannot generally be calculated from one another, particularly if all you have are static measurements.

The main reason for this is that a material's permittivity is a purely static property, while its conductivity involves its dynamics. (One way to see this is in their units: $[\sigma/\varepsilon]=1\textrm{ s}^{-1}$, so any connection between them must involve a specific timescale.) As Purcell makes quite clear, conductance is a time-dependent phenomenon, and the distinction between insulators and conductors depends on what timescale you're considering, much like distinguishing solids and liquids depends on the timescale.

That said, a material's permittivity and its conductivity can be very productively integrated into a single, complex quantity. (Confusingly, this is often called its permittivity, with little distinction.) It is important to note that this is only ever done at one specific frequency $\omega$. In this case we write $$\epsilon(\omega)=\epsilon'(\omega)+j\epsilon''(\omega)=\epsilon_r(\omega)\epsilon_0+j\frac{\sigma(\omega)}{\omega},\tag{1}$$ where $j^2=-1$. This is very useful for oscillating fields, even in the quasi-static regime, as it integrates and simplifies a good number of formulae, but it is not very useful for DC fields. (Why the complex numbers? Oscillating fields are often represented as $\mathbf E=\mathbf E_0 e^{j\omega t}$, which puts $j$ into the game. If $\omega=0$ then the game is over.)

It is important to stress that equation $(1)$ is the correct one. For DC fields or where $\sigma$ can be neglected, we usually shorten it to $\varepsilon=\varepsilon_r\varepsilon_0$, but the latter is only an approximation of the former.

Having said all of that, I must add that there is one way to calculate a material's conductivity from its permittivity. This relies on the fact that, because of causality considerations, $\varepsilon(\omega)$ must be an analytic function for complex $\omega$ in the upper half-plane. Because of that, its real and imaginary parts must obey what are known as Kramers-Kronig relations, which let you calculate $\sigma(\omega)$ as $$ \sigma(\omega)= -\frac{\varepsilon_0\omega}{\pi} \mathcal P\int_{-\infty}^\infty \frac{\varepsilon_r(\omega')}{\omega'-\omega}\mathrm{d}\omega', $$ where $\mathcal P$ denotes the Cauchy principal value of the integral must be taken at the singularity at $\omega'=\omega$. This is, however, rarely a practical option, as you need to know the electrostatic response $\varepsilon_r(\omega)$ at all frequencies $\omega$, or be in a position where you know you can neglect all frequencies outside some interval. If all you have is a single, static measurement of $\varepsilon_r$, then this can't possibly work.

I hope this is enough to point you in the right direction (which is, of course, to go and measure the conductivity directly!).

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The permittivity $ \epsilon(\omega)$ could be a complex quantity, and you can write it :

$\epsilon(\omega) = \epsilon'(\omega) + j\epsilon''(\omega)$, where j is the imaginary unit.

In the case of a material with conductivity, $\epsilon''(\omega)$ comes from conductivity : $\epsilon''(\omega)= \frac{\sigma(\omega)}{\omega}$

So, if you know the imaginary part of the permittivity, you know the conductivity.

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