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I have a question in which I need to show that sea water is effectively a "good conductor", when considering the propagation of radio waves of frequency $< 10^9$. We're given that the conductivity of sea water is around $5 Sm^{-1}$ and has a refractive index of around $9$.

It is my understanding that in order to show that the sea water is a "good conductor", you would need to show that: $\sigma>>\omega\epsilon_r\epsilon_0$

$\sigma, \omega, \epsilon_0$ are trivial but I am not sure how you would get a value of $\epsilon_r$.

Edit: I managed to solve the problem using Gilbert's help.

Here's my proof:

As, $${v=\frac{1}{\sqrt{\mu\epsilon}}},$$ $$v=\frac{c}{n},$$ $$\mu_r=1,$$ $$\implies v=\frac{1}{\sqrt{\mu_0\epsilon_r\epsilon_0}}=\frac{c}{n},$$

$$\implies \mu_0\epsilon_r\epsilon_0=\frac{n^2}{c^2},$$

$$\implies \mu_0\epsilon_r\epsilon_0=\frac{n^2}{c^2},$$

$$\implies \epsilon_r=\frac{n^2}{c^2\mu_0\epsilon_0},$$

As $\epsilon_0=\frac{1}{\mu_0c^2}$,

$$\implies \epsilon_r=n^2$$

Oliver Wilkins
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  • P.66 of your notes? – ProfRob Dec 09 '18 at 23:41
  • I see this now... I also enjoyed the final problem of your sheet. After I had finished it, I applied its concepts to radio waves and electrical wiring - which is in itself, an interesting area. With a bit of digging in the general area I came across this. I was wondering if you would make an appearance with this question here... – Oliver Wilkins Dec 10 '18 at 06:24

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For non-magnetic materials, $n=\sqrt{\epsilon_r}$.

Gilbert
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  • Thanks for your answer Gilbet, do you have some resource in which I can read on this? – Oliver Wilkins Dec 09 '18 at 19:01
  • @HomelessSandwich check out Wikipedia: https://en.m.wikipedia.org/wiki/Electromagnetic_wave_equation. The refractive index is defined by the speed of light in a medium v=c/n. Then from Maxwell’s equations, you can see that the speed of propagation of electromagnetic radiation is 1/(\epsilon\mu). – Gilbert Dec 09 '18 at 19:21
  • Thank you very much. I managed to prove what you are saying with the following. $v=\frac{1}{\sqrt{\mu\epsilon}}$, $v=\frac{c}{n}$ and as $\mu_r=1$, $\implies v=\frac{1}{\sqrt{\mu_0\epsilon_r\epsilon_0}}=\frac{c}{n}$ , $\implies \mu_0\epsilon_r\epsilon_0=\frac{n^2}{c^2}$, $\implies \mu_0\epsilon_r\epsilon_0=\frac{n^2}{c^2}$, $\implies \epsilon_r=\frac{n^2}{c^2\mu_0\epsilon_0}$, As $\epsilon_0=\frac{1}{\mu_0c^2}$, $\implies \epsilon_r=n^2$ – Oliver Wilkins Dec 09 '18 at 20:27
  • @HomelessSandwich that’s right! I also have a typo in my above comment: it’s missing the \sqrt. – Gilbert Dec 09 '18 at 23:49