# Dielectric constant of water

I need the dielectric constant of water from $10^{-2}$ Hz to $10^4$ Hz. As stupid as it may seem, I cannot find much info. I've googled for days. All I can find is close to GHz. And the only info close to Hz, ($100$ Hz) shows a great variation. A relative dielectric constant at $100$ Hz of about $4000$. So, I cannot interpolate back in frequency (I put a link to the paper at the end). Does anyone have any info about where I could find this data? I know that for constant current and about $20$ C the constant is $80.1$. What about at $50$ Hz?

I need the complex dielectric constant to test a program. Any lead would be really appreciated.

http://arxiv.org/abs/1010.4089

• I suspect the problem is that in this range of frequencies, the value depends greatly on contamination in the water - so unless you have really pure deionized water, currents will flow at these low frequencies that will dominate any effects you are trying to measure. Commented Jul 6, 2015 at 2:02
• How much do you care about the imaginary component? If you are only testing a program, is 5% "good enough"? The value of 78 from the paper @akhmeteli found certainly seems valid - and not inconsistent with the 80.1 you found for DC. Do you need it better than 3%? Commented Jul 6, 2015 at 2:17

Sometimes "absence of evidence" leads to "safe to extrapolate" . My bet is that the dielectric value is, to more precision than you could use, constant from 0 to 1 MHz. I notice the wikipedia entry under permittivity suggests at least 0 to 1kHz.

However, your search-fu may be wanting, grasshopper. I found this calculator:

http://www.random-science-tools.com/electronics/water_dielectric.htm

and this paper: http://arxiv.org/pdf/1010.4089.pdf

• At low frequencies you can bet that the conductivity is independent of frequency. But that means that the imaginary part of permittivity is inversely proportional to frequency, i.e. goes to infinity as the frequency approaches zero. en.wikipedia.org/wiki/Mathematical_descriptions_of_opacity Commented Apr 22, 2014 at 14:03
• I would hesitate to recommend using a calculator at "random-science-tools.com". Commented Jul 3, 2015 at 20:40
• @DanielSank really - and why is that? Judging a web page by its URL is pretty much the 21st-century equiv. of judging abook by its cover. Remember books? :-) Commented Jul 4, 2015 at 1:37
• @CarlWitthoft you're kidding, right? If the URL started with something like $\text{www.nist.gov}$ I might feel otherwise. This is not like judging a book by its cover by rather by author. If a book is authored by "Random Physics Guy McGee" (and published by ACME Comics Inc) I'd trust it less than if it were written by Lev Landau and published by McGraw Hill. C'mon, get rid of the shady calculator link. Commented Jul 4, 2015 at 1:39
• @DanielSank that shady calculator link, contains a link to a paper on which it is based. Whether you trust those authors is another question. I can understand your doubts, but its not just a random calculator without any possibility to see where it comes from. Commented Jun 27, 2017 at 9:55

There is some data in http://www.nist.gov/data/PDFfiles/jpcrd487.pdf (J. Phys. Chem. Ref. Data, vol. 24, No. 1, 1995, p. 33) See, e.g., Table 2 there. Looks like dielectric permittivity of water is about 78.

• Excellent link. Graph 3c shows a very small effect of frequency below 20 kHz (note the vertical scale: 4540 to 4560). Note the observation in this paper (last paragraph of page 41): [...]specific conductance of the water used by Vidulich et a. which was a factor of ten less that that used by Malmberg and Maryott . I think that for the imaginary part of the dielectric constant (losses), the exact composition of the water will play a big role. Commented Jul 6, 2015 at 2:13
• @Floris: Actually, measurement of water conductivity is the leading method of water purity measurement, AFAIK:-) And ultrapure water's conductivity very strongly depends on the temperature iccontrols.com/files/4-2.pdf Commented Jul 6, 2015 at 2:47
• We agree completely. Indeed, you can talk about "18 MOhm water" when you mean water that has been fully deionized (resistivity 18 MOhm-cm). And the moment it is exposed to air, the resistance drops as CO2 dissolves... it will even suck ions out of its glass container. Tricky stuff. Commented Jul 6, 2015 at 3:06
• @Floris: No disagreement was implied in my comment:-) And thank you for the interesting info. Commented Jul 6, 2015 at 4:42

EDIT#2:

• I'm now made aware that you need wavelengths that are much larger than those presented here(a bit of an oops from reading this question quickly). This approach is still valid, but what you need cannot be obtained from these data. I'm going to leave this here however to collect downvotes and if anyone needs $\epsilon_r$ as it depends on $10^{7}$ to $10^{16}$ Hz

You could find the imaginary(absorption) and real parts of the complex refractive index of water

$$\bar{n} = n + i \kappa$$

and relate to the relative permitivitty where

$$\epsilon_r = n^2 - \kappa^2$$

at a given frequency. Some information on the frequency dependent absorption and refractive index are certainly available.

EDIT: See www.philiplaven.com/p20.html -- Figure 6: Complex refractive index of water at different wavelengths.

so it is clear the data exists. This is from D. Segelstein, "The Complex Refractive Index of Water", M.S. Thesis, University of Missouri, Kansas City (1981). You can download the data at this page. From this data, you can construct the relative dielectric permitivitty using the formula above.

• These wavelengths correspond to much higher frequencies than is asked for in the question. 10 m wavelength of light is 30 MHz (even higher if this is wavelength in water - not clear from the plot). OP is looking for data that is a few orders of magnitude lower. Commented Jul 6, 2015 at 2:15