I am wondering about a specific question regarding the speed at which an electrical current traverses through salt-water / saline.
By this I do not mean the electron drift speed - I mean, at what speed would a current travelling from an anode to a cathode immersed in salt water be?
A ball park figure will do. Thank you.
The transmission speed of a signal in anything, saline, water or vacuum is given by:
$$ v = \frac{c}{\sqrt{\epsilon}} $$
where $\epsilon$ is the relative permittivity. See this article for a little more info, or Google for many related articles. I couldn't find a figure for saline, but the relative permittivity of water at STP is about 80 (at 1kHz), so that suggests the signal transmission speed in saline would be of order $c/\sqrt{80}$ or about $0.11c$.
To expand on John's answer, it all depends on the conditions. The relative permittivity that he mentioned is based on the properties of the current and the solution. The concentration of the solution would affect the transmission speed. The properties of the electicity such as the voltage, current, and frequency (if AC). Unfortunately, I don't know an equation that will get you directly to the result, and I think the only way to get that information would be to experiment and see what kind of results you get.
As another note on John's answer, I believe that equation is only applicable to electromagnetic waves, and not current flow of electricity. I may be wrong there, though.
UPDATE:
I have found out that John's equation $v = \frac{1}{\sqrt{\epsilon}}$ is indeed correct. In a more complete form, there is also a factor of $\mu$ for the permeability, but in water that is basically 1, so for most purposes it can be ignored. The issue is finding the permittivity ($\epsilon$); typically this is done by experimentation and finding it mathematically is very challenging. I did manage to find a way to calculate the conductivity of a saline solution, assuming the solution only consists of $Na^+$ and $Cl^-$ ions. The conductivity is as follows:
$$
\kappa = \frac{\Lambda_m^\circ - K\sqrt{c}}{c}
$$
Where:
$\kappa$ is the conductivity
$\Lambda_m^\circ$ is the molar conductivity (12.645 for saline solution with parameters above) Wikipedia link to calculation
$K$ is the Kohlrausch coefficient, which depends on the ions in the solution. Unfortunately, I could not find any calculations for this.
$c$ is the concentration of the solution.
I have searched for relations of conductivity to wave speed. permittivity, and susceptibility, but I have not found anything. The closest thing that I have found to relate conductivity with is Ohm's law: $$ \vec{J} = \sigma\vec{E} $$ Where $\vec{J}$ is the current density, $\sigma$ is the conductivity and $\vec{E}$ is the electric field. If you were able to find the current density, you would then have the electric field and from the should be able to calculate the speed. Unfortunately, I have not found a way to calculate the current density as every way that I know of deals with an area, which you just don't have a figure for when you're dealing with an open space. I'll keep looking but I haven't found anything yet. If anyone has any ideas, please let me know.
As an additional note, this is a halfway-related question that is interesting at the least.
