# What is the speed of electrical current in salt water?

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.

• So do you mean... how long it will take for a current to be established once a voltage difference is applied? The speed of current is by definition the electron drift speed, is it not? – Kyle Oman Jul 16 '13 at 18:56
• @Kyle Maybe... sorry for not being clear, I am really asking a very simple question - I have a electrical current that is being propagated through salt water. What is its speed? For example, I understand that the speed of a current travelling in copper wire is on the order of the speed of light. So similarly, what is the speed of electrical current through salt water? – The Grape Beyond Jul 16 '13 at 19:03
• I think you're trying to ask this: If there is a resistive mass of saline, with two widely-separated electrodes in it initially at ground potential, and one electrode is subject to a voltage step at time 0, how much later will the voltage be felt at the other electrode? If that's your question, I would vote for John's answer. – Mike Dunlavey Jul 16 '13 at 21:20
• @MikeDunlavey Yeah, I see his answer... sooo.. 10% the speed of light then? I am confused as to why he is using the permittivity of pure water in the salt-water calculation though. Perhaps it doesnt matter? – The Grape Beyond Jul 17 '13 at 15:21
• It will be something very fast, analogous to the speed of sound in a cloud of electrons (as in a metal), where the wave of force is transmitted by photons at the speed of light. Relative permitivity sounds like a complex subject, but for various other ionic solutions, like acids or alakalis, 80 is not an unusual number. – Mike Dunlavey Jul 17 '13 at 16:00

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 $0.11c$.

• Thanks - so I take it then that the permittivity is (heavily?) influenced by the frequency of the transmission?... Would your answer change if say, this was at 20 Khz? – The Grape Beyond Jul 16 '13 at 20:12
• @TheGrapeBeyond: to be honest I don't know what the frequency dependance of the permittivity is. Before I posted I spent some time Googling to see what data I could find, but all I found was water at 1KHz, along with some comments in forums suggesting that saline wouldn't be that different to water. If you're really interested some further Googling might strike gold. – John Rennie Jul 17 '13 at 6:28
• Thank you, last confusing point, you are using the permittivity for water, (80), but then applying it for computation of speed for salt-water... I take it they are the same? – The Grape Beyond Jul 17 '13 at 14:52
• @TheGrapeBeyond: I only used the permittivity of water because I couldn't find the permittivity of saline documented anywhere. I'm guessing addition of salt won't make that much difference, i.e. not an order of magnitude, but this is just a guess. – John Rennie Jul 17 '13 at 16:27

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.

• Re your final paragraph, after I'd posted I started wondering whether what I'd said was applicable. On reflection, I think it is because we are talking about the propogation of EM waves. In effect the OP is asking about the wave group velocity in the salt solution. – John Rennie Jul 17 '13 at 6:24
• @danielu, you are absolutely right. What JohnRennie described is the maximum speed with which information can be transmitted in a medium. In Electrolytic solutions, speed of information transfer is called ionic mobility. It is like the drift velocity of ions. – udiboy1209 Jul 17 '13 at 10:49
• @udiboy: Aren't we talking about two different things? Take the analogy of a water hose of some length, full of water, with a valve at the far end. If the valve is suddenly turned on, water starts coming out of the near end almost immediately, but the actual velocity of water molecules need not be very fast. – Mike Dunlavey Jul 17 '13 at 13:51
• @MikeDunlavey, almost instantaneously doesn't mean infinite velocity, nor does it mean max possible velocity. Talking about your analogy, the water should travel the distance of the hose before it comes out, with the same velocity with which it comes out, not with the speed of light. Same is with ions in an electrolyte. – udiboy1209 Jul 17 '13 at 14:06
• From what I understand and comparing this question to the water hose analogy, the maximum transmission rate of a signal would be analogous to a water hose full of water when the valve is turned on, and almost instantaneous signal transfer, but I understand the original question to be like a water hose with no water, and the water has to travel the entire length of the hose before the signal is transferred. Is that correct, or will the ions in the water essentially act like a wire or conducting material. I know they will conduct, but I was thinking it would be in a different manner. – Daniel Underwood Jul 17 '13 at 14:10