Max current densities in a conducting medium

[Toodles]

Hi,
I am looking for some information regarding maximum current density in conducting mediums, such as a conducting fluid. Say that we have a setup like the figure that I have attached. We are looking down on a conducting medium with a uniform magnetic field coming out of the page and the electrodes attached as shown. I know that theoretically, the E field (and thus current) generated by the external load will be irrotational. I am wondering what will happen, in reality, at the singular locations between the electrodes where the magnitude of the E field will tend to infinity. Of course, the currents here will exceed the maximum current density that a physical conductor is capable of accommodating. What will the result look like, in general? Will this cause the field to become rotational? Divergent? I haven't been able to find much information about such a process, so any thoughts or recommendations for related articles or studies would be extremely helpful.

Thanks a lot.

 

[berkeman]

I am wondering what will happen, in reality, at the singular locations between the electrodes where the magnitude of the E field will tend to infinity.

Why would the E-field be high there? Can you add to your sketch to show these regions? The way you've drawn the circuit, the 2 sources of E-fields are not related to each other (they are floating)...

 

[Toodles]

Why would the E-field be high there? Can you add to your sketch to show these regions? The way you've drawn the circuit, the 2 sources of E-fields are not related to each other (they are floating)...

Sorry, I should change the figure. The electrodes are supposed to be touching the conducting medium. So there are infinitely large current right at the joining point of the electrodes (essentially a dipole here).

 

[berkeman]

So the horizontal centerline of the cell will tend to be at zero volts, and the voltage sources bias the ends to +/- V/2, I guess. You are asking about where the opposite polarity plates contact the conducting medium, but are only separated by an arbitrarily small distance. You can model this with a network of resistors, where the resistance does lower as the current density goes up. That should probably model the real physical situation, at least for normal conductors.

EDIT/ADD -- Note that for semiconductors, you can get an increase in conductivity with rising temperature, at least until the material melts...

https://ecee.colorado.edu/~bart/book/intrinsi.htm

 

[Nidum]

@Toodles : Could you please give us a much better explanation of what this problem is all about ?

 

[Nidum]

(1) For a highly conductive fluid all I see is two shorted out batteries where current in the external loops will be limited by the internal resistance of the batteries .

(2) For a weakly conducting fluid all I see conceptually is four resistances in a square . One resistance each linking top to bottom pair of electrodes and one resistance each linking side to side pairs of electrodes . The side to side resistances are relatively low value and the top to bottom resistances relatively high value . Current flowing in the external loops is determined by the internal resistances of the batteries and the values of the resistances between the electrodes .

(3) If accurate numerical results are needed for fluid with any level of conductivity the four resistance model could easily be expanded into a mesh of resistances as advised by @berkeman or a quasi continuum model could be set up using finite difference or finite element methods on computer .

(4) The finite element model could be set up to handle non linear resistance and possibly external influences like the magnetic field which you mentioned but did not explain the influence or purpose of .

 

[Nidum]

Only a five minute try out with FEA . I obviously need to have a denser mesh around the split line between electrodes to get better accuracy in that area .

 

[Toodles]

Hi everyone,
Thanks for the replies. First off, I unfortunately can not give out a much better explanation of the problem nor can I elaborate on the rule of the magnetic field, but for my question here, it can be considered to not exist. Sorry!

I understand what the current should do theoretically and have modeled this myself in FEA; however, not with nonlinear resistance, which is what I am interested in. Obviously based on my question, I am not an expert in electrodynamics, so any good resources on max current density or nonlinear resistance would be appreciated. I have searched with no real success.

I am just trying to find out what happens in practice due to the large voltage jump between the two electrodes on either side. This is basically a dipole, so there should be an electric field here that tends to infinity, right? This would in turn, create a current that exceeds the max current density that is supported by the conducting fluid. I am wondering if, due to this current being limited, you end up with a rotational current. This would have to be the case if, say, the max current density acted in such a way where it just limited all exceeding currents and didn't affect those that didn't exceed it; however, I doubt that is how it works.

Let me ask this: if I was TRYING to get a current that is rotational by merely supplying voltages, would the nonlinear resistance help me achieve that in any way? Obviously, in electrostatics the current is, theoretically, always irrotational. You would need to change the magnetic flux to induce the rotational current. I guess the main point I'm wanting to know is, in practice, when you are dealing with far-from-ideal conductors, is it possible to get a rotational current field (e.g. eddy currents) due to the nonlinear resistance with some type battery setup?

Thanks all so much.

 

[berkeman]

if I was TRYING to get a current that is rotational

What's rotational current?

I know that theoretically, the E field (and thus current) generated by the external load will be irrotational.

Which EM equations are you referring to? Are you hoping to create a geometry that violates classical EM theory?

 

[Toodles]

What's rotational current?

Which EM equations are you referring to? Are you hoping to create a geometry that violates classical EM theory?

Rotational current would be one with nonzero curl:
$$\nabla \times \mathbf{j} \neq 0$$

And I am referring to the standard Maxwell equations; in particular Faraday's law
$$-\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \mathbf{E} $$
and Ohm's law
$$\mathbf{j} = \sigma \mathbf{E} $$.

If we assume that there is no change in magnetic flux, we get
$$\nabla \times \frac{1}{\sigma} \mathbf{j} = 0. $$

Thus, there is no way to make the current rotational without a change in magnetic flux. However, what if we relax the assumption of steady magnetic flux. Then, is it possible that the large currents in between the +/- electrodes on both ends will get limited by the max current density, which will then produce a nonzero curl here. Or does the current just "readjust" to keep itself irrotational? Obviously, this is all based on the fact that the only thing driving the current is the supplied voltages.

View attachment 209714

Only a five minute try out with FEA . I obviously need to have a denser mesh around the split line between electrodes to get better accuracy in that area .

Was this simulation done with a linear resistivity?

 

[Toodles]

(2) For a weakly conducting fluid all I see conceptually is four resistances in a square . One resistance each linking top to bottom pair of electrodes and one resistance each linking side to side pairs of electrodes . The side to side resistances are relatively low value and the top to bottom resistances relatively high value . Current flowing in the external loops is determined by the internal resistances of the batteries and the values of the resistances between the electrodes .

Thanks for this reply. Why do you say the top to bottom resistances are high and the side to side resitances are low? I would expect due to nonlinear effects that since the current is so large (assumedly exceeding the max current density of the conductor) in the side to side electrodes, that you would have much higher resistances here than in the top to bottom resistance. Is this not the case?