# Electrodes and Current Density

Discussion in 'General Electronics Chat' started by Mazaag, Jul 16, 2009.

1. ### Mazaag Thread Starter Senior Member

Oct 23, 2004
255
0
Hey guys,

I have a general question about Electromagnetic fields and metal electrodes.

Suppose 2 flat circular metal electrodes connected to a voltage source (say a function generator). The two electrodes are placed side by side flat down onto a homogeneous medium with known properties.

Now, what do I need to know/do to calculate the current density around the electrodes? I'm assuming its going to depend on the voltage and the conductivity of the medium, but what equations would I need to solve to be able to map out this field..

Thanks guys

Last edited: Jul 16, 2009
2. ### russ_hensel Well-Known Member

Jan 11, 2009
818
47
Maxwell's equations should fit the bill. Not easy.

3. ### steveb Senior Member

Jul 3, 2008
2,433
469
No, it doesn't look easy and would probably need numerical solutions based on the geometry described.

One simplification is if you can make what is called a quasi-static approximation. If the frequency is low enough, which is testable based on the source frequency and material permitivities/conductivities of the media involved, the problem can be calculated as if it is an electro-static problem. Basically, magnetic fields will be ignored and Maxwell's Equations are simplified. I think you will end up with Poisson's electrostatic equation, boundary conditions for the electric fields at current densities at interfaces and the point form of Ohms's law relating electric field and currrent density. You will end up solving for the electric field distribution in space, the surface charge distribution at the interface of any conductors, and the current density distribution within the conductors.

A further simplifications is to work in the frequency domain via Fourier transforms. Consider sinusoidally varying voltage and use the complex conductivity/permeability to describe the media. EDIT: This last suggestion is of course only valid for linear media.

Last edited: Jul 16, 2009
4. ### KL7AJ AAC Fanatic!

Nov 4, 2008
2,040
287

Indeed, it's not easy...but, as a first order approximation...

The current density will follow a basically elliptic function as you depart from a straight line between the electrodes.

You can test this by putting some electrodes in a hunk of dirt, then placing a couple of VOLTAGE probes into the dirt at different distances from the center line.

Eric

5. ### Mazaag Thread Starter Senior Member

Oct 23, 2004
255
0

I'm actually not interested in practically solving the problem, but rather the theoretical tools I need in order to solve it..

I understand it involves Maxwell's equations and so on..but where does the material of the electrodes and its dimensions come into play? How do I use maxwell's equations to come up with the solution..?

6. ### steveb Senior Member

Jul 3, 2008
2,433
469
The material conductivity and permittivity can be used to specify a frequency dependent complex conductivity for the electrode. This assumes you have a linear material for the field amplitudes you are dealing with, and that you are using a Fourier transform approach, which just means looking at sinusoidal source and response functions. It really depends on what the material is. If it is a very good conductor at low frequency, you can simplify and just assume frequency independent constant conductivity for the electrode. So far you've left things ambiguous by not specifying the frequency, the material or the properties of the metal electrodes.

The dimensions come into play when you set up the geometry for whatever solution method you use. The problem sounds like a three dimensional problem with insufficient symmetry to reduce the problem to two or one dimensions. Hence, you need to use the real geometry when solving Maxwell's equations, and most likely this needs to be done numerically on a computer.

Maybe an approximate method can be used as described KL7AJ but I can't be sure unless you are more clear on stating the exact problem details.

This is a very broad question, even though you've narrowed it down by outlining a real problem. There are still many details that could be specified about your problem, and then one would still need to creatively explore the best approach to solve the problem.

Assuming you need a numerical solution, Maxwell's equations need to be simplified as much as possible based on the details and then the simplified equations need to be expressed in a discrete mathematical form according to an established numerical method such as finite element methods, finite difference etc.

If your question is directed toward analytical solutions for cases where there is enough symmetry, then you would try to match boundary conditions for the the general solutions valid in each separate material (air, conductive medium, electrode). The general solutions for each material depend on the material details and the coordinate system which is appropriate based on the symmetry.

You need to understand the derivation of boundary conditions for electric fields and current density as derived from Maxwell's equations. You also need to know how to start with the most general form of Maxwell's equations valid for any media (and general coordinates) and then simplify based on the known constraints and symmetries.

Before tackling a problem as difficult as the one you specified, try a simplified case. For example, you could study a something similar to a parallel plate capacitor with very large plates, closely spaced, with your electrode material for the plates and your medium material as the dielectric between the plates. Of course this may not be a very good capacitor since your medium properties are not specified. But this problem has some symmetry since you can ignore fringing effects just as is done to calculate the approximate capacitance of a parallel plate capacitor. You could also look at a case with concentric spheres, although this requires spherical coordinates for symmetry to match the boundary conditions in a simple way.