Charge density distributions over a conductor surface

[DonChubby]

So I've just posted this same thing over on reddit, but I thought I'd give it a go here too, since I know you're all a clever lot.

So at the moment I am doing a project for school, that I have chosen myself. The basic idea is this:

On the surface of a conductor there are a lot of free charges, that are essentially free to move on the surface. In the electrostatic case, these charges will arrange themselves in such a way that they will be in equilibrium. That is, in such a way that they all remain stationary.
Now purely mathematically it is incredibly difficult, if not downright impossible, to determine how the charge distribution will look, for all but the simplest of cases. So I thought it would be interesting to do some computer simulations to determine these, with good results so far. However, I've been asked this question:
What is the application of this? What can we do with this knowledge?

And to be honest, I am stumped. I can't for the life of me think how one would apply this knowledge. I was myself driven by pure curiosity, I just thought it would be interesting to see how the distributions would turn out.
Obviously we can tell how the fields look from the distribution, but that doesn't seem to me to lead anywhere. We can only tell the distribution for a single, well or several, special cases. Not how they would look in general, under influence of arbitrary charges.

So if anyone has any ideas, I'd be really grateful for your input.

[mfb]

With numerical simulation, you are not limited to special cases any more. Or, at least you can look at much more cases. Even under influence of other charges or fields.
The special case of equilibrium without other charges can be interesting if you want to look at the field. For example, to calculate its influence on other charges (small enough to not disturb the field much) and their flight path or to calculate its strength at the surface (do you have the risk of discharges at geometric spikes?). To calculate capacitors, coils, ...

[Sourire]

This is a fairly interesting problem, I think. However, it's also worth mention that I think your knowledge of EM is slightly greater than my own.

First-a purist argument, applications are fun, but as a project I think intellectual merit is enough.

Second, I'd be interested to see if this type of modeling had any synonyms with social behavior (urban sprawl, if that's the correct term?)

Third, if you could find a way to allow a slight perturbation (like a current being introduced), and model the system afterwards, you could get some cool concept of how electronics work. Potentially even do it better. Yes, I'm aware this goes on to electrodynamics, but I enjoyed the idea too much not to share.

[Iferius]

I can't think of a cool application or visualisation in the electrostatics case. It would be cool to see the relative speeds in electrodynamics - pulse diffusion in an alternating current, for example.

[Deduska]

Hi.

I stumbled upon this subject while looking the answer to the question of charge distribution question.

I can also say what would be the point of calculating charge distributions - for example this:
http://en.wikipedia.org/wiki/Electrohydrodynamic_thruster



Ofcourse the page gives relatively easy instructions how to build a EHD, but to raise efficiency there is need to build a mathematical model of the EHD thruster which includes charge distribution calculations.

I personally would need answeres to two questions:
1: the charge distribution and induced field of a infinite plate placed to a electric field under an specified angle
This should be easy problem, but still I am stuck.
2: Charge distribution on a ideal aerodynamical body.

I think that any app that would help calculate the charge distribution would be very helpful - especially if also effects of external electric field can be calculated.

[Tass]

HA! YES!

I have to turn in my phd-thesis in three weeks. A part of it was an experiment with an electrode (can be modeled as an long, vertical, conducting cylinder of radius r, terminating at height h) above a sample (infinite, horizontal, conducting plane), with a voltage applied between them.

I would like to have known the exact electric field at a given position of the sample, but I couldn't solve it analytically, I couldn't find any software that would solve it numerically, and I didn't have time to code it myself.

Theres an application right there. If you can give me an answer within two weeks I will be very thankful and you will be included in the acknowledgements.

[mfb]

Hmm... quick&dirty in Excel:



The idea: Use a cylindrical coordinate system with the obvious orientation. The system is invariant under phi-rotation, therefore it is sufficient to model the two-dimensional (rho,z)-plane.
Determine the potential phi at each point of this plane with some finite resolution. The sample and the electrode get a fixed phi of 0 and 1, respectively. For all other points, the Maxwell equations give 0 = \nabla^2 \phi = \frac{1}{\rho} \partial_\rho \left( \rho\; \partial_\rho \phi \right) + \partial^2_z \phi. The electric field is then given by the gradient of the potential.

To apply this to a grid, the derivatives have to be transformed to differences of grid cells i: \partial f(i+1/2) = f(i+1) - f(i)

This allows to calculate the second derivatives as

\partial^2 f(i) = \partial f(i+1/2)-\partial f(i-1/2) = f(i+1) + f(i-1) - 2f(i)

and

\frac{1}{i} \partial \left(i\; \partial f(i)\right) = \frac{1}{i} \left[ \left(i+\frac{1}{2}\right) \partial f(i+1/2) - \left(i-\frac{1}{2}\right) \partial f(i-1/2) \right] = \frac{1}{i} \left[ \left(i+\frac{1}{2}\right) \left(f(i+1) - f(i)\right) - \left(i-\frac{1}{2}\right) \left(f(i) - f(i-1)\right) \right]

Putting everything together, solving for a specific cell and using the index i for the rho-coordinate and j (columns) for z, I get

\phi(i,j)==\frac{1}{2}\left((\phi(i,j-1)+\phi(i,j+1))/2+\frac{1}{2i}\left((i+1/2)*\phi(i+1,j)+(i-1/2)\phi(i-1,j)\right)\right)

I hope that the prefactors are right.

As the simulated area is finite, there are 4 borders to consider:
- the sample border is simple, as it has a fixed potential.
- rho=0 is a coordinate singularity and cannot be calculated using the formula given above. But there, it is easy to get the appropriate formula from cartesian coordinates:

\phi(0,j)=\frac{1}{6}\left(\phi(0,j+1)+\phi(0,j-1)+4\phi(1,j)\right)

- far away from the sample. This needs a bit more thought. Here, I just assumed that the field is nearly constant with z, and therefore \phi(i,0)=\phi(i,1). This is a good approximation for the region close to the electrode, but not so good far away from both sample and electrode.
- far away from the electrode. This is the most tricky case. Depending on the precision you need, it might be sufficient to just assume a nearly constant potential here or a constant first derivative in rho-direction or something similar.


The component of the electric field in one direction is then given by

E = \partial \phi(i) \approx \frac{1}{2} \left(\partial \phi(i+1/2) + \partial \phi(i-1/2)\right) = \frac{1}{2}\left(\phi(i+1)-\phi(i-1)\right)

Here is the magnitude of the E-field as graph.

Every reasonable programming language which supports arrays can calculate this, and as all calculations are quite basic it should be easy to implement it.
There are certainly better ways to calculate the electric potential and field, and it might be useful to decrease the grid size close to the electrode (especially the corner of it, as it has the highest field strength). Smaller grid cells everywhere can do the same job, they just need more computing power.


Is that useful for you?

[Tass]

Ah, yes, it can be done that simple. I do too little math now-a-days.

I should be able to do it myself in matlab now. Thanks a lot.

I only need the electric field at the sample (and rather close to the electrode) so this should be sufficient. In a paper where they did something similar they actually just assumed uniform charge on the electrode rather than constant potential, this will be way more accurate.

Edit: Given that each cell is dependent on the ones around it, you obviously need to iterate to convergence. How did you do that in excel? It looks like you simply copied the formula into each cell and the program did the rest. Mine would return an error for circular referencing. Anyway this it mostly for my curiosity, I can certainly write a matlab script that will iterate it.

Edit2: I threw this together, and it seems to be working:

Code: Select all

phi=zeros(1000);
phi(1:r,h:1000)=1;
r=50;
h=30;

for n=1:2000
    for z=2:h-1
        phi(1,z)=1/6*(phi(1,z+1)+phi(1,z-1)+4*phi(2,z));
    end
    for rho=r+1:999
        phi(rho,1000)=phi(rho,999);
    end
    for rho=2:999
        A=1/(2*(rho-1));
        for z=2:999
            if rho>r || z<h
                phi(rho,z)=0.25*(phi(rho,z-1)+phi(rho,z+1)+(1+A)*phi(rho+1,z)+(1-A)*phi(rho-1,z));
            end
        end
    end
end

It is very quick and dirty brute force, since it spends a lot time on cells of zeros. The potential change doesn't move faster than one cell per iteration, so updating thousands of cells of zeroes for hundreds of iterations is a bit of a waste. Anyway it still goes fast so I think it is not worths it to improve the code. I'll just add some more iterations and it should be fine.

Edit3: No wait, it will actually spread much faster outwards since I don't use the updated cells in the same iteration rather than waiting for one to complete and then updating all. It is the kind of things that can make artifacts in some simulations I think. Ah well as long as it converges it should be fine.

Edit4: Added graphs of the potential and the field in the area close to the electrode tip. Looks about as you'd intuitively expect. Nearly uniform field under the electrode, 1/r drop of horizontally away from it, and high field close to the corner.