Conformal Mappings: A Powerful Tool for Solving Electrostatic Problems

[tommyers]

Hi,

Why do fields fringe at the edges of plates i.e the eletric field at the edges of a parallel capacitor arrangement?

Is there a way of calculating how far these fields fringe outwards?

And finally, what effect they have on the overall capacitance.

Answers to ANY part of this post will be greatly appreciated.

Regards

Tom

 

[pallidin]

The edges of a capacitor "plate" defines and forces charge-distribution boundaries. In other words, the charges at the edge have no-where to go, unlike, say, the "middle" of a plate.
In addition, since like charges repel, one can find a greater percentage of charge at those "boundary" layers. Classically, this is called the "skin-effect"

 

[Claude Bile]

Essentially, the charge distribution near the edges is somewhere between a continuous surface charge density and a dipole (In a very hand-wavey qualitative manner), which is why the field fringes outward.

This fringing can be calculated quantitatively using Maxwell's equations.

Capacitance is a measure of how much charge you can pile onto a pair of plates per volt. I don't think the E field ought to have much of an effect at all, unless the electric permittivity varies across the region the field is fringing into.

Claude.

 

[berkeman]

And finally, what effect they have on the overall capacitance.

They can increase the actual capacitance of a parallel plate capacitor significanly above the simple equation that we use often:

C = \frac{\epsilon A}{d}

The larger the separation d is with respect to the linear dimensions of the plates, the larger the contribution of the fringe field to the total capacitance above the capacitance listed in the equation above. Some air-gap parallel plate caps that I've made for various experiments had capacitances that were 20-30% above what the forumula said they would have (if I remember correctly -- that was long ago).

 

[tommyers]

Hi Guys,

This is all good stuff - if anyone could provide any real simple examples using maxwells equation, I would be eternally in your debt!

Are if they could explain methods of showing charge distribution visually i.e. some form of breaking the plates up into sqaures and representing the charge in each square like boundary element method.

Regards

Tom

PS My cofidence is starting to grow again

 

[lalbatros]

Why do fields fringe at the edges of plates i.e the eletric field at the edges of a parallel capacitor arrangement?

A point charge creates a spherical field: the electric field is the same in all directions. The edge of a plate is just like a charged point: electric field in all directions. The presence of the opposites charges on the other plate does not cancel everything out. Note that the details may be complicated to calculate: the charge distribution on the plates will not be uniform, but will be determined by the nullity of the electric field parallel to the plates.

Is there a way of calculating how far these fields fringe outwards?

Yes, try to figure out what the charge distribution on the plates is, and then summ up the fields from these charges. Easier to say than to do ! There are old methods to do this, with "image" or "ghost" charges.

The idea is to assume a first approximation: uniform charge density. This first approximation will already display fringing of the field. Eventually this may suffice to your need, depending on the required precision. The effect of fringing on the capacity is then obtained by comparing the calculated capacity to the capacity of an infinite plate condensator (per m² of course). The problem is: the voltage that you will calculate will depend on the points chosen on the plates. This because the field parallel to the plates will not vanish. Then, choose the points of symmetry as the most representative. I would be interrested to know your result: does it have an effect on the capacity, and why?

The second approximation consist of a correction to the charge distribution. This correction should reduce or nullify the field parallel to the plates.

Another way to solve the problem is to use a numerical solver. You could use a finite element method or a boundary element method.

The boundary element method fits better to the nature of the problem as I explained it. You could eventually program that yourself easily for a 2D model: a linear solve is all that is needed, if you discretize your plates by 1000 elements, this will bring you 1000 equations to solve (each charge in these point) - 1 (one charge should be fixed). Observe that you will not need to calculate the details of the fringing fields to calculate the capacity. Why is that so?

Maybe there are also analytical solution for the circular plates. To be checked.

Note that the "quality" of the edge might play a role: is it sharp or rounded, is it infinitely thin or not ... ? But I have no idea if it affects the capacity of only the edge fields.

Finally, I remember, when I was student, I had to study that experimentally, by drawing field lines on a special paper. This was a conducting paper, a black paper. The conducting elements were represent with metallic ink. Voltage were applied as needed for the model. We could then measure the voltage anywhere on the sheet of paper and draw the equi-potential lines and the field lines (perpendicularly). It is easy to establish a link between electrostatics and this resistivity-based experiment. But it is 2D, of course.

 

[lalbatros]

Classically, this is called the "skin-effect"

The skin effect is related to high frequencies.
High frequency currents cannot penetrate deep inside conductors.
The high frequency magnetic fields these currents generate inside the conductor create an emf that reduce (shield) the currents inside the conductor.

I thought the question was about electrostatics.

 

[tommyers]

Hi lalbatros,

Yes. The question is aimed at electrostatics and therefore it relates to fringing fields as opposed to 'skin-effects'.

I have attached a photgraph of my 'simple' apparatus and also attached is an excel file showing some basic measurements.

I would be exstatic if someone could provide me with some theory as to predict the measured capacitance more accurately, or whether they can spot some wild mistake I am making with my apparatus or calculations.

Many thanks

Tom

 

[tommyers]

Please note - my attachments will not upload - if anyone is interested in my photo and excel spreadsheet then please email me.

Thanks

Tom

 

[vanesch]

Hi Guys,

This is all good stuff - if anyone could provide any real simple examples using maxwells equation, I would be eternally in your debt!

For most cases, you'll need to have numerical solutions.
However, for some simple systems in 2 dimensions (that is: prismatic systems), one can often use conformal mappings to find an exact solution to the electrostatic field problem.

Look for instance at:

http://physics.usask.ca/~bradley/pll_plate_edge.pdf

cheers,
Patrick.

 

[tommyers]

Thank you for the recommendation Patrick.

I have actually come across this document before, but I couldn't find a path through it which I could use and understand.

My advanced mathematics is not fully up to speed yet - but I am working on it. If you could provide any further support regarding this document such as converting some of the 'symbol mathematics' into 'number mathematics' for a simple geometery then that would be a great start.

Regards

Tom

 

[vanesch]

The trick of conformal mappings is simply the following:

If u = f(x,y) ; v = g(x,y) is a conformal mapping of the (x,y) plane into the (u,v) plane, and if we have a function V1(u,v) which is a solution of the Poisson equation (solution to the problem of electrostatic potential in vacuum), then V2(x,y) = V1 ( f(x,y), g(x,y) ) is also a solution of the Poisson equation.

This means: from one solution, you can produce another, by substitution of the coordinates.

Now, a conformal mapping is a mapping that preserves locally the angles, but where are you going to find such a thing ? Complex analytic functions !

It turns out that a complex analytic function, seen as real and imaginary functions of real and imaginary coordinates, are conformal mappings.

And you can write a solution to the Poisson equation ALSO as a complex analytic function, of which the real part is the genuine electrostatic potential.

For instance, in the plane, a point charge in point (x0,y0) or z0 = x0 + i y0, has the potential q/(2 pi eps) Log(z - z0)
(or something similar, check).

Now, there are situations in which we KNOW the electrostatic potential. For instance, in the case of an infinite parallel plate capacitor. Or for a coaxial capacitor. Or for a set of point charges. Or...

The trick is then to try to find a conformal mapping (a complex function) which will map one of these known geometries onto the geometry you're interested in. There are books around (for instance, Schaum's Complex Variables, but many others) where there are tables (with pictures :-) of what complex function maps what part of the complex plane on what.
You then simply have to find one (or combine several) of these, in order to find a way to map your specific case onto one of the "solved" cases.
It's a bit out of habit, but there used to be big books with existing solutions of electrostatic configurations, and with conformal mappings.

It can sound overwhelming at first sight, but after some practice this really becomes a fun game. I often do this, and I often find analytic solutions where my collegues run a monster of a finite element program

As to the document, it simply SHOWS you that the solution is correct, but doesn't indicate how it was obtained. There is usually no general technique: you have to puzzle yourself your mapping together, by looking at an atlas of conformal mappings... (but one quickly gets the hang of it).