Capacitance calculation by using conformal mapping

[asmani]

Hi all

How to calculate the capacitance of a sphere-plane system by using conformal mapping?

Thanks

 

[jasonRF]

asmani,

just wondering why you would want to use conformal mapping for this. Conformal mapping applies to 2D problems. Your problem looks 3D, although with azimuthal symmetry it can be made 2D. You would have to work out a lot of details to make this work, if it works at all. Sounds painful to me.

An easier approach would be to use images. Now, this is pretty challenging for an image problem, but some textbooks get you most of the way there. Once you understand the solution for a point charge near a conducting sphere, you are most of the way there.

good luck,

jason

 

[tiny-tim]

hi asmani!

i don't know much about this …

in goldfish college, we don't have electricity labs, for obvious reasons …

but looking only at the geometry, i wonder whether it would help to make a conformal map onto a pair of concentric spheres? ​

 

[asmani]

Thanks a lot for the replies.

but looking only at the geometry, i wonder whether it would help to make a conformal map onto a pair of concentric spheres? ​

Yes, I think it helps.

 

[asmani]

Apparently there is a theorem saying that the capacitance of a 2D system is unaffected by a conformal mapping. I'm not sure if this holds in 3D.

I have seen the solution by using the method of images and apparently the capacitance calculated by this method has not a closed form.

As we know, the capacitance of a pair of concentric spheres with radius R1 and R2 has a closed form expression in terms of R1 and R2.

Let's assume the radius of the sphere is R, the distance between its center and the plane is d and the above theorem holds in 3D. If the sphere-plane system is mapped by a conformal mapping to a pair of concentric spheres with radius R1 and R2, then R1 and R2 have not a closed form expression in terms of R and d, unless one of the above assumptions is not true.

 

[Dmytry]

The thing is, there's not a lot of useful conformal mappings in 3D . Rotations, translations, uniform scaling, and that's it. Well, you may also include mirroring.

 

[Bob S]

The sphere-plane capacitance is solved on page 131 in Smythe Static and Dynamic Electricity (Third edition).

Bob S

 

[asmani]

Thanks Bob, I'm looking for the solution by using conformal mapping.

but looking only at the geometry, i wonder whether it would help to make a conformal map onto a pair of concentric spheres? ​

Do you know such a mapping?

 

[tiny-tim]

Do you know such a mapping?

sorry, no …

i can't even find anywhere to look it up ​

 

[jasonRF]

asmani,

just wondering why you would want to use conformal mapping for this. Conformal mapping applies to 2D problems. Your problem looks 3D, although with azimuthal symmetry it can be made 2D. You would have to work out a lot of details to make this work, if it works at all. Sounds painful to me.

An easier approach would be to use images. Now, this is pretty challenging for an image problem, but some textbooks get you most of the way there. Once you understand the solution for a point charge near a conducting sphere, you are most of the way there.

good luck,

jason

obviously I was wrong ... I learned conformal mapping in complex analysis and have only used for solving 2D problems. This thread, and google, have set me straight. Thanks! I haven't figured out how to solve your problem, though ...

jason

 

[jsgruszynski]

Did you folks not learn about integration of a http://en.wikipedia.org/wiki/Surface_of_revolution" ? Particularly in this case of sphere-plane you have the right symmetry for that. So yes, of course you can use conformal mapping for 3D. Just revolve the result of the conformal transform for the area and use the normals for the distance.

Just go back to the derivation of capacitance for all the standard geometric cases (parallel plate, two wires, etc). The common feature for all: you have integration over area S for the charge and integration of over distance d for the voltage. Then it's just C = Q/V

A little math but not that much really. Should be a no brainer.

 

[jasonRF]

Did you folks not learn about integration of a http://en.wikipedia.org/wiki/Surface_of_revolution" ?

Of course we did.

Just revolve the result of the conformal transform for the area and use the normals for the distance.

I have no idea what this means ... Could you please explain?

Just go back to the derivation of capacitance for all the standard geometric cases (parallel plate, two wires, etc). The common feature for all: you have integration over area S for the charge and integration of over distance d for the voltage. Then it's just C = Q/V

A little math but not that much really. Should be a no brainer.

Of course, once you know the potential (or equivalently the electric field) then you just follow the recipe you mentioned. The hard part is finding the field, which is of course the hard part of the OP question. The parallel plate problem is "easy" if you ignore edge effects (which I am guessing is what you are talking about), but if you want to truly solve for the capacitance of two finite parallel plates it is quite difficult to determine the field, and hence the capacitance.

jason