Electrical Resistance of a Sphere?

[cabraham]

I am re-examining my stacked disk model, & it looks like I will have to deep six it. I presumed at first that current would laterally diffuse & distribute evenly throught each disk cross section. That is approximately true if we are near the center of the sphere & equipotential surfaces are almost flat. But near the poles, the curve is too pronounced for even distribution. I think the "UD" model (uniform density) results in too low an R value.

I will try to solve the Laplace equation as soon as I figure out how. It will require general curvilinear coordinates. I will post when I have the solution. BR.

Claude

 

[Redbelly98]

Not sure if this would help in the modeling, but I was thinking along the lines of assuming small spherical electrodes, each centered on opposite sides of the conducting sphere. The E-field must be parallel to the edge of the conducting sphere everywhere***, so there would be a surface charge on the sphere's surface to force that to happen.

I'm hard-pressed to spend time on solving this, but if I did I'd be more inclined to go the numerical approximation route rather than grinding it out algebraically.

*** If the E-field is not parallel to the surface, then it is either pushing charge into the surface or removing charge from the surface, thereby changing the surface charge density. In steady state the surface charge doesn't change in time, so E must be parallel to the conductor's surface everywhere, in steady state.

 

[CarlosSoli]

I wish to stress that my work, "The notion of electrical resistance" is a general discussion of the concept, where the calculation of the resistance of a sphere is only an illustration. I will gladly discuss here or elsewhere any details of it, though I point out that I have not followed the details of the previous discussions. If anyone finds any error in my calculations, I will be glad to correct them and acknowledge the contribution in the following editions of the document, which is published under a Noncommercial-Share Alike 3.0 Unported license (free use, modification and distribution, acknowledging original author).

Carlos Solivérez

 

[Menaus]

If you guys want even more mathematical fun, you can try calculating the resistance if this sphere is under resonance, let's make it pulsed DC to make things simpler. :p