 Quote by gvjt
I understand how to calculate the capacitance of an isolated conductor, say a sphere, and I know that the associated formula involves the surface area of the conductor. Does all of this make the assumption that the surface in question is convex, or at least "non-concave"?
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Nope - you can make a capacitor out of any shaped materials - consider two bi-concave lenses facing each other.
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More specifically, what if the surface is extremely complex, like a fractal where the surface area might be extremely large,...
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etc.
It will depend on the details of how the surface is arranged. You'll have to do the math.
Most complicated surfaces can be understood in terms of a network of regular capacitors.
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Put another way, if I was to fashion an extremely convoluted surface, could I wind up with an extremely high capacitance for the volume the object is occupying?
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Yes you can. The simplest example, I think, would be wrapping a regular parallel plate cap into a spiral.
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Or perhaps here's a more concrete example: What if I fashioned an object consisting of hundreds of thin metal plates all connected electrically at their centers by a rod, but with an air gap between each.
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A network of capacitors, in other words.
How do you have to join two capacitors together to increase the overall capacitance?
But I think you should be using, as your basic model of a capacitor, something involving
paired conductors ... one providing charge and the other receiving it. The concept of capacitance does not really make sense if there is only one conductor.