Two of us were batting around the following problem:
You have two spherical conductors. Each has radius a and their center-to-center separation is d. Find the mutual capacitance in the limit d >> a.
Apparently the answer is C/k = a/2.
My friend’s approach goes as follows: One isolated sphere has capacitance C/k = a. So put two of them in series to get the answer.
My objection is the formula for an isolated sphere assumes the other plate is a concentric sphere at infinity. (That’s how one derives the formula for spherical capacitance after all.) But that means the other sphere is located “inside” the first capacitor. How can one say the two spheres are in series in that case?
As you can easily check yourself, that exact series formula reduces to the answer given above for d >> a, ie. for large D.
So I’m now back to trying to understand if my friend’s approach is somehow okay after all. There are two infinities in the problem. One infinity is the separation between the isolated sphere and its “partner” plate at infinity. The other infinity is the separation between the two isolated spheres. Maybe the rationalization is to start with two finite spherical capacitors separated by a large distance. Then let each of those finite capacitors become isolated spheres.