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Re: [Phys-L] spherical capacitor problem



Better ==>
We don't need the conducting plane. Each sphere forms a capacitor with the surrounding shell at infinity. This shell is a common electrode. The capacitance measured sphere to sphere is a series view of the combination of C1 and C2.

See my crude drawing at
www.sciamanda.com/spheres.docx.

-----Original Message----- From: Bob Sciamanda
Sent: Monday, June 29, 2015 11:40 AM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] spherical capacitor problem

1.) Surround the two sphere system with a conducting sphere of infinite
radius.
2.) Midway between the spheres insert an infinite conducting plane,
perpendicular to the line joining the sphere centers.
3.) This plane was, and still is, at zero potential (the potential at
infinity). Inserting the conductor has changed nothing electrical.
4.) Ostensibly, you now have two capacitors in series.

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
treborsci@verizon.net
www.sciamanda.com

-----Original Message----- From: Carl Mungan
Sent: Monday, June 29, 2015 8:42 AM
To: PHYS-L
Subject: [Phys-L] spherical capacitor problem

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?

So I didn’t buy his answer. However, I then discovered an exact formula for
any a and d at:
http://www.quickfield.com/advanced/non-concentric_spheres_capacitance.htm
<http://www.quickfield.com/advanced/non-concentric_spheres_capacitance.htm>

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.

What do you all think?

-----
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/

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