Conductor that can completely surround another conductor?

[merrypark3]

Homework Statement

"demonstrate that the capacitance of any conductor is always smaller than or equal to the capacitance of a conductor which can completely surround it"

Homework Equations

Multipole expansion?

The Attempt at a Solution

Tried to compare each terms of pole-moments of the distribution...

 

[kuruman]

Usually, capacitance is defined in terms of two conductors, one with charge +Q and the other with charge -Q. The capacitance then is the ratio Q/ΔV, where ΔV is the potential difference between the two conductors. I assume that by the "capacitance of a conductor" is meant one of the conductors is at infinity. So look at two conductors, one larger than the other, bearing the same charge Q and compare the ratio Q/ΔV, where ΔV is the potential difference between a given conductor and infinity.

 

[merrypark3]

I think the capacitance in this problem is defined in terms of one conductor.
If a conductor of spherical shell of radius R is charged with Q, then the potential is Q/R so the capacitance is R.
So If we limit the case to a concentric spherical shell, then obviously the small one has small capacitance. But for any shape of conductor? How can I demonstrate? I've tried with mutipole expansion, but ,...
Please help me

 

[kuruman]

I will help you, but you have to read and understand what I post. Imagine an irregular conductor, call it A, that has charge Q on it. It is an equipotential, but there is potential difference ΔV_A between it and infinity. Its capacitance is C_A = Q/ΔV_A. Now imagine a second conductor, B, that can completely surround conductor A if called upon to do so. Put the same amount of charge Q on conductor B. It too is an equipotential and its capacitance will be C_B = Q/ΔV_B. If you can demonstrate that ΔV_B is less than ΔV_A, then you have answered the question.

 

[merrypark3]

Yes, that's right. But, How can I demonstrate that ΔV_B is less than ΔV_A..

 

[kuruman]

What happens to the potential on conductor B in the limit that its size becomes infinitely large?

 

[merrypark3]

zero. I have thought about that. But considering only that case is not enough?

 

[kuruman]

No it is not. What will happen if you bring the boundaries of the huge conductor closer in from infinity. Will its potential rise or fall? What if after that you brought the boundaries in even closer?

 

[merrypark3]

the more closer the potential increases. but that process preserves the shape..

 

[kuruman]

the more closer the potential increases. but that process preserves the shape..

I don't know what you mean by this or why it is relevant. You have enough information (and help) to answer the question. Just put it together.