Re: [Phys-l] defn of capacitance

[Carl Mungan <mungan@usna.edu>]

I'll respond to what was in today's digest on this topic:

> [JohnD] The relevant equation is
>     Qi = Cij Vj
>
> This equation and the underlying physics is discussed at
>   http://www.av8n.com/physics/laplace.html#q_of_v

An assumption of global charge neutrality appears to be embedded in 
there. I have two isolated conductors with Qa not equal to -Qb. 
Charge isn't overall neutral (excluding charges at infinity - 
mentioned at the end of this message).

In any case, that aside, I would be glad if you could actually 
demonstrate how to use the formula to calculate a standard 
two-conductor capacitor. I'm interested in learning how capacitance 
matrices work, as I don't fully get it yet. If my request is too 
trivial, feel free to do a simple three-conductor example.

> [Brian] Capacitance is the property of a two terminal device to store
> external electrical charge whose magnitude is measured by the reciprocal
> of the rate of change of voltage difference  between
> the terminals,  which is proportional to the amplitude of a constant current
>  impressed between the terminals.

Sounds reasonable. Can you actually lay out the calculations for the 
two example cases I proposed? Show me the money.

> [JohnM] There was a discussion on Phys-L back in March of 1997 that 
> began  with a question from Ludwik titled "How many volts?" and led 
> into  this same territory.
>
> <https://carnot.physics.buffalo.edu/archives/1997/03_1997/msg00476.html>

WOW! How can you remember something like that from over a decade ago? 
I'm impressed.

> [Alphonsus] Actually, I guess your question is related to the 
> "Physics Challenge".

If you mean the one on page 119 of TPT Feb 2008, I admit I wasn't 
relating my question to that. When I solved that Challenge, I used 
the standard infinite-plate approximations.

> [Mark] In your spherical capacitor example, you have to distinguish 
> between the charge on the inner surface of the outer sphere, and the 
> charge on the outer surface.  The charge on the inner surface of the 
> outer sphere will adjust itself to be equal to -Qa, since there can 
> be no electrostatic field within the metal of the outer sphere.  The 
> capacitance between the inner surface of the outer sphere and the 
> inner sphere, which is a purely geometric parameter, won't change.
>
> But, since in this case the outer sphere is "floating" and has an 
> excess charge on its outer surface, there also will be a capacitance 
> between the outer surface of the outer sphere and infinity.

GREAT, a concise answer and evidently consistent with the 1997 PHYS-L 
discussion. Thank you!

Following Mark's logic, here are what I come up with as the answers 
for the two examples I cited. If anything is wrong, please chime in:

1. Two parallel plates, charge Qa on the left plate and Qb on the 
right plate each of area A. Say Qa>Qb for specificity.

between the plates E = Qa/2*A*e - Qb/2*A*e where e=1/4*pi*k
so delta(V) = (Qa-Qb)*d/2*A*e where d=distance between plates
choose +/- Q=(Qa-Qb)/2 on the inner faces of the plates so that E=0 
inside either plate [this leaves Q' = (Qa+Qb)/2 on the outer face of 
each plate]
conclude C=Q/delta(V)=A*e/d as usual

2. Two concentric spheres, an inner one of radius a and an outer one 
of radius b. Place charge Qa on the inner one and Qb on the outer 
one. Again say Qa>Qb for specificity.

between the spheres E = k*Qa/r^2 independent of Qb
so delta(V) = k*Qa*(b-a)/a*b
choose +/- Q=Qa on the facing surfaces of the spheres [this leaves 
Qb+Qa on the outer face of the outer sphere]
conclude C=Q/delta(V)=a*b/k*(b-a) as usual

In both cases, there are thus additional capacitances between the 
outer surfaces (of the plates or sphere) and infinity, assuming we 
wish to imagine opposite charge "at infinity" to satisfy overall 
charge neutrality as per JSD.

I guess the only final detail I wish to sew up to close this topic in 
my mind is whether there is a simple formula that gives Q in general. 
We notice that in case 1, my assumption that Q is half the charge 
difference between the plates is correct. My mistake was trying to do 
the same in case 2. I'm now wondering whether there could be cases 
where it's not obvious what Q to use. If I'm worried about nothing, 
feel free to tell me as much! Thanks to everyone, Carl
--
Carl E Mungan, Assoc Prof of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/