Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: funny capacitor



OOPS!
The first equation under the "Bonus" heading should read:
"Note that V2 - V1 = Q1*k/(1/b - 1/a), . . ."
Blame it on the wee hour :)

Bob

----- Original Message -----
From: "Bob Sciamanda" <trebor@VELOCITY.NET>
To: <PHYS-L@lists.nau.edu>
Sent: Saturday, March 10, 2001 5:23 AM
Subject: Re: funny capacitor


A simple, transparent illustration of the traditional interpretation of
the
equation set Qi = Cij Vj and its inverse (the Einstein summation
convention
is implied):

Consider the isolated system of two concentric, thin conducting spherical
shells. The smaller sphere has radius a, charge Q1 and potential V1
relative to infinity. The larger sphere has radius b, charge Q2 and
potential V2 relative to infinity.
Since we know the fields and potentials of a uniform, spherical shell of
charge, we can quickly write, using k = 1/(4*PI*epsilon):

V1 = k ( Q1/a + Q2/b )

V2 = k ( Q1/b + Q2/b ) These are easily inverted to:


Q1 = {ab/(k(b-a))} * {V1 - V2}

Q2 = {b/(k(b-a))} * {-aV1 + bV2)

Note that neither coefficient matrix is singular. (multiply the two
matrices
and you will get the identity matrix.)

Note that the total charge Q1 + Q2 = (b/k)V2 is not fixed, since V2 is a
freely adjustable variable. This allows us to consider all possible
charge
states of the system. Q1 and Q2 are freely adjustable, without
constraint,
and will determine the V's. Or, the V's are adjustable and will determine
the Q's.

Bonus:
Note that V2 - V1 = Q1*k/(1/a-1/b), a result taken advantage of by the
van
de Graff machine (existing fields will always tend to drive any non-zero
Q1
outward to sphere 2, however great is Q2). (Or more generally, internal
unbalanced charges are driven to the surface of a conductor.) By taking
Q1
= -Q2, this also yields the ordinary capacitance: C = ab/(k(b-a)).

Exercise: Add a third spherical shell - there will be no surprises.

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
www.velocity.net/~trebor