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] |
A simple, transparent illustration of the traditional interpretation ofthe
equation set Qi = Cij Vj and its inverse (the Einstein summationconvention
is implied):matrices
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
and you will get the identity matrix.)charge
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
states of the system. Q1 and Q2 are freely adjustable, withoutconstraint,
and will determine the V's. Or, the V's are adjustable and will determinevan
the Q's.
Bonus:
Note that V2 - V1 = Q1*k/(1/a-1/b), a result taken advantage of by the
de Graff machine (existing fields will always tend to drive any non-zeroQ1
outward to sphere 2, however great is Q2). (Or more generally, internalQ1
unbalanced charges are driven to the surface of a conductor.) By taking
= -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