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Re: Kirchhoff's rules and linear dependence



I changed the spelling of the Subject: line
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Kirchhoff.html

Quoting Justin Parke <FIZIX29@AOL.COM>:

I would like to talk to the students about this tomorrow but
a) I can't convince myself why the three loop rule equations are not
linearly independent

At the statement-of-fact level, I agree with the answer given
by STRatliff.

At the next level of detail, you can have some fun with this;
see below.

b) I am not sure how to (or if I should) explain this to HS juniors who
probably haven't learned much about linear algebra.

It's not an algebra issue so much as a physics issue.

The physics is most clearly seen in the case where Kirchhoff's
so-called laws do not apply, specifically in the case where
there are induced voltages due to time-dependent magnetic fields.

Draw a fairly coarse section of graph-squares on the board.
There will be a circulatory induced voltage around each little
square. Now pick two adjacent squares, i.e. two little squares
that share a common side. Let the circulation around one
square be called "a" and the circulation around the other square
be called "b". Mark the two squares accordingly. Now erase
the common side between the two squares. Argue that the
circulation around the newly formed rectangle is a+b. That's
because the thing you erased made equal-and-opposite contributions
to the two ancestral squares, so it makes no contribution to the
total. If the total was a+b before you erased the divider, it is
a+b afterward.

Feynman volume II, figure 3-9

If you carry this argument far enough, you can re-invent Stokes's
law. (I'm not suggesting you carry it that far ... but that
gives you some idea of the importance of the principle!)

In its simplest form, the principle is that there is some notion
of circulation per unit area. Among other things, the circulation
around any "big loop" is the sum of the circulation around the "little
loops", no matter how you divide the big loop into (non-overlapping)
little loops.

The most elegant way of saying this is that we are seeing the
effects of the law of conservation of flux (magnetic flux).

This sum-rule guarantees that any circuit complicated to have a
topological genus greater than one will have multiple linearly
*dependent* Kirchhoff laws that can be written down. That's
interesting, but only the tip of the iceberg of interesting
things that happen when you have a circulation per unit area,
since Kirchhoff only considers the case of circulation=0.

======================

Presumably nobody will be surprised to hear that the Kirchhoff
node rules are overcomplete i.e. linearly dependent.

The story is mostly the same, but not quite word-for-word the
same. No notion of circulation is involved. And this time
the sum rule is based on conservation of charge (whereas the
loop law was based on conservation of flux).