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Re: [Phys-l] RC Disharge Analysis



Bob LaMontagne wrote:

I don't see this as being about conventions at all.

Huh? I stand by what I wrote previously. But it's not "just"
about conventions ... it's about the _meaning_ of equations.

The q in dq/dt is a
different q than the one in q = CV.

Yes! That _supports_ the main point I was making.

They are different physical locations.
For example, I is the same everywhere except between the plates

Making I the same everywhere is at best a convention, and is
in general impossible, i.e. unphysical.

I've been through this with students many times. I consider it
just a particular form of the familiar "plug and chug" disease:
they have a formula, and they want to apply it without thinking
about what it _means_.

The whole notion of "the" current (I) is a non-starter. This
may come as a surprise when your attention is restricted to
simple two-component circuits; the point I'm making is more
obvious if you consider a real circuit such as (say) the innards
of an op-amp:
http://www.ibiblio.org/obp/electricCircuits/Semi/03323.png

It should be obvious that there is no such thing as "the" current
in the op-amp circuit. Rather there is current I1 flowing through
wire 1, current I2 flowing through wire 2, etc. etc. etc.

Returning to the ultra-simple two-component circuit, we can
identify current I1 flowing into the top of the capacitor,
current I2 flowing into the top of the resistor, and current
I3 flowing through the wire (in some direction of your choosing).
There is _some_ simple relationship between these three currents,
but if you just blissfully drop the subscripts and set them all
equal to "the" current, you run a serious risk of getting the wrong
answer. If you draw the circuit in the way that M.E. seems to
prefer, I1 will be opposite to I2. If you call them both "the"
current, you're doomed; you've already lost a minus sign.

I also find that students are often neither patient nor methodical;
they want to solve every problem in one swell foop. They are so
accustomed to being judged by multiple-choice tests with 40
questions in 30 minutes that they assume every question in the
world can be answered in about 45 seconds.

I tell 'em slow down: draw the diagram. Don't rush to write down
one big equation; write down N little equations that you know are
right. Then massage them according to the rules of algebra. The
equations aren't enough; you need some sort of legend (typically
a diagram and a legend) to tell you what the variables _mean_.

General philosophical sermon:
Starting with N little equations that are right is better than
starting with one big equation that is just a guess. Getting the
right answer slowly is better than getting the wrong answer quickly.
Beginners don't learn to ski on the double-black-diamond slopes;
if you try that you'll crash every time and never learn anything.

There's no point in knowing a formula unless you know what it _means_.
That includes having some clue about the _limits_ of its validity.

===========

To recap: You must look at the _meaning_
I = C Vdot refers to the current flowing into the capacitor
I = V / R refers to the current flowing into the resistor

Maybe those are the same current; maybe they're not. You can't
blithely set them equal without identifying some clear physical
reason why they must be equal.

Also: You can't assume there will be any meaning to the notion of
"the" current in "the" circuit.

Maybe it would help to write lots of subscripts, such as
I1 = C1 V1dot
I2 = V2 / R2
but alas textbooks very commonly suppress the subscripts when
stating general laws such as Ohm's law. Students simply *must*
learn to attach their own subscripts before applying the law
to specific cases.

This is a very general principle; the same principle applies
in mechanics where (F = m a) and (F = G m M / r^2) ... but the
meaning of F may differ from one equation to the other (e.g. if
the gravitational force is not the only force acting on the object).

it's just bad pedagogy.

I agree the typical books are just terrible at teaching these concepts.
This stuff is soooo easy if you think about it the right way, and sooo
confusing if you don't think about it the right way. One might hope
the books would bend over backwards to explain it........