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Re: [Phys-l] RC Discharge



Bob Sciamanda wrote:

Nowhere do I mention the plate "with the positive charge"

I guess not. Sorry. The convention that "+" means "with the
positive charge" is so firmly established that I simply could
not imagine that it meant anything else.

The + sign does not imply that this plate will be positively charged. The + and - charge signs merely serve to define the charge configuration to be represented by a positive value of Q, even though Q can be + or -.

This is horrible notation. It is highly nonstandard. I find
it confusing, and I imagine students would find it confusing
(although I can't say for sure, because I have never exposed
students to anything so horrible).

This is the same approach that we use when we draw a directed arrow to define the direction of a current to be represented by a positive value of i, even though i can be + or -.

No, it is not the same.

It is permissible to label one plate as "Q", or equivalently "+Q" in
contrast to the other plate which is "-Q", but a plus sign by itself
(not multiplying the Q variable) means something else entirely.

The + and - on an electrolytic capacitor mean the same thing as the
+ and - on a battery; the + side must be at a positive voltage w.r.t
the - side.

I am willing to give authors considerable scope to define their own
terms, but we have to draw the line somewhere; it makes no sense
to define "yellow" to be the color of the midight sky and define
"black" to be the color of the sun.

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

we draw a directed arrow to define the direction of a current to be represented by a positive value of i, even though i can be + or -.

Yes, as regards current, that is fine.

It must be emphasized that the arrow in this situation does not
represent i itself; the arrow represents a _basis vector_. The
actual current is some multiple of this basis vector.

This is confusing to students because
-- sometimes we want to draw a basis vector, and
-- sometimes we want to draw the actual physically-meaningful
vector.

One could make a case for labelling the unit vector "i hat"
rather than just "i", especially in introductory situations,
but I've never seen a textbook do this. For that matter, I've
never seen a textbook discuss the "unit vector" issue clearly.

In the general, complex circuit case one does not know ahead of time the directions of currents or the charge configurations of capacitors. This is true even in simple, one-loop circuits if one does not yet know the initial conditions. The equations should be written for the general case: as a computer algorithm applicable to any and all initial conditions. One therefore needs a general set of defining conventions for correlating current directions and capacitor charge configurations with the algebraic signs of the quantities i and q, respectively.

Yes indeed.

These conventions are firmly established, and do *not* include using
"+" as a shorthand for "+Q".

One safe way to do this is to:
1) For each current, choose (and indicate) a definite direction to be represented by a positive value of the signed variable i(t).

Yes. This is the "unit vector" mentioned above.

2) For each capacitor, choose (and indicate) one of its two plates to be that whose absolute charge is represented by the (signed) value of a variable q(t). The other plate of this capacitor will carry the absolute charge - q(t).

Yes.

All of these choices are completely arbitrary. However, once chosen, they will force one or the other of the relations i=dq/dt or i=-dq/dt in the wires directly connected to each capacitor.

Yes.