I don't see this as being about conventions at all. The q in dq/dt is a
different q than the one in q = CV. They are different physical locations.
For example, I is the same everywhere except between the plates - so choose
an arbitrary place to evaluate it - maybe through a cross section of the
resistor where it becomes j times A. In this case, j is certainly not
referring to the charges, Q, residing on the capacitor, but to the local
charges passing through the resistor. Numerically, however, jA = -dQ/dt (as
John states in a previous posting).
I also find Serway inconsistent. In Serway & Jewett, Physics for Scientists
and Engineers, 6th ed., he attempts to use the same equation for the
charging of a capacitor when an emf, E, is in the circuit, and for the
discharging of the capacitor when E is removed by setting it to 0. However,
in his figures 28.19 and 28.21, he reverses the direction of the current
without reversing the direction in which the potential increases across the
resistor, coming up with what I consider a faulty Eqn. 28.16 when referred
to Figure 28.21. He then seems to try to patch this up with I = +dq/dt. To
me, it's just bad pedagogy. If he really wants to use the same equation and
then let E go to 0, then he shouldn't reverse the direction of the current.
Either way, his use of the q in dq/dt as the same physical charge q on the
capacitor without some sort of explanation is just plain confusing to the
reader.
Bob at PC
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Saturday, February 18, 2006 10:58 PM
To: PHYS-L Maillist
Subject: Re: [Phys-l] RC Disharge Analysis
There are conflicting conventions here.
a) Take any two-terminal device (notably including resistors and
capacitors) and let the voltage across it be V. Without loss
of generality label the terminals A and B such that V = V_A - V_B.
Then by convention, the current flows into terminal A and flows out
of terminal B.
b) Kirchhoff's loop law requires the current to flow the same way
(e.g. "clockwise") through all elements.
In the version that M.E. suggests, these two conventions conflict.
Something's got to give. Pick one element, reverse the convention,
and accordingly reverse the sign in the corresponding equation.