I thought I would enter the fray. Nothing really new to say, just some
of my thoughts and experiences.
I've been caught on this before and yes it can get particularly
confusing for students. I usually analysize the circuit in a way that
gives rise to I = -dq/dt
I think Tim F.'s post is the best way to try to explain this to
students, with subscripts on the q's, as was mentioned the q in dq/dt
and the q in v=q/c are different variables.
This brings up a teachable moment, the chance to emphasize the point of
"define your variables" or "know what your variables mean"
A rather similar problem can occur in multi-body Newton's 2nd law
problems. After drawing FBD for say 4 bodies and setting up Newton II
for them; I will sometimes have on the r.h.s for one of the bodies
something like Sum_of_forces on Object 3 = -ma, which can seem odd,
until you realize that object 3's acceleration was the negative for some
other objects acceleration and the variable "a" was defined as positive
for that object. Again bringing up the chance to emphasize "know what
your variables mean".
For the case at hand perhaps it would be helpful for students to
subscript the q variable
I.e. I = dq/dt = -d (q_c)/dt
And V = q_c/C
If analysing via Sciamanda's diagram 1 (what I would think is the most
natural way to choose directions, though not required)
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I would emphasize the sequence to the student:
A) general formula or definition I= dq/dt
B) apply definition to this problem using the defined variable q_c, then
formula reads upon substitution
I = -d (q_c)/dt
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PS,
Useful discussion, I'll start capacitors in about a week. I think I'll
try to practice what I preach in the above.
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Joel Rauber
Department of Physics - SDSU