Differentiating Kirchoff's voltage law expression

[And123]

Homework Statement

Differentiate V0 - iR - q/C = 0 to prove that di/dt = -i/RC.

Homework Equations

V0 - iR - q/C = 0
^ derived from previous question for a circuit that had one battery with emf V0, a resistor of resistance R and a capacitor of capacitance C (all in series).
di/dt = -i/RC

The Attempt at a Solution

V0 - iR - q/C = 0
V0 - (dq/dt)R - q/C = 0
V0C - (dq/dt)RC - q = 0
(dq/dt)RC = V0C - q
d/dt((dq/dt)RC) = d/dt(V0C - q)
RC(d/dt(dq/dt) = -dq/dt
RC(di/dt) = -i
di/dt = -i/RC

^ not sure if my maths makes sense (not very good at doing dif/integration when there are multiple variables). I did get the "answer" (which is easy, since its given), but obviously the method's what matters.

Thanks a lot!

 

[aralbrec]

It's right but you may have gotten there quicker if you just differentiated this:

V0 - iR - q/C = 0

with respect to time right away. You'll notice the answer is in terms of di/dt so you don't really want to substitute i=dq/dt into that equation.

 

[And123]

Thanks for the reply.

Do you mean like this?:

V0 - iR - q/C = 0
d/dt(V0 - iR - q/C) = d/dt(0)
d/dt(V0) + d/dt(-iR) + d/dt(-q/C) = 0
0 - R(di/dt) - (1/C)(dq/dt) = 0
-R(di/dt) - (i/C) = 0
di/dt = -i/(RC)

^ I pull out the R and C like they're constants (which I think is correct), and got rid of V0 like you would when you dif a constant.

Edit: I realize the 3rd step isn't required, but just to step it out for myself, I included it.

 

[aralbrec]

Yes that's right too. I wouldn't have shown so many steps but it's not too different anyway.

V0 - iR - q/C = 0
d/dt (V0 - iR - q/C) = d/dt (0)
-R di/dt - i/C = 0 ;; maybe they want you to say i = dq/dt here
di/dt = - i/(RC)

 

[Nickie]

Your solution appears to be correct. To differentiate the given expression, you first need to use the chain rule to differentiate the term (dq/dt)R. Then, using the product rule, you can differentiate the term (dq/dt)(RC) and equate it to the derivative of V0C - q. From there, you can solve for di/dt and arrive at the given expression of -i/RC. Your method is correct and shows a good understanding of differentiation. Keep up the good work!