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So this is a homework question and I am having trouble figuring out what they are asking.

'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The quantity RC has the dimensions of time and is often called the time constant for the circuit. How many time constants does it take for a capacitor to charge to 90% of the applied voltage, V0? Justify your answer'

So change in V, or $V_C$, or $\delta{V}$ is 90%. In other words we have $0.9V=q(t)/C$?

I have found in a previous question that $q(t)=V_0C(1-e^{-\frac{t}{CR}})$ so

$$0.9V=\frac{V_0C(1-e^{-\frac{t}{CR}})}{C}$$

However I am not sure where to go from here, if I am even on the right track at all.

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  • $\begingroup$ No I only have the above info $\endgroup$ – user88720 Apr 30 '14 at 1:48
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If you look at the capacitor voltage curve, you notice that somewhere between $2$ and $3$ time constants, we have $90\%$ charge. We should be able to figure this out generally when not given the resistor and capacitor value.

We have the unknown:

$$\mbox{Time Constant} = \tau = RC$$

Using:

$$\large 0.9V=\frac{V_0C(1-e^{-\frac{t}{CR}})}{C} = V_0(1-e^{-\frac{t}{CR}}) = V_0(1-e^{-\frac{t}{\tau}})$$

We want to solve for $t$, so we have:

$$t = - \tau \ln \left(-\frac{0.9 V-V_0}{V_0}\right)$$

However, the voltage across the capacitor is $.9$ of the the voltage source $V_0$, so we can rewrite this as:

$$t = - \tau \ln \left(-\frac{0.9 V-V_0}{V_0}\right) = - \tau \ln \left(-\frac{0.9 V_0-V_0}{V_0} \right) = - \tau \ln (0.1) = -(-2.30259)\tau = 2.30259 \tau$$

In other words, it will take $2.30259$ time constants to charge to $90\%$.

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  • $\begingroup$ You're an EE as well? $\endgroup$ – Robert Lewis Apr 30 '14 at 2:16
  • $\begingroup$ What happened to the /C part. Isn't $0.9V=q(t)/C$? $\endgroup$ – user88720 Apr 30 '14 at 2:27
  • $\begingroup$ Did you read my profile? I've got that math/CS/EE thing going as well! Plus RS! As I said, degree'd out! $\endgroup$ – Robert Lewis Apr 30 '14 at 2:35
  • $\begingroup$ As for RS, I didn't much like writing sociology papers which were part of the core general requirement--every grad student had to take some "general RS stuff", you know--psychology, sociology, anthropology, philosophy; I found that part of it BORING!!! When I began to focus on what I wanted, which was ancient Hebrew literature and the history of early Judea-Christianity, I was so into it that it was easy. I think it's hard for "techies" to get the humanities mindset! The logic is the same, but some of the arguments make mathematical subtlety pale! It's like . . more . . . $\endgroup$ – Robert Lewis Apr 30 '14 at 3:18
  • $\begingroup$ Next: trying to get the precision Euclid's axioms out of ordinary writing/speech; it can be done, but you really have to read stuff very carefully! $\endgroup$ – Robert Lewis Apr 30 '14 at 3:20

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