Express Laplace transform of voltage across a capacitor in terms of charge
I'd be interested to hear folks' opinions.
Express Laplace transform of voltage across a capacitor in terms of charge
I'd be interested to hear folks' opinions.
Disclaimer: I have written an answer to the question discussed here, so one might consider my opinion in this matter biased. I did not vote on the closure of the question in any way.
I think the question should have the homework-and-exercises tag, but should not be closed. Clearly, this is essentially an exercise in computing the behaviour of a specific circuit in a specific situation, and the question asks to understand a specific step in the derivation, clearly not because it is interested in the end result since the end result is already known. So "primarily for educational value" applies as well as being about a specific exercise, and therefore I think the tag should be applied.
However, I don't think it should be closed - it shows some effort to work through the problem (the equation it asks about in the end was derived by that effort in the first place) and the question of why the voltage is computed in this way is conceptual. There is no straightforward "plug-and-chug" derivation that yields eq. ($\star$), it must be arrived at mainly by physical reasoning about the circuit, not computation.
As supporting evidence that this is not (or should not be) off-topic as homework-like, consider the 3:0 Leave Open review this question received in the queue. Homework-like questions that are clearly off-topic usuaslly receive quick and almost always unanimous close reviews, so this at least shows it is not clearly off-topic.
As our current policy says, the homework-and-exercises tag should be applied to questions which arise in the context of doing "educational exercises". The defining property of an educational exercise is that the purpose of solving the exercise is learning to use (or gaining further practice with) the method by which it is solved.
I don't believe that is the case in this question. It's been claimed that this is the exercise:
Consider a circuit consisting of a capacitor $C$, an impedance $\tilde{Z}(\omega)$, and a constant voltage source $V$ connected in a loop. Suppose in equilibrium the charge across the capacitor is $Q_e$. Now at time $t=0$ we drop some charge on the capacitor, making the total capacitor charge $Q_0$. We'd like to solve for the time dependent charge $Q(t)$ on the capacitor.
The purpose of solving this problem is not for you to learn the method by which it can be solved, I presume. If I understand correctly, the entire purpose of this problem is to provide context for the underlying conceptual question,
The paper I'm reading writes an equation which leads me to believe that $$\hat{V}(p) = \frac{Q_e}{pC} - \frac{\hat{Q}(p)}{C} \, , \tag{$\star$}$$ but I don't understand why that's the case.
Why is equation $(\star)$ correct?
It's completely irrelevant (again if I understand correctly) whether you actually make any further progress in solving the exercise. That would not be the case if your purpose were to learn the method of solution. Therefore, I conclude the homework-and-exercises tag is inappropriate for this question.
As further justification, another property of almost all educational exercises is that you get them from some other source: a textbook, a homework assignment, an old contest, a problem book, or so on. As I understand it, this is not the case here. Now, it is possible to have a question which does not include an exercise from an external source but is nevertheless a homework-and-exercises question, e.g. if the poster has made up an exercise for themselves to get practice with a certain technique. But this is quite rare. Rare enough that, when a question is not based on an exercise from some external source, we should probably avoid applying the homework-and-exercises tag unless we know that the poster's intent is to practice the method. I certainly don't believe that is the case here.