[Physics] RC circuit theory and voltage in the capacitor vs Ohm’s Law

capacitanceelectric-circuitselectrical-resistance

Consider a simple series RC circuit at steady state (capacitor is full).

I've been told that once the capacitor is full we can literally "cut" the circuit because no current can flow.

That argument makes sense, but because the capacitor is full, doesn't that also mean it has a potential difference across it? In other words, the capacitor is a new battery? Then we should apply Kirchhoff Law in the circuit if we want to find the current? (even though I just said it was zero).

What is going on? What is this contradiction?

There is no current in the circuit yet clearly the potential difference in the capacitor is NOT zero.

EDIT

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For part (c), why do I have to use $I = I_0 e^{-t/ \tau}$? Clearly at steady state, the current of the right circuit is just the potential difference across the capacitor (which now can be treated as a battery) and the resistance. As pointed out before I can only apply Ohm's Law to resistors and batteries, but at steady state the capacitor can be thought of as a battery. So the contradiction lies here.

On the left circuit the current is simply $10V = I_1(50k \Omega)$, and on the right it should be $10V = I_2(100k \Omega)$ only because this is steady state

EDIT2 Let me clarify my question again. I am also confused why $I = I_0 e^{-t/ \tau}$ is used? Why does the capacitor discharge? Yes it's considered an isolated circuit once the switch is shut, but does that mean even though part of the circuit is still hooked up to the battery, that the current doesn't continue to build to prevent discharge?

Best Answer

You are right, there is a potential difference across capacitor according to the expression

$$\Delta V = \frac{Q}{C},$$

where $Q$ is charge on the capacitor and $C$ is capacity of the capacitor. And yes, capacitor can be thought of as being a tiny rechargeable battery (see below).

Edit:

  • When you close the switch you are no longer in steady state.
  • Ideal battery is taken to have infinitive charge $Q = \infty$ and constant voltage $\Delta V = \text{const}$ . So $\Delta V = \frac{Q}{C}$ is not valid for ideal batteries.
  • Capacitor on the other hand can change voltage according to $\Delta V = \frac{Q}{C}$.
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