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title: "In the circuit shown, an ideal battery of potential difference \\(\\Delta V_0 = 12\\text{ V}\\) is connected in series with an open switch \\(S\\) and resistor \\(R_1 = 10\\text{ }\\Omega\\). A parallel combination consisting of resistor \\(R_2 = 20\\text{ }\\Omega\\) and an initially uncharged capacitor \\(C = 50\\text{ }\\mu\\text{F}\\) is connected in series with \\(R_1\\). At time \\(t = 0\\), switch \\(S\\) is closed, and the potential difference \\(V_C\\) across the capacitor is recorded as a function of time \\(t\\), producing the graph shown. Which of the following statements correctly interprets the features of the graph regarding the circuit’s behavior?"
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url: "https://nerd-notes.com/ubq/117308/"
date_modified: "2026-08-04T06:45:28+00:00"
---

# In the circuit shown, an ideal battery of potential difference \(\Delta V_0 = 12\text{ V}\) is connected in series with an open switch \(S\) and resistor \(R_1 = 10\text{ }\Omega\). A parallel combination consisting of resistor \(R_2 = 20\text{ }\Omega\) and an initially uncharged capacitor \(C = 50\text{ }\mu\text{F}\) is connected in series with \(R_1\). At time \(t = 0\), switch \(S\) is closed, and the potential difference \(V_C\) across the capacitor is recorded as a function of time \(t\), producing the graph shown. Which of the following statements correctly interprets the features of the graph regarding the circuit’s behavior?

In the circuit shown, an ideal battery of potential difference \(\Delta V_0 = 12\text{ V}\) is connected in series with an open switch \(S\) and resistor \(R_1 = 10\text{ }\Omega\). A parallel combination consisting of resistor \(R_2 = 20\text{ }\Omega\) and an initially uncharged capacitor \(C = 50\text{ }\mu\text{F}\) is connected in series with \(R_1\). At time \(t = 0\), switch \(S\) is closed, and the potential difference \(V_C\) across the capacitor is recorded as a function of time \(t\), producing the graph shown. Which of the following statements correctly interprets the features of the graph regarding the circuit's behavior?

![A schematic of a DC circuit containing an ideal battery labeled \Delta V_0 = 12 \text{ V}, an open switch S, a resistor R_1 = 10 \ \Omega, and a parallel combination of resistor R_2 = 20 \ \Omega and capacitor C = 50 \ \mu\text{F}. The battery, switch S, and resistor R_1 are in series in the main branch. The main branch then splits into two parallel branches: top branch contains resistor R_2 and bottom branch contains capacitor C. All components are labeled with variable names and values. No other labels appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785825927-lPxlvq.jpg)

![A graph of potential difference V_C across the capacitor in volts versus time t in microseconds. The vertical axis ranges from 0 to 12 V with grid lines at 0, 2, 4, 6, 8, 10, and 12 V. A horizontal dashed line at V_C = 8.0 \text{ V} is labeled '8.0 V'. The horizontal axis ranges from 0 to 1000 microseconds. A smooth curve begins at (0,0), rises steeply initially, and asymptotically approaches the 8.0 V line. At time t = 333 microseconds, a point on the curve is highlighted with a vertical dashed line down to the axis labeled \tau = 333 \ \mu\text{s} and a horizontal dashed line to the vertical axis at 5.0 V. No other labels appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-2-1785825927-jtsnCc.jpg)

- **A.** The asymptotic potential difference of \(8.0\text{ V}\) indicates that internal resistance in the battery reduces the available voltage, so the time constant \(\tau\) represents the time required for \(V_C\) to reach \(63\%\) of the ideal battery voltage \(\Delta V_0\) (\(7.6\text{ V}\)).
- **B.** The initial potential difference of \(0\text{ V}\) occurs because the uncharged capacitor acts as an open circuit at \(t = 0\), causing all initial current to bypass the parallel network and flow only through \(R_1\).
- **C.** The asymptotic potential difference of \(8.0\text{ V}\) represents the steady-state voltage across \(R_2\) when the fully charged capacitor acts as an open circuit, and the time constant \(\tau\) represents the time required for \(V_C\) to reach \(63\%\) of this asymptotic value (\(5.0\text{ V}\)).
- **D.** The potential difference approaches \(8.0\text{ V}\) because current through \(R_1\) drops to zero at steady state, and the time constant \(\tau\) represents the time required for \(V_C\) to reach \(37\%\) of the battery voltage \(\Delta V_0\) (\(4.4\text{ V}\)).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117308/*