RC Circuits Problem Set
AP Physics 2 • Unit 11: Electric Circuits
Topic 11.8: RC Circuits Problem Set
Transient Analysis, Time Constants, Multi-Loop Circuits, & Graphical Representations
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Date: Period:
Directions: Select the best answer for Questions 1–12. For all problems, assume ideal batteries and meters unless noted. \( \tau = RC \quad | \quad U_C = \frac{1}{2}CV^2 \)
An uncharged capacitor of capacitance \( C \) and a resistor of resistance \( R \) are connected in series with an ideal battery of emf \( \mathcal{E} \) and an open switch. Immediately after the switch is closed (\( t = 0^+ \)), which pair of values correctly represents the current \( I_0 \) from the battery and the potential difference \( V_C \) across the capacitor?
(A) \( I_0 = 0 \quad \text{and} \quad V_C = \mathcal{E} \)
(B) \( I_0 = \frac{\mathcal{E}}{R} \quad \text{and} \quad V_C = 0 \)
(C) \( I_0 = \frac{\mathcal{E}}{R} \quad \text{and} \quad V_C = \mathcal{E} \)
(D) \( I_0 = 0 \quad \text{and} \quad V_C = 0 \)
A capacitor is charged through a resistor by a constant-voltage source. The time constant of the circuit is \( \tau \). At time \( t = 2\tau \) after charging begins, the charge accumulated on the capacitor is approximately what percentage of its maximum final charge \( Q_{\max} \)?
(A) \( 50.0\% \)
(B) \( 63.2\% \)
(C) \( 86.5\% \)
(D) \( 95.0\% \)
An uncharged capacitor of capacitance \( C \) is connected in series with a resistor \( R \) and an ideal battery of emf \( \mathcal{E} \). The switch is closed, and the capacitor becomes fully charged. During the entire charging process, the total work done by the battery is \( W_B \), the final electrostatic energy stored in the capacitor is \( U_C \), and the total thermal energy dissipated by the resistor is \( E_R \). Which statement correctly relates these quantities?
(A) \( W_B = U_C \) and \( E_R = 0 \)
(B) \( W_B = 2U_C \) and \( E_R = U_C = \frac{1}{2} C \mathcal{E}^2 \)
(C) \( W_B = U_C \) and \( E_R = \frac{1}{2} U_C \)
(D) \( E_R \) depends on the value of \( R \), with larger \( R \) dissipating more energy
In the circuit shown, the capacitor \( C \) is initially uncharged. The resistors have values \( R_1 = 20\,\Omega \) and \( R_2 = 30\,\Omega \), and the battery has an emf of \( \mathcal{E} = 12\,\text{V} \). Immediately after the switch \( S \) is closed, what is the current through resistor \( R_2 \)?
Ε R₁ R₂ C S
(A) \( 0\,\text{A} \)
(B) \( 0.24\,\text{A} \)
(C) \( 0.40\,\text{A} \)
(D) \( 0.60\,\text{A} \)
AP Physics 2 • Unit 11: Electric Circuits Topic 11.8: RC Circuits Practice Page 1 of 5
AP Physics 2 • Section I: Multiple Choice (Continued) Topic 11.8
An uncharged capacitor is connected in series with a resistor and an ideal battery. The graph to the right shows curves \( X \) and \( Y \) as functions of time after the switch is closed. Which set of identifications correctly matches each curve with its physical quantity?
Time t Value Curve X Curve Y
(A) \( X \): Current \( I(t) \); \( Y \): Capacitor Voltage \( V_C(t) \)
(B) \( X \): Resistor Voltage \( V_R(t) \); \( Y \): Stored Energy \( U_C(t) \)
(C) \( X \): Capacitor Charge \( q(t) \); \( Y \): Current \( I(t) \)
(D) \( X \): Resistor Power \( P_R(t) \); \( Y \): Capacitor Charge \( q(t) \)
A circuit consists of a \( 12\,\mu\text{F} \) capacitor initially charged to a potential difference of \( 24\,\text{V} \). At time \( t = 0 \), it is connected across a \( 50\,\text{k}\Omega \) resistor to discharge. At what time \( t \) will the potential difference across the capacitor decrease to \( 6.0\,\text{V} \)?
(A) \( 0.42\,\text{s} \)
(B) \( 0.83\,\text{s} \)
(C) \( 1.20\,\text{s} \)
(D) \( 2.40\,\text{s} \)
A capacitor of capacitance \( C \) is fully charged to voltage \( V_0 \) and then discharged through two identical resistors of resistance \( R \) connected in parallel. A second identical capacitor, also charged to \( V_0 \), is discharged through the same two resistors connected in series. Let \( \tau_p \) and \( \tau_s \) be the time constants of the parallel and series discharge circuits, respectively. What is the ratio \( \tau_p / \tau_s \)?
(A) \( 1/4 \)
(B) \( 1/2 \)
(C) \( 1 \)
(D) \( 4 \)
A parallel-plate air capacitor of capacitance \( C_0 \) is connected in series with a resistor \( R \) and an ideal battery. The circuit reaches steady state. With the battery still connected, a slab of dielectric material with dielectric constant \( \kappa = 3.0 \) is slowly inserted to completely fill the space between the capacitor plates. Which of the following correctly describes what occurs immediately after the dielectric is inserted?
(A) No current flows because the circuit was already in steady state
(B) Current flows in the original charging direction until charge triples
(C) Current flows in the reverse direction to discharge the excess electric field
(D) The time constant decreases by a factor of 3 during the new transition
AP Physics 2 • Unit 11: Electric Circuits Topic 11.8: RC Circuits Practice Page 2 of 5
AP Physics 2 • Section I: Multiple Choice (Continued) Topic 11.8
In the circuit shown, the switch has been closed for a very long time. The values are \( \mathcal{E} = 18\,\text{V} \), \( R_1 = 6.0\,\Omega \), \( R_2 = 12\,\Omega \), \( R_3 = 4.0\,\Omega \), and \( C = 5.0\,\mu\text{F} \). What is the steady-state potential difference across the capacitor \( C \)?
Ε R₁ R₂ R₃ C
(A) \( 0\,\text{V} \)
(B) \( 6.0\,\text{V} \)
(C) \( 12\,\text{V} \)
(D) \( 18\,\text{V} \)
A student collects potential difference data across a discharging capacitor over time. To obtain a linear graph whose slope can be used directly to calculate the time constant \( \tau \), which of the following quantities should the student plot on the vertical and horizontal axes?
(A) Vertical: \( V_C \); Horizontal: \( 1/t \)
(B) Vertical: \( \ln(V_C) \); Horizontal: \( t \)
(C) Vertical: \( V_C^2 \); Horizontal: \( t \)
(D) Vertical: \( \ln(t) \); Horizontal: \( V_C \)
Two identical capacitors are connected in series with a resistor \( R \) and a battery \( \mathcal{E} \) in Circuit 1. In Circuit 2, the same two capacitors are connected in parallel with the same resistor \( R \) and battery \( \mathcal{E} \). How do the effective time constant \( \tau \) and the total energy stored \( U_{\text{total}} \) in steady state compare between the two circuits?
(A) \( \tau_1 > \tau_2 \) and \( U_{\text{total},1} > U_{\text{total},2} \)
(B) \( \tau_1 < \tau_2 \) and \( U_{\text{total},1} < U_{\text{total},2} \)
(C) \( \tau_1 < \tau_2 \) and \( U_{\text{total},1} = U_{\text{total},2} \)
(D) \( \tau_1 > \tau_2 \) and \( U_{\text{total},1} < U_{\text{total},2} \)
A light bulb (acting as an ohmic resistor) is placed in series with an uncharged large capacitor and a DC battery. When the switch is closed, what happens to the brightness of the light bulb over time?
(A) It glows brightly at first, then dims exponentially until it goes out
(B) It is off initially, then brightens exponentially to a constant maximum
(C) It maintains a constant steady brightness from the instant the switch closes
(D) It blinks periodically as the capacitor repeatedly charges and discharges
AP Physics 2 • Unit 11: Electric Circuits Topic 11.8: RC Circuits Practice Page 3 of 5
Section II: Free-Response
Question 1: Qualitative/Quantitative Translation
Suggested Time: 15 minutes • 8 Points
The circuit shown contains an ideal battery of electromotive force \( \mathcal{E} \), two resistors with resistances \( R_1 \) and \( R_2 \) (where \( R_2 = 2R_1 \)), an uncharged capacitor of capacitance \( C \), and a two-way switch \( S \).
Initially, the switch is in the neutral position. At time \( t = 0 \), switch \( S \) is moved to contact position A.
Ε A B S R₁ C R₂
(a) [2 points] In terms of given quantities, determine the current \( I_0 \) through the battery immediately after the switch is thrown to position A (\( t = 0^+ \)), and the charge \( Q_f \) on the capacitor after a very long time (\( t \to \infty \)). Briefly justify your answers.
(b) [2 points] After remaining at position A for a long time, switch \( S \) is quickly moved from position A to position B at a new time \( t' = 0 \). Write an expression for the time constant \( \tau_{\text{discharge}} \) of this discharge circuit in terms of \( R_1 \) and \( C \). Explain why the discharge time constant differs from the charging time constant.
(c) [2 points] On the axes below, sketch the potential difference across the capacitor \( V_C(t) \) from \( t = 0 \) to a long time after the switch was moved to position A, and then from \( t' = 0 \) after being switched to position B. Clearly label key asymptotic values.
Time \(V_C\) \(\mathcal{E}\) Switch at A (Charging) Switch at B (Discharging)
(d) [2 points] A student claims: "Because \( R_2 \) is twice as large as \( R_1 \), the total thermal energy dissipated by \( R_2 \) during discharge will be twice as large as that dissipated by \( R_1 \)." Is the student's claim correct? Explain your reasoning using physical principles.
AP Physics 2 • Unit 11: Electric Circuits Topic 11.8: RC Circuits Practice Page 4 of 5
Section II: Free-Response
Question 2: Experimental Design & Data Analysis
Suggested Time: 25 minutes • 12 Points
Students are tasked with experimentally determining the unknown capacitance \( C_x \) of a cylindrical capacitor. They are provided with an ideal DC power supply of known emf \( \mathcal{E} = 10.0\,\text{V} \), a precision resistor with resistance \( R = 50.0\,\text{k}\Omega \), a single-pole double-throw switch, a digital voltage sensor with negligible current draw, and a computer interface that logs voltage every \( 0.05\,\text{s} \).
(a) [3 points] In the space below, draw a complete schematic diagram of an RC circuit that allows the capacitor to be charged by the power supply and then discharged through resistor \( R \). Clearly indicate where the voltage sensor is connected to measure the discharging voltage across the capacitor.
Draw circuit schematic here:
(b) [3 points] Starting from the discharging voltage relationship \( V_C(t) = V_0 e^{-t/\tau} \), show how this equation can be algebraically manipulated into a linear relationship of the form \( y = mx + b \). Clearly identify what quantities should be plotted on the vertical and horizontal axes, and state what the slope \( m \) represents in terms of \( R \) and \( C_x \).
(c) [3 points] The students linearize their discharge data and obtain a best-fit line with the equation: \[ \ln\left(\frac{V_C}{1\,\text{V}}\right) = -0.416\,t + 2.30 \] where \( t \) is in seconds. Using the given value \( R = 50.0\,\text{k}\Omega \), calculate the experimental capacitance \( C_x \). Show your work with units.
(d) [3 points] Suppose the voltage sensor used in the experiment was not ideal, but instead had an internal resistance \( R_V = 200\,\text{k}\Omega \) connected in parallel with the capacitor.
- Would the measured time constant be greater than, less than, or equal to the true time constant \( R C_x \)?
- Would the experimental value of capacitance calculated in part (c) be an overestimate, an underestimate, or unaffected? Justify your answers.
AP Physics 2 • Unit 11: Electric Circuits Topic 11.8: RC Circuits Practice Page 5 of 5
