Resistor–Capacitor (RC) Circuits - AP Physics C Electricity and Magnetism

1. What is the effect of doubling resistance on $\tau$ in an RC circuit?

   $\tau$ doubles. Time constant is directly proportional to resistance.

2. Identify the unit of resistance.

   Ohm ($\Omega$). Named after Georg Ohm, measures opposition to current flow.

3. Calculate voltage across a capacitor at $t = 4\tau$ when discharging.

   $V(t) \approx 0.018 V_0$. At $t=4\tau$: $e^{-4} \approx 0.018$ of initial voltage remains.

4. State the formula for current in a discharging RC circuit.

   $I(t) = I_0 e^{-t/\tau}$. Current decays exponentially from initial value $I_0$ during discharge.

5. What is the initial charge on a capacitor in an RC circuit?

   Zero when starting to charge. Uncharged capacitor acts like open circuit initially.

6. What fraction of the initial voltage remains after $5\tau$ in an RC circuit?

   Less than 1%. After $5\tau$: $e^{-5} \approx 0.007$ or about 0.7% remains.

7. What happens to the current in an RC circuit over time when charging?

   The current decreases exponentially. Current starts at maximum and decreases as capacitor voltage increases.

8. What is the expression for $\tau$ in terms of resistance and capacitance?

   $\tau = R \times C$. Direct product of circuit resistance and capacitance values.

9. Identify the relationship between current and voltage in a discharging RC circuit.

   Both decrease exponentially. Both follow same exponential decay pattern with time constant $\tau$.

10. Find the time required to reach half the maximum charge in an RC circuit.

    $t = \tau \ln(2)$. Natural logarithm of 2 times the time constant gives half-charge time.

11. What does an RC circuit model in terms of natural and forced response?

    First-order linear system. RC circuits exhibit characteristic first-order exponential response behavior.

12. What is the final charge on a capacitor in an RC circuit after full charge?

    $Q = C \times V_0$. Final charge equals capacitance times applied voltage.

13. What is the charge on a capacitor at any time $t$ during charging?

    $Q(t) = C V_0 (1 - e^{-t/\tau})$. Charge accumulates exponentially approaching final value $CV_0$.

14. What is the effect of halving capacitance on $\tau$ in an RC circuit?

    $\tau$ is halved. Time constant is directly proportional to capacitance.

15. State the formula for the voltage across a discharging capacitor in an RC circuit.

    $V(t) = V_0 e^{-t/\tau}$. Exponential decay from initial voltage $V_0$ with time constant $\tau$.

16. Express $V(t)$ in terms of $V_0$ for $t = 0$ in a charging RC circuit.

    $V(t) = 0$. Initially uncharged capacitor has zero voltage at start of charging.

17. What is the initial current in an RC circuit when the capacitor starts charging?

    $I_0 = \frac{V_0}{R}$. Using Ohm's law: initial voltage across resistor equals supply voltage.

18. What is the time constant $\tau$ for $R = 100 \Omega$ and $C = 10 \mu\text{F}$?

    $\tau = 1 \text{ms}$. Converting units: $100 \times 10 \times 10^{-6} = 1 \times 10^{-3}$ s.

19. What is the voltage across a capacitor in an RC circuit after infinite time?

    Equals supply voltage $V_0$. After infinite time, capacitor fully charges to source voltage.

20. Find $Q(t)$ at $t = \tau$ for a charging capacitor if $C = 1 \text{F}$, $V_0 = 5 \text{V}$.

    $Q(t) = 3.16 \text{C}$. At $t=\tau$: $Q = 1 \times 5 \times (1-e^{-1}) = 3.16$ C.

21. Calculate $I(t)$ at $t = \tau$ for a discharging circuit if $I_0 = 2 \text{A}$.

    $I(t) = 0.736 \text{A}$. At $t=\tau$: $I = I_0 e^{-1} = 2 \times 0.368 = 0.736$ A.

22. State the formula for the voltage across a charging capacitor in an RC circuit.

    $V(t) = V_0(1 - e^{-t/\tau})$. Exponential approach to final voltage $V_0$ with time constant $\tau$.

23. Calculate the energy stored in a capacitor with $C = 4\text{F}$ and $V = 3\text{V}$.

    $U = 18\text{J}$. Using energy formula: $U = \frac{1}{2} \times 4 \times 3^2 = 18$ J.

24. Calculate the voltage across a capacitor at $t = \tau$ when charging.

    $V(t) = 0.632 V_0$. At one time constant: $1 - e^{-1} = 0.632$ of final voltage.

25. Calculate the voltage across a capacitor at $t = 0$ when discharging.

    $V(t) = V_0$. At $t=0$ for discharging, capacitor starts at full voltage.

26. What does the time constant $\tau$ represent in an RC circuit?

    Time to reach approximately 63.2% of final value. At $t=\tau$, exponential function yields $1-e^{-1} \approx 0.632$ or 63.2%.

27. What is the energy stored in a capacitor at any time $t$ during charging?

    $U(t) = \frac{1}{2} C V(t)^2$. Energy formula using instantaneous voltage across capacitor plates.

28. What is the relationship between $V(t)$ and $V_0$ at $t = 2\tau$ when charging?

    $V(t) = 0.865 V_0$. At $t=2\tau$: $1 - e^{-2} = 1 - 0.135 = 0.865$ of final voltage.

29. Determine $I(0)$ for an RC circuit with $R = 10 \Omega$ and $V_0 = 10 \text{V}$.

    $I(0) = 1\text{A}$. Using Ohm's law: $I = V/R = 10/10 = 1$ ampere.

30. What is the final voltage across a fully charged capacitor in an RC circuit?

    Equal to the supply voltage $V_0$. Capacitor voltage approaches source voltage as charging completes.

31. What is the formula for the time constant in an RC circuit?

    $\tau = RC$. Product of resistance and capacitance defines the exponential decay/growth rate.

32. What percentage of the charge remains after $2\tau$ in a discharging RC circuit?

    About 13.5%. At $t=2\tau$: $e^{-2} \approx 0.135$ or about 13.5% remains.

33. Find the time at which $V(t) = 0.9V_0$ in a charging RC circuit.

    $t \approx 2.303\tau$. Solving $0.9 = 1-e^{-t/\tau}$ gives $t = -\tau \ln(0.1) = 2.303\tau$.

34. What is the relationship between $V(t)$ and $V_0$ at $t = 3\tau$ when discharging?

    $V(t) = 0.05 V_0$. At $t=3\tau$: $e^{-3} = 0.05$ of initial voltage remains.

35. Find $\tau$ for an RC circuit with $R = 5 \Omega$ and $C = 2 \text{F}$.

    $\tau = 10 \text{s}$. Using $\tau = RC$: $5 \times 2 = 10$ seconds.

36. Identify the unit of capacitance.

    Farad (F). Named after Michael Faraday, measures charge storage ability.

37. Determine the time constant if $R = 250 \Omega$ and $C = 20 \mu\text{F}$.

    $\tau = 5 \text{ms}$. Converting units: $250 \times 20 \times 10^{-6} = 5 \times 10^{-3}$ s.

38. What is the charge on a capacitor at any time $t$ during discharging?

    $Q(t) = Q_0 e^{-t/\tau}$. Charge decays exponentially from initial value $Q_0$ during discharge.

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