Resistor–Capacitor (RC) Circuits - AP Physics 2

1. Identify the initial current through a capacitor when discharging begins.

   $\frac{V_0}{R}$. Same as charging current but in opposite direction.

2. What is the charge on a capacitor at $t = 0$ when charging begins?

   $Q = 0$. Capacitor starts with no charge when charging begins.

3. State the unit of the time constant $\tau$ in an RC circuit.

   Seconds (s). Time unit from $R$ (ohms) × $C$ (farads) = seconds.

4. What is the initial voltage across a capacitor after a switch is closed in an RC circuit?

   Zero volts (0 V). Capacitor starts uncharged when switch closes.

5. What happens to the time constant if capacitance is reduced by half?

   Time constant is halved. $\tau = RC$, so halving $C$ halves the time constant.

6. Identify the expression for the voltage across a charging capacitor over time.

   $V(t) = V_0 (1 - e^{-t/RC})$. Exponential approach to supply voltage $V_0$.

7. State the relationship between voltage and charge for a capacitor.

   $V = \frac{Q}{C}$. Fundamental capacitor equation relating charge and voltage.

8. Which component stores energy in an RC circuit?

   Capacitor. Capacitor stores electrical energy in electric field.

9. Determine the time required for a capacitor to discharge to half its initial voltage.

   $t = RC \ln(2)$. Half-life formula using natural logarithm of 2.

10. Calculate the time constant if $R = 5 \text{ k}\Omega$ and $C = 2 \text{ \muF}$. Provide in milliseconds.

    $\tau = 10 \text{ ms}$. $\tau = RC = (5000)(2 × 10^{-6}) = 0.01$ s = 10 ms.

11. State the expression for the current in an RC circuit during charging.

    $I(t) = \frac{V_0}{R} e^{-t/RC}$. Current decays exponentially from initial value $\frac{V_0}{R}$.

12. Calculate the charge on a capacitor at $t = \tau$ during charging.

    $Q = 0.632Q_0$. At one time constant, charge reaches 63.2% of maximum.

13. Identify the effect of a short circuit across a charged capacitor.

    Discharges immediately. Short circuit provides zero resistance discharge path.

14. What is the voltage across the capacitor after five time constants during charging?

    Approximately equal to $V_0$. After $5\tau$, capacitor reaches 99.3% of final voltage.

15. What is the time to reach 99% of the final charge in an RC circuit?

    Approximately $5\tau$. Practical time for essentially complete charging/discharging.

16. What is the current in an RC circuit immediately after a switch is closed?

    $\frac{V_0}{R}$. Maximum current flows when capacitor acts like short circuit.

17. Determine the energy dissipated by the resistor during capacitor discharge.

    Equal to initial capacitor energy. All stored energy converts to heat in resistor.

18. Find the time constant if the resistance is $2 \text{ k}\Omega$ and capacitance is $10 \text{ \muF}$.

    $\tau = 20 \text{ ms}$. $\tau = RC = (2000)(10 × 10^{-6}) = 0.02$ s = 20 ms.

19. What happens to the time constant if the resistance is doubled?

    Time constant doubles. $\tau = RC$, so doubling $R$ doubles the time constant.

20. Identify the formula for the voltage across a discharging capacitor.

    $V(t) = V_0 e^{-t/RC}$. Exponential decay from initial voltage $V_0$.

21. What is the voltage across a discharging capacitor after one time constant?

    About 36.8% of initial voltage. After $\tau$, voltage decays to $e^{-1} \approx 0.368$ of initial.

22. What percentage of the final voltage is reached at one time constant?

    About 63.2%. At $t = \tau$, $e^{-1} \approx 0.368$, so $1 - e^{-1} \approx 0.632$.

23. State the voltage across a capacitor at $t = \infty$ when discharging.

    Zero volts (0 V). Capacitor fully discharges to zero voltage eventually.

24. How does capacitance affect the time constant in an RC circuit?

    Directly proportional. Larger capacitance increases time constant linearly.

25. What is the initial voltage across a capacitor in an uncharged RC circuit?

    Zero volts (0 V). Uncharged capacitor has no stored energy or voltage.

26. How does decreasing the capacitance affect the charging time of an RC circuit?

    Decreases charging time. Smaller capacitance reduces time constant, faster charging.

27. What is the current through the resistor at $t = \infty$ in a charging RC circuit?

    Zero amperes (0 A). No current flows when capacitor is fully charged.

28. What is the behavior of the current in a discharging RC circuit over time?

    Exponential decay. Current follows exponential decay pattern during discharge.

29. Which law applies to the sum of voltages in an RC circuit loop?

    Kirchhoff's Voltage Law. Voltage drops must sum to zero around any closed loop.

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

    $\tau = RC$. Product of resistance and capacitance determines RC circuit timing.

31. State the voltage across a capacitor when it is fully charged in an RC circuit.

    Equal to the supply voltage. Fully charged capacitor voltage equals source voltage.

32. Find the voltage across the capacitor after three time constants during discharge.

    Approximately 5% of initial voltage. After $3\tau$, voltage is $e^{-3} \approx 0.05$ of initial.

33. In an RC circuit, what does a larger resistor value do to the charging time?

    Increases charging time. Higher resistance increases time constant, slower charging.

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

    Equal to the supply voltage. No current flows when capacitor reaches maximum voltage.

35. What is the expression for the charge on a capacitor during charging?

    $Q(t) = Q_0 (1 - e^{-t/RC})$. Charge builds exponentially toward maximum value $Q_0$.

36. What is the energy stored in a fully charged capacitor?

    $E = \frac{1}{2}CV^2$. Energy formula for charged capacitor's electric field.

37. What is the effect of adding a resistor in parallel to the existing one in an RC circuit?

    Decreases total resistance. Parallel resistors reduce equivalent resistance, decreasing $\tau$.

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