Resistors and Capacitors in Series and Parallel (4C) - Chem & Physics

1. What is the current during capacitor discharging through $R$ starting from $V_0$?

   $I(t)=\frac{V_0}{R}e^{-t/\tau}$. Discharging current begins at V₀/R and decreases exponentially as stored charge depletes.

2. In a parallel capacitor circuit, which quantity is the same across each capacitor: $Q$ or $V$?

   Voltage $V$ is the same across all parallel capacitors. Parallel capacitors are connected across the same potential difference, equalizing voltage on each.

3. What is the equivalent capacitance relation for capacitors $C_1$ and $C_2$ in series?

   $\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}$. For series capacitors, reciprocals add since charges are equal and total voltage is the sum of individual voltages.

4. What is the current during capacitor charging in an RC circuit with battery $V$ and resistance $R$?

   $I(t)=\frac{V}{R}e^{-t/\tau}$. Charging current starts at maximum V/R and decays exponentially as the capacitor voltage rises.

5. If $V_{tot}=12\ \text{V}$ across series resistors $2\ \Omega$ and $4\ \Omega$, what is $V$ across $4\ \Omega$?

   $V=8\ \text{V}$. Voltage divides proportionally; the 4Ω resistor takes 4/6 of total voltage.

6. Find $C_{eq}$ for capacitors $2\ \mu\text{F}$ and $3\ \mu\text{F}$ in series.

   $C_{eq}=\frac{6}{5}\ \mu\text{F}$. For two series capacitors, equivalent capacitance is their product divided by their sum.

7. What is the total charge relation for parallel capacitors in terms of capacitor charges $Q_i$?

   $Q_{tot}=\sum Q_i$. With identical voltage, total charge on parallel capacitors sums the charges on each.

8. In a series resistor circuit, which quantity is the same through every resistor: $I$ or $V$?

   Current $I$ is the same through all series resistors. In a series circuit, current has only one path, remaining constant through each resistor by conservation of charge.

9. If two series capacitors are $2\ \mu\text{F}$ and $4\ \mu\text{F}$ with charge $Q$, what is $\frac{V_{2\mu F}}{V_{4\mu F}}$?

   $\frac{V_{2\mu F}}{V_{4\mu F}}=2$. With equal charge, voltage ratio is inverse to capacitance ratio, so $V_{2μF}$ is twice $V_{4μF}$.

10. In a parallel resistor circuit, which quantity is the same across each branch: $I$ or $V$?

    Voltage $V$ is the same across all parallel branches. Parallel branches connect across the same points, experiencing identical potential difference by definition.

11. Find $C_{eq}$ for capacitors $2\ \mu\text{F}$ and $3\ \mu\text{F}$ in parallel.

    $C_{eq}=5\ \mu\text{F}$. Parallel capacitances sum directly to yield the equivalent capacitance.

12. What is the voltage division relation for series resistors: $V_i$ in terms of $R_i$ and $V_{tot}$?

    $V_i=V_{tot}\frac{R_i}{\sum R}$. Voltage drops proportionally to each resistor's share of the total resistance in a series circuit.

13. Find $C_{eq}$ for two series capacitors $C$ and $C$ (identical capacitors).

    $C_{eq}=\frac{C}{2}$. Two identical series capacitors sum reciprocals, yielding half the individual capacitance.

14. What is the equivalent resistance for resistors $R_1$ and $R_2$ in series?

    $R_{eq}=R_1+R_2$. Resistances in series add directly as current flows through each sequentially, increasing total opposition to flow.

15. In a series capacitor circuit, which quantity is the same on each capacitor: $Q$ or $V$?

    Charge $Q$ is the same on all series capacitors. Series capacitors share the same current, resulting in equal charge accumulation on each plate.

16. What is the charge division relation for series capacitors: $V_i$ in terms of $C_i$ and $Q$?

    $V_i=\frac{Q}{C_i}$. With constant charge on series capacitors, voltage on each is inversely proportional to its capacitance.

17. Find $R_{eq}$ for resistors $2\ \Omega$ and $3\ \Omega$ in parallel.

    $R_{eq}=\frac{6}{5}\ \Omega$. For two parallel resistors, equivalent resistance is their product divided by their sum.

18. Identify the correct discharging equation for a capacitor: $V_C(t)$ starting from $V_0$.

    $V_C(t)=V_0e^{-t/\tau}$. During discharge, capacitor voltage decays exponentially from its initial value over time constant τ.

19. Find $R_{eq}$ for two parallel resistors $R$ and $R$ (identical resistors).

    $R_{eq}=\frac{R}{2}$. Two identical parallel resistors double the conductance, halving the equivalent resistance.

20. Identify the correct charging equation for a capacitor: $V_C(t)$ in an RC circuit with battery $V$.

    $V_C(t)=V\left(1-e^{-t/\tau}\right)$. Capacitor voltage approaches the battery voltage exponentially during charging in an RC circuit.

21. What is the RC time constant formula for a circuit with equivalent values $R_{eq}$ and $C_{eq}$?

    $\tau=R_{eq}C_{eq}$. The time constant in RC circuits is the product of equivalent resistance and capacitance, governing exponential behavior.

22. What is the equivalent resistance relation for resistors $R_1$ and $R_2$ in parallel?

    $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$. In parallel, conductances add, so the reciprocal of equivalent resistance equals the sum of reciprocals of individual resistances.

23. What is the equivalent capacitance for capacitors $C_1$ and $C_2$ in parallel?

    $C_{eq}=C_1+C_2$. Capacitances in parallel add because they share the same voltage, and total charge is the sum of individual charges.

24. What is the total current relation for parallel resistors in terms of branch currents $I_i$?

    $I_{tot}=\sum I_i$. By Kirchhoff's current law, total current entering a parallel junction equals the sum of branch currents.

25. Find $R_{eq}$ for resistors $2\ \Omega$ and $3\ \Omega$ in series.

    $R_{eq}=5\ \Omega$. Series resistances add directly to give the total equivalent resistance.

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