Electric Circuits — Hard Practice Quiz

A Physics cheat sheet for Electric Circuits — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Ohm's Law: Voltage \(V\) equals current \(I\) times resistance \(R\).

$$V = IR$$

Resistance: Resistance \(R\) of a wire depends on resistivity \(\rho\), length \(L\), and cross-sectional area \(A\).

$$R = \rho \frac{L}{A}$$

Electric Power: Power \(P\) dissipated in a resistor or delivered by a source.

$$P = IV = I^2 R = \frac{V^2}{R}$$

Series Resistors: Equivalent resistance \(R_S\) is the sum of individual resistances.

$$R_S = R_1 + R_2 + \dots$$

Parallel Resistors: Reciprocal of equivalent resistance \(R_P\) is sum of reciprocals of individual resistances.

$$\frac{1}{R_P} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$$

Terminal Voltage: Voltage across battery terminals equals EMF \(\mathcal{E}\) minus voltage drop across internal resistance \(r\).

$$V_{term} = \mathcal{E} - Ir$$

Series Capacitors: Reciprocal of equivalent capacitance \(C_S\) is sum of reciprocals of individual capacitances.

$$\frac{1}{C_S} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$$

Parallel Capacitors: Equivalent capacitance \(C_P\) is sum of individual capacitances.

$$C_P = C_1 + C_2 + \dots$$

RC Time Constant: Time constant \(\tau\) characterizes the charging/discharging rate of an RC circuit.

$$\tau = RC$$

Capacitor Charging: Charge \(q\) on a capacitor at time \(t\) as it charges towards equilibrium charge \(q_0\).

$$q = q_0 [1 - e^{-t/\tau}]$$

Capacitor Discharging: Charge \(q\) on a capacitor at time \(t\) as it discharges from initial charge \(q_0\).

$$q = q_0 e^{-t/\tau}$$

Practice quiz

  1. A cylindrical wire of length $L$ and diameter $D$ is made of a material with resistivity $\rho$. It is connected to a voltage source $V$. If the wire's length is doubled to $2L$ and its diameter is halved to $D/2$, how does the power dissipated in the wire change, assuming the voltage source $V$ remains constant?

    • Power decreases by a factor of $8$.
    • Power decreases by a factor of $4$.
    • Power increases by a factor of $2$.
    • Power remains unchanged.

    Answer: Power decreases by a factor of $8$.

  2. Two resistors, $R_1$ and $R_2$, are connected in series to a voltage source $V$. The total power dissipated is $P_S$. If these same two resistors are then connected in parallel to the same voltage source $V$, the total power dissipated is $P_P$. What is the ratio $\frac{P_P}{P_S}$ in terms of $R_1$ and $R_2$?

    • $\frac{(R_1 + R_2)^2}{R_1 R_2}$
    • $\frac{R_1 R_2}{(R_1 + R_2)^2}$
    • $\frac{R_1 + R_2}{R_1 R_2}$
    • $1$

    Answer: $\frac{(R_1 + R_2)^2}{R_1 R_2}$

  3. A battery with EMF $\mathcal{E}$ and internal resistance $r$ delivers power $P$ to an external resistor $R$. Which of the following expressions correctly relates $R$ to $\mathcal{E}$, $P$, and $r$?

    • $P = \frac{\mathcal{E}^2 R}{(R+r)^2}$
    • $P = \frac{\mathcal{E}^2 R}{R^2+r^2}$
    • $P = \frac{\mathcal{E} R}{R+r}$
    • $P = \frac{\mathcal{E}^2}{R+r}$

    Answer: $P = \frac{\mathcal{E}^2 R}{(R+r)^2}$

  4. An RC circuit consists of a capacitor $C$ and a resistor $R$. The capacitor is initially charged to a voltage $V_0$. At time $t=0$, it begins to discharge through the resistor. At what time $t$ will the voltage across the capacitor be exactly $1/e$ of its initial value $V_0$?

    • $\tau = RC$
    • $2\tau$
    • $\frac{\tau}{2}$
    • $0.5 RC$

    Answer: $\tau = RC$

  5. Two capacitors $C_1$ and $C_2$ are connected in series to a voltage source $V$. The total charge stored is $Q_S$. If these same two capacitors are then connected in parallel to the same voltage source $V$, the total charge stored is $Q_P$. What is the ratio $\frac{Q_P}{Q_S}$ in terms of $C_1$ and $C_2$?

    • $\frac{(C_1 + C_2)^2}{C_1 C_2}$
    • $\frac{C_1 C_2}{(C_1 + C_2)^2}$
    • $\frac{C_1 + C_2}{C_1 C_2}$
    • $1$

    Answer: $\frac{(C_1 + C_2)^2}{C_1 C_2}$

  6. A cylindrical wire of length $L$ and radius $r$ is made of a material with resistivity $\rho$. If a current $I$ flows uniformly through the wire, what is the magnitude of the electric field $E$ inside the wire?

    • $\frac{I \rho}{\pi r^2}$
    • $\frac{I \rho L}{\pi r^2}$
    • $\frac{I \rho}{L}$
    • $\frac{I \pi r^2}{\rho L}$

    Answer: $\frac{I \rho}{\pi r^2}$

  7. Two resistors $R_1$ and $R_2$ are connected in series to a voltage source. The power dissipated in $R_1$ is $P_{1S}$ and in $R_2$ is $P_{2S}$. If the same two resistors are then connected in parallel to the same voltage source, the power dissipated in $R_1$ is $P_{1P}$ and in $R_2$ is $P_{2P}$. Which of the following statements is true regarding the power dissipation?

    • In series, $\frac{P_{1S}}{P_{2S}} = \frac{R_1}{R_2}$; in parallel, $\frac{P_{1P}}{P_{2P}} = \frac{R_2}{R_1}$.
    • In series, $\frac{P_{1S}}{P_{2S}} = \frac{R_2}{R_1}$; in parallel, $\frac{P_{1P}}{P_{2P}} = \frac{R_1}{R_2}$.
    • In both series and parallel, $\frac{P_1}{P_2} = \frac{R_1}{R_2}$.
    • In both series and parallel, $\frac{P_1}{P_2} = \frac{R_2}{R_1}$.

    Answer: In series, $\frac{P_{1S}}{P_{2S}} = \frac{R_1}{R_2}$; in parallel, $\frac{P_{1P}}{P_{2P}} = \frac{R_2}{R_1}$.

  8. An uncharged capacitor $C$ is connected in series with a resistor $R$ and a battery with EMF $\mathcal{E}$. How long does it take for the voltage across the resistor, $V_R$, to drop to $1/e$ of its initial value?

    • $\tau = RC$
    • $2\tau$
    • $\frac{\tau}{e}$
    • $0.5 RC$

    Answer: $\tau = RC$

  9. A battery with EMF $\mathcal{E}$ and internal resistance $r$ is connected to an external resistor $R$. What value of $R$ maximizes the power delivered to the external resistor, and what is the maximum power delivered?

    • $R=r$, and $P_{max} = \frac{\mathcal{E}^2}{4r}$.
    • $R=r$, and $P_{max} = \frac{\mathcal{E}^2}{2r}$.
    • $R=0$, and $P_{max} = 0$.
    • $R \to \infty$, and $P_{max} = \frac{\mathcal{E}^2}{r}$.

    Answer: $R=r$, and $P_{max} = \frac{\mathcal{E}^2}{4r}$.

  10. Consider a circuit with three identical resistors, each with resistance $R$. Two of these resistors are connected in parallel, and this parallel combination is then connected in series with the third resistor. If the total voltage across the entire circuit is $V$, what is the voltage across the single resistor that is in series with the parallel combination?

    • $\frac{2V}{3}$
    • $\frac{V}{3}$
    • $\frac{V}{2}$
    • $V$

    Answer: $\frac{2V}{3}$

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