Alternating Current Circuits — Hard Practice Quiz

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

Formulas & key concepts

Capacitive Reactance: Opposition to current flow by a capacitor \(C\) at frequency \(f\).

$$X_C = \frac{1}{2 \pi f C}$$

Inductive Reactance: Opposition to current flow by an inductor \(L\) at frequency \(f\).

$$X_L = 2 \pi f L$$

Impedance (Series RLC): Total opposition to current flow in a series RLC circuit.

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

Ohm's Law for AC: Relationship between RMS voltage, RMS current, and impedance.

$$V_{rms} = I_{rms} Z$$

Phase Angle: Angle \(\phi\) by which voltage leads (positive) or lags (negative) current.

$$\tan \phi = \frac{X_L - X_C}{R}$$

Average Power: Power consumed by the circuit. \(\cos \phi\) is the power factor.

$$P = I_{rms} V_{rms} \cos \phi$$

Resonant Frequency: Frequency at which inductive and capacitive reactances are equal ($X_L = X_C$).

$$f_0 = \frac{1}{2 \pi \sqrt{LC}}$$

Practice quiz

  1. A series RLC circuit is designed to resonate at a frequency $f_0$. If the inductance $L$ is quadrupled and the capacitance $C$ is simultaneously reduced to one-fourth of its original value, what is the new resonant frequency $f_0'$?

    • $f_0$
    • $2f_0$
    • $f_0/2$
    • $4f_0$

    Answer: $f_0$

  2. For a series RLC circuit operating at its resonant frequency $f_0$, what is the power factor $\cos \phi$?

    • $0$
    • $0.5$
    • $1$
    • Depends on the value of $R$.

    Answer: $1$

  3. A series RLC circuit has a resistance $R = 50 \text{ \Omega}$, an inductance $L = 0.3 \text{ H}$, and a capacitance $C = 50 \text{ \mu F}$. If the circuit is connected to an AC source with $V_{rms} = 120 \text{ V}$ at a frequency $f = 60 \text{ Hz}$, what is the RMS current $I_{rms}$ flowing through the circuit?

    • $1.25 \text{ A}$
    • $1.54 \text{ A}$
    • $1.80 \text{ A}$
    • $2.40 \text{ A}$

    Answer: $1.54 \text{ A}$

  4. A series RLC circuit has $L = 0.8 \text{ H}$ and $C = 2 \text{ \mu F}$. At what frequency $f$ will the circuit behave purely resistively?

    • $79.6 \text{ Hz}$
    • $125.8 \text{ Hz}$
    • $159.2 \text{ Hz}$
    • $251.6 \text{ Hz}$

    Answer: $125.8 \text{ Hz}$

  5. A series RLC circuit is initially operating at its resonant frequency $f_0$, where $X_L = X_C = 100 \text{ \Omega}$ and $R = 100 \text{ \Omega}$. If the frequency is then doubled to $2f_0$ while $V_{rms}$ remains constant, how does the average power $P$ consumed by the circuit change?

    • It increases to $2P_0$.
    • It remains the same.
    • It decreases to approximately $0.31 P_0$.
    • It decreases to approximately $0.5 P_0$.

    Answer: It decreases to approximately $0.31 P_0$.

  6. An AC circuit contains only a resistor $R = 60 \text{ \Omega}$ and an inductor $L$ in series. If the RMS voltage across the circuit is $V_{rms} = 150 \text{ V}$ and the RMS current is $I_{rms} = 2 \text{ A}$ at a frequency $f = 50 \text{ Hz}$, what is the inductance $L$?

    • $0.095 \text{ H}$
    • $0.143 \text{ H}$
    • $0.191 \text{ H}$
    • $0.239 \text{ H}$

    Answer: $0.143 \text{ H}$

  7. In a series RLC circuit, if the resistance $R$ is exactly equal to the magnitude of the net reactance $|X_L - X_C|$, what is the magnitude of the phase angle $\phi$ between the voltage and current?

    • $0$
    • $\pi/6$
    • $\pi/4$
    • $\pi/2$

    Answer: $\pi/4$

  8. A series RLC circuit consists of a resistor $R = 100 \text{ \Omega}$, an inductor $L = 0.4 \text{ H}$, and a capacitor $C = 10 \text{ \mu F}$. At what frequency will the RMS current through the circuit be maximum for a constant RMS voltage source?

    • $50.0 \text{ Hz}$
    • $60.0 \text{ Hz}$
    • $79.6 \text{ Hz}$
    • $100.0 \text{ Hz}$

    Answer: $79.6 \text{ Hz}$

  9. In a series RLC circuit, the phase angle $\phi$ between the RMS voltage and RMS current is measured to be $60^\circ$. If the resistance $R = 75 \text{ \Omega}$, what is the magnitude of the net reactance $|X_L - X_C|$?

    • $43.3 \text{ \Omega}$
    • $75.0 \text{ \Omega}$
    • $129.9 \text{ \Omega}$
    • $150.0 \text{ \Omega}$

    Answer: $129.9 \text{ \Omega}$

  10. An AC circuit draws an RMS current of $I_{rms} = 4 \text{ A}$ from an RMS voltage source of $V_{rms} = 200 \text{ V}$. If the average power consumed by the circuit is $P = 400 \text{ W}$, what are the impedance $Z$ and the power factor $\cos \phi$ of the circuit?

    • $Z = 50 \text{ \Omega}$, $\cos \phi = 0.5$
    • $Z = 50 \text{ \Omega}$, $\cos \phi = 0.8$
    • $Z = 80 \text{ \Omega}$, $\cos \phi = 0.5$
    • $Z = 80 \text{ \Omega}$, $\cos \phi = 0.8$

    Answer: $Z = 50 \text{ \Omega}$, $\cos \phi = 0.5$

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