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Chapter 5 Quiz — Passive Components

Test your understanding of capacitors, inductors, energy storage, series and parallel combinations, mutual inductance, and signal fundamentals including RMS and decibels. Each question includes a detailed explanation.

Question 1

A 100 µF capacitor is charged to 20 V. How much energy is stored in it?

  1. 2 mJ
  2. 10 mJ
  3. 20 mJ
  4. 40 mJ
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Correct Answer: C

Using \(E = \frac{1}{2}CV^2 = \frac{1}{2} \times 100 \times 10^{-6} \times (20)^2 = \frac{1}{2} \times 100 \times 10^{-6} \times 400 = 0.020 \text{ J} = 20 \text{ mJ}\). Notice the voltage-squared relationship: if you double the voltage to 40 V, the stored energy quadruples to 80 mJ. This is why high-voltage capacitors (like those in camera flashes) store substantial energy.

Concept Tested: Capacitor Energy \(E = \frac{1}{2}CV^2\)

Question 2

Three capacitors — 10 µF, 20 µF, and 30 µF — are connected in series. What is the equivalent capacitance?

  1. 60 µF
  2. 10 µF
  3. 5.45 µF
  4. 1.83 µF
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Correct Answer: C

Series capacitors use the reciprocal formula: \(\frac{1}{C_T} = \frac{1}{10} + \frac{1}{20} + \frac{1}{30} = \frac{6}{60} + \frac{3}{60} + \frac{2}{60} = \frac{11}{60}\), so \(C_T = 60/11 \approx 5.45 \text{ µF}\). Series capacitors reduce the total capacitance (opposite of resistors!), and the result is always less than the smallest individual capacitor (10 µF).

Concept Tested: Capacitors in Series

Question 3

An inductor with \(L = 50 \text{ mH}\) carries a steady current of 400 mA. How much energy is stored in its magnetic field?

  1. 20 mJ
  2. 4 mJ
  3. 8 mJ
  4. 10 mJ
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Correct Answer: B

Using \(E = \frac{1}{2}LI^2 = \frac{1}{2} \times 0.05 \times (0.4)^2 = \frac{1}{2} \times 0.05 \times 0.16 = 0.004 \text{ J} = 4 \text{ mJ}\). The current-squared relationship for inductors mirrors the voltage-squared relationship for capacitors. When this inductor's current is suddenly interrupted, this 4 mJ must go somewhere — usually as a dangerous voltage spike, which is why flyback diodes are essential.

Concept Tested: Inductor Energy \(E = \frac{1}{2}LI^2\)

Question 4

The voltage across a 10 µF capacitor is changing at a rate of 500 V/s. What current flows through the capacitor?

  1. 50 µA
  2. 500 mA
  3. 5 mA
  4. 50 mA
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Correct Answer: C

Using the capacitor I-V relationship: \(I = C \frac{dV}{dt} = 10 \times 10^{-6} \times 500 = 0.005 \text{ A} = 5 \text{ mA}\). Current only flows through a capacitor when voltage is changing. If the voltage were constant (DC), no current would flow at all — this is why capacitors block DC signals.

Concept Tested: Capacitor I-V Relationship \(I = C \, dV/dt\)

Question 5

Two inductors, \(L_1 = 4 \text{ mH}\) and \(L_2 = 12 \text{ mH}\), are connected in parallel (with no mutual inductance). What is the equivalent inductance?

  1. 16 mH
  2. 8 mH
  3. 4 mH
  4. 3 mH
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Correct Answer: D

Inductors in parallel combine like resistors in parallel (unlike capacitors). Using the product-over-sum formula: \(L_{eq} = \frac{L_1 \cdot L_2}{L_1 + L_2} = \frac{4 \times 12}{4 + 12} = \frac{48}{16} = 3 \text{ mH}\). Remember: inductors follow resistor rules (series adds, parallel reciprocal), while capacitors follow the opposite.

Concept Tested: Inductors in Parallel

Question 6

A sinusoidal voltage has a peak value of 14.14 V. What is its RMS value?

  1. 14.14 V
  2. 20 V
  3. 10 V
  4. 7.07 V
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Correct Answer: C

For a sinusoid: \(V_{RMS} = V_{peak} / \sqrt{2} = 14.14 / 1.414 \approx 10 \text{ V}\). RMS is the "equivalent DC" voltage for power delivery — a 10 V RMS AC signal dissipates the same power in a resistor as a 10 V DC signal. The US wall outlet is 120 V RMS, meaning its peak is \(120 \times \sqrt{2} \approx 170 \text{ V}\).

Concept Tested: RMS Value \(V_{RMS} = V_{peak}/\sqrt{2}\)

Question 7

Two coupled inductors have \(L_1 = 9 \text{ mH}\) and \(L_2 = 4 \text{ mH}\) with a coupling coefficient \(k = 0.5\). What is their mutual inductance \(M\)?

  1. 6.5 mH
  2. 6 mH
  3. 3 mH
  4. 1.5 mH
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Correct Answer: C

Using the coupling coefficient formula: \(M = k\sqrt{L_1 \cdot L_2} = 0.5 \times \sqrt{9 \times 4} = 0.5 \times \sqrt{36} = 0.5 \times 6 = 3 \text{ mH}\). The coupling coefficient \(k\) ranges from 0 (no coupling) to 1 (perfect coupling). A coupling coefficient of 0.5 means the inductors share 50% of the maximum possible mutual flux linkage. This principle underlies transformer operation.

Concept Tested: Mutual Inductance \(M = k\sqrt{L_1 L_2}\)

Question 8

An audio amplifier stage has a voltage gain of +26 dB. By what factor does it multiply the voltage?

  1. 26 times
  2. 10 times
  3. 20 times
  4. 200 times
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Correct Answer: C

For voltage gain in dB: \(A_V = 10^{dB/20} = 10^{26/20} = 10^{1.3} \approx 20\). You can also use the additive shortcut: \(+20 \text{ dB} = 10\times\) and \(+6 \text{ dB} = 2\times\), so \(+26 \text{ dB} = 10 \times 2 = 20\times\). This is why decibels are used: cascaded gain stages simply add their dB values instead of multiplying their linear gains.

Concept Tested: Decibel Gain Conversion \(dB = 20\log_{10}(A_V)\)

Answers Summary

Question Answer Concept Tested
1 C Capacitor Energy \(E = \frac{1}{2}CV^2\)
2 C Capacitors in Series
3 B Inductor Energy \(E = \frac{1}{2}LI^2\)
4 C Capacitor I-V Relationship
5 D Inductors in Parallel
6 C RMS Value
7 C Mutual Inductance
8 C Decibel Gain Conversion