Electromagnetic Waves — Hard Practice Quiz

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

Formulas & key concepts

Speed of Light: Relationship between speed \(c\), frequency \(f\), and wavelength \(\lambda\). \(c \approx 3.00 \times 10^8 m/s\).

$$c = f \lambda$$

Speed of Light (Maxwell): Speed of light related to permittivity \(\epsilon_0\) and permeability \(\mu_0\) of free space.

$$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$

Electric and Magnetic Fields: Relationship between magnitudes of electric field \(E\) and magnetic field \(B\) in an EM wave.

$$E = cB$$

Total Energy Density: Sum of electric and magnetic energy densities. \(u_{avg} = \epsilon_0 E_{rms}^2\).

$$u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2 \mu_0} B^2$$

Intensity: Average power per unit area carried by the wave. \(S = c \epsilon_0 E_{rms}^2\).

$$S = c u_{avg}$$

Doppler Effect (EM Waves): Observed frequency \(f_o\) for source frequency \(f_s\) with relative speed \(v_{rel}\). (+ for approaching, - for receding).

$$f_o = f_s (1 \pm \frac{v_{rel}}{c})$$

Malus' Law: Intensity \(S\) of polarized light after passing through an analyzer at angle \(\theta\) to polarization direction.

$$S = S_0 \cos^2 \theta$$

Practice quiz

  1. An electromagnetic wave has its frequency doubled. If its average intensity remains constant, how does its peak magnetic field amplitude ($B_{peak}$) change?

    • It doubles.
    • It halves.
    • It remains constant.
    • It quadruples.

    Answer: It remains constant.

  2. Consider a hypothetical scenario where the permittivity of free space $\epsilon_0$ is doubled, while the permeability of free space $\mu_0$ remains unchanged. If an electromagnetic wave maintains a constant peak magnetic field amplitude ($B_{peak}$) and constant frequency ($f$), how would its peak electric field amplitude ($E_{peak}$) and wavelength ($\lambda$) be affected?

    • $E_{peak}$ decreases by a factor of $\sqrt{2}$, $\lambda$ decreases by a factor of $\sqrt{2}$.
    • $E_{peak}$ increases by a factor of $\sqrt{2}$, $\lambda$ increases by a factor of $\sqrt{2}$.
    • $E_{peak}$ remains constant, $\lambda$ decreases by a factor of $\sqrt{2}$.
    • $E_{peak}$ decreases by a factor of $2$, $\lambda$ decreases by a factor of $2$.

    Answer: $E_{peak}$ decreases by a factor of $\sqrt{2}$, $\lambda$ decreases by a factor of $\sqrt{2}$.

  3. An electromagnetic wave has its peak electric field amplitude ($E_{peak}$) doubled. How does its average intensity ($S$) change?

    • It doubles.
    • It halves.
    • It remains constant.
    • It quadruples.

    Answer: It quadruples.

  4. A spaceship is receding from Earth at a speed of $0.2c$. If it emits a radio signal with a wavelength of $20 \text{ m}$, what is the observed wavelength on Earth?

    • $16 \text{ m}$
    • $20 \text{ m}$
    • $25 \text{ m}$
    • $30 \text{ m}$

    Answer: $25 \text{ m}$

  5. Unpolarized light of initial intensity $S_{unpol}$ passes through a polarizer and then an analyzer. If the average energy density of the light after the analyzer is $u_{final}$, and the angle between the transmission axes of the polarizer and analyzer is $30^\circ$, what was the initial intensity $S_{unpol}$?

    • $\frac{4}{3} c u_{final}$
    • $\frac{8}{3} c u_{final}$
    • $\frac{3}{4} c u_{final}$
    • $\frac{3}{8} c u_{final}$

    Answer: $\frac{8}{3} c u_{final}$

  6. An electromagnetic wave has an average energy density of $u_{avg}$. What is the peak magnetic field amplitude ($B_{peak}$) in terms of $u_{avg}$, $\epsilon_0$, and $\mu_0$?

    • $\sqrt{\frac{u_{avg}}{\epsilon_0}}$
    • $\sqrt{2 u_{avg} \mu_0}$
    • $\frac{1}{c} \sqrt{\frac{u_{avg}}{\epsilon_0}}$
    • $c \sqrt{2 u_{avg} \mu_0}$

    Answer: $\sqrt{2 u_{avg} \mu_0}$

  7. A source of electromagnetic waves is moving towards an observer. If the observed frequency is $1.5$ times the source frequency, what is the ratio of the observed wavelength to the source wavelength ($\lambda_o / \lambda_s$)?

    • $1.5$
    • $0.5$
    • $2/3$
    • $3/2$

    Answer: $2/3$

  8. An electromagnetic wave has a peak electric field amplitude of $E_{peak}$. If the wave's intensity is $S$, what is the peak magnetic field amplitude ($B_{peak}$) in terms of $S$, $E_{peak}$, and fundamental constants?

    • $\frac{2S}{\epsilon_0 E_{peak}^2}$
    • $\frac{\epsilon_0 E_{peak}^3}{2S}$
    • $\frac{S}{c E_{peak}}$
    • $\frac{E_{peak}}{c}$

    Answer: $\frac{\epsilon_0 E_{peak}^3}{2S}$

  9. If the permeability of free space $\mu_0$ were to decrease, how would the speed of light $c$ and the ratio of electric to magnetic field amplitudes ($E/B$) in an electromagnetic wave be affected? Assume $\epsilon_0$ remains constant.

    • $c$ increases, $E/B$ increases.
    • $c$ decreases, $E/B$ decreases.
    • $c$ increases, $E/B$ remains constant.
    • $c$ remains constant, $E/B$ increases.

    Answer: $c$ increases, $E/B$ increases.

  10. An unpolarized light beam with initial intensity $S_{initial}$ passes through a polarizer. The transmitted light then passes through an analyzer whose transmission axis is oriented at an angle $\theta$ relative to the polarizer's transmission axis. If the final average energy density of the light is $u_{final}$, what is the relationship between $S_{initial}$ and $u_{final}$?

    • $S_{initial} = 2 c u_{final} \cos^2 \theta$
    • $S_{initial} = \frac{c u_{final}}{2 \cos^2 \theta}$
    • $S_{initial} = \frac{2 c u_{final}}{\cos^2 \theta}$
    • $S_{initial} = \frac{c u_{final}}{\cos^2 \theta}$

    Answer: $S_{initial} = \frac{2 c u_{final}}{\cos^2 \theta}$

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