Chapter 32 — Electromagnetic Waves

14 points · 3 extra credit

Chapter 32 — Electromagnetic Waves

Core: 14 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (1 pt •) Starting from the free-space result $c = 1/\sqrt{\mu_0\varepsilon_0}$, evaluate the speed of light using $\mu_0 = 4\pi\times10^{-7}\,\text{T}\cdot\text{m/A}$ and $\varepsilon_0 = 8.85\times10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2)$, and confirm it agrees with $3.00\times10^{8}\,\text{m/s}$.

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2. 2. (2 pts ••) A sinusoidal plane electromagnetic wave travels through vacuum in the $+x$ direction. Its magnetic field has amplitude $B_0 = 1.5\times10^{-8}\,\text{T}$ and points along $+z$ at the instant the electric field is measured at a certain point.

   (a) Find the amplitude $E_0$ of the electric field.

   (b) In which direction does $\vec{E}$ point at that instant, so that $\vec{E}\times\vec{B}$ gives the propagation direction?

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3. 3. (2 pts ••) At a point in space, a plane wave in vacuum has an electric-field amplitude $E_0 = 60\,\text{V/m}$.

   (a) Find the peak energy density $u_E$ stored in the electric field.

   (b) Find the peak energy density $u_B$ stored in the magnetic field, and comment on how it compares to $u_E$.

   (c) Find the total time-averaged energy density of the wave.

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4. 4. (2 pts ••) A plane electromagnetic wave in vacuum has electric-field amplitude $E_0 = 1000\,\text{V/m}$.

   (a) Find the time-averaged intensity (average magnitude of the Poynting vector), $I = \tfrac12\varepsilon_0 c E_0^2$.

   (b) Find the amplitude $B_0$ of the magnetic field.

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5. 5. (2 pts ••) Just above Earth's atmosphere, sunlight arrives with an average intensity of $I = 1.4\times10^{3}\,\text{W/m}^2$. Treat it as a single plane wave.

   (a) Find the amplitude $E_0$ of the electric field.

   (b) Find the amplitude $B_0$ of the magnetic field.

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6. 6. (2 pts ••) A laser pointer emits a beam of power $P = 5.0\,\text{mW}$ in a circular spot of radius $0.50\,\text{mm}$.

   (a) Find the intensity of the beam.

   (b) Find the radiation pressure the beam exerts on a surface that completely absorbs it.

   (c) Find the total force the beam exerts if the surface instead perfectly reflects it straight back.

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7. 7. (2 pts ••) A flat solar sail of area $A = 10\,\text{m}^2$ is oriented face-on to sunlight of intensity $I = 1.4\times10^{3}\,\text{W/m}^2$ and perfectly reflects it.

   (a) Find the radiation force on the sail.

   (b) Find the total momentum the sunlight delivers to the sail in one hour.

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8. 8. (1 pt •) A microwave oven sets up a standing electromagnetic wave between its walls. Measurements of where food heats fastest show that adjacent "hot spots" (antinodes of intensity) are $6.1\,\text{cm}$ apart. Using the fact that adjacent antinodes are half a wavelength apart, find the wavelength and the operating frequency of the oven.

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9. Extra Credit (3 pts •••)

   Consider a linearly polarized plane wave traveling in the $+x$ direction in vacuum, with

   $$\vec{E}(x,t) = E_0\cos(kx-\omega t)\,\hat{\jmath}, \qquad \vec{B}(x,t) = B_0\cos(kx-\omega t)\,\hat{k}.$$

   (a) Write the instantaneous Poynting vector $\vec{S} = \dfrac{1}{\mu_0}\,\vec{E}\times\vec{B}$ and show it points along $+x$.

   (b) By integrating over one full period $T = 2\pi/\omega$, evaluate the time average $\langle S\rangle = \dfrac{1}{T}\displaystyle\int_0^T S\,dt$, and use $B_0 = E_0/c$ together with $c = 1/\sqrt{\mu_0\varepsilon_0}$ to reduce your result to $I = \tfrac12\varepsilon_0 c E_0^2$.

   (c) A perfectly absorbing surface of area $A$ is placed face-on to this wave. Argue from the fact that electromagnetic energy $U$ carries momentum $p = U/c$ that the time-averaged radiation pressure on the surface is $\langle p_\text{rad}\rangle = I/c$, and explain why a perfect reflector feels twice this pressure.

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