PHYS212 : Homework


Homework Assignment 8 (due 4/6)

Doppler Shift of Sound, Polarization, Energy, and Intensity of Electromagnetic Waves

\includegraphics{train.eps}
1.
A locomotive and a car approach an intersection with perpendicular velocities as shown in the diagram. The locomotive and the car travel with constant speeds of 30 m/s and 25 m/s, respectively. At time $t=0$, the car is 100 m to the south of the intersection, and the locomotive is 50 m to the west of the intersection. The locomotive blows its whistle. What is the Doppler correction factor $\frac{f'}{f}$ for the observer in the car (a) at $t=0$, (b) as the car crosses the intersection? (c) The sound detected by the observer in the car at $t=0$ was actually emitted at an earlier time, when the locomotive was some distance greater than 50 m from the intersection. Hence, 50 m is not the source position you should have used in part (a) to determine the source and observer angles. Determine how much of a difference correcting for this makes in your result for part (a). Is this correction important in this situation?

\scalebox{0.5}{
\includegraphics{mach.eps}
}
2.
An object traveling faster than the speed of sound in air generates a conical surface of wave fronts interfering constructively, called a shock wave, illustrated in the diagram. (a) Find an expression for $\theta$, the angle the shock wave surface makes with the source velocity $\vec{v}_s$, called the (Mach angle). Hint: the velocities of the wave fronts along the shock wave are perpendicular to its conical surface. (b) The air speed record is 979 m/s. If a plane travels at this speed at an altitude of 10 km, roughly how long after the plane passes directly overhead do observers on the ground hear the sonic boom? Ignore variation in the speed of sound with altitude, and assume the STP sound velocity of 331 m/s.

Note: These results are not only approximate due to the variation of the speed of sound with altitude. We also neglect the fact that the large amplitude of the shock wave violates the acoustic approximation.

3.
A 10.0 cm linear antenna connected to an oscilloscope is used to pick up an oscillating voltage signal induced by the electric field of a radio wave. Adjustment of the orientation of the antenna reveals that a maximum signal of 10.0 mV is measured when the antenna is oriented along $\hat{k}$.

(a) What are the amplitudes $E_o$ and $B_o$ of the electric and magnetic fields of the wave?

(b) What can we say about the directions of $\vec{B}$ and $\vec{k}$ on the basis of the above information?

(c) What is the intensity of the wave?

(d) The radio station has a 1 MW transmitter. Assuming the radio station emits its signal isotropically, and ignoring the effects of obstructions, how far from the station is the pickup antenna? (A real transmitter with a linear antenna actually emits maximum power in the plane perpendicular to it.)

4.
The average electromagnetic power flux from the sun incident on the upper atmosphere of the earth is about 1.3 kW/m$^2$.

(a) If the average earth-sun distance is $1.50\mathrm{e}{11}$ m, and the sun emits a uniform angular distribution of electromagnetic radiation, what is its electromagnetic power output?

(b) We have a 1000 kg spaceship which can unfurl a sail which reflects about 50% of solar radiation and transmits the rest. Reflection reverses the momentum of the incident light and hence yields twice the light pressure. The ship is equipped with conventional thrusters to ensure that its sail always faces the sun. What area must the sail have in order to balance the gravitational force exerted on it by the sun? (Canceling the sun's gravitational force would give the ship a linear trajectory tangent to its initial orbit around the sun.)

Hints: Both forces depend on the inverse square of the distance from the sun, so the answer does not depend on position.

5.
A quarter wave plate is an optical element which introduces a phase difference of $\frac{\pi}{2}$ (one quarter wave) between perpendicular components of incident electromagnetic waves. One way to accomplish this is to use a material which exhibits double refraction, meaning that it has different indices of refraction $n_\parallel$ and $n_\perp$ for light with its electric field oriented along two perpendicular axes. Show that the thickness $d$ of a quarter wave plate made of such a material must satisfy the equation

\begin{displaymath}
d = \frac{m\lambda}{4} \frac{1}{\vert n_\parallel - n_\perp
\vert} \hspace{1cm} (m = 1,3,5,...)
\end{displaymath}

A linear polarizer and a quarter wave plate can be used to produce elliptically polarized light.


Copyright © 2003-2009, Lewis A. Riley Updated Tue Apr 14 02:04:45 2009