Question 1

1A) A loud speaker emits sound at a frequency of $200~{\rm Hz}$ uniformly into space. A person traveling with a speed of $20~{\rm m/s}$ carrying the loud speaker moves directly toward a vertical wall that reflects the sound. Assume that there are no reflections from the ground and that there are no other reflecting surfaces behind the source. Take the speed of sound to be $340~{\rm m/s}$.

i)[6 pts] Find the frequency of the sound as heard by a climber on the vertical wall.

We have a stationary observer and a moving source. The doppler shifted frequency heard by the climber on the wall is

$$\nu_1~=~\nu_0 \left({c\over c-v}\right)~=~200\ {340\over 320}~=~212.5~{\rm Hz}$$

ii)[8 pts] What is the beat frequency heard by the person carrying the loud speaker?

The frequency calculated above is the frequency of the sound reflected by the wall. We now have the situation of a stationary source and a moving observer, for which we know that

$$\nu_2~=~\nu_1 \left({c+v\over c}\right)~=~212.5 {360\over 340}~=~225.0~{\rm Hz}$$

This means that the beat frequency heard by the person holding the source, moving toward the wall is

$$\nu_{\rm beat}~=~\vert\nu_2-\nu_0\vert~=~25~{\rm Hz}$$

1B) A light bulb produces $200~{\rm J}$ of energy in monochromatic light of wavelength $600~{\rm nm}$in a time interval of $2~{\rm s}$. The light is emitted uniformly into space. The bulb is at the center of spherical shell of radius $R~=~1~{\rm m}$that absorbs all of the light.

i)[5 pts] What is the intensity of light at the spherical shell?

The power emitted in monochromatic light is

$$P~=~{E\over t}~=~{200\over 2}~=~100~{\rm W}$$

The intensity at the spherical shell is

$$I~=~{P\over A}~=~{100\over 4\pi r^2}~=~{25\over\pi}~=~7.96~{\rm W/m^2}$$

ii)[8 pts] What is the radiation pressure on the spherical shell?

Pressure is defined to be force per unit area, hence

$$P~=~{\vert{\bf F}\vert\over A}~=~{I\over c}~=~{25\over\pi c}~=~2.65\times10^{-8}~{\rm N/m^2}$$

iii)[6 pts] What is the maximum value of the electric and magnetic fields associated with this electromagnetic radiation on the spherical shell?

$$I~=~\vert{\bf\overline{S}}\vert~=~{1\over\mu_0 c} \vert{\bf E_{\rm RMS}}\vert^2$$

from which we obtain

$$\vert{\bf E_{\rm RMS}}\vert~=~\sqrt{I\mu_0 c}~=~54.8~{\rm V/m}$$

which gives

$$\vert{\bf E_{\rm max}}\vert~=~\sqrt{2}\vert{\bf E_{\rm RMS}}\vert~=~\sqrt{I\mu_0 c}~=~77.4~{\rm V/m}$$

and

$$\vert{\bf B_{\rm max}}\vert~=~{ \vert{\bf E_{\rm max}}\vert\over c}~=~2.58\times10^{-7}~{\rm T}$$