Homework X
due Tuesday, April 7, 2009 Chaps. 38-39
1.) As usual, please supply a proposed test problem with solution on the material in these two chapters.
2.) This is a waveguide problem. First, a short tutorial (cribbed from the web at
)
a.) the idea is to transmit/transfer electromagnetic radiation with minimal losses. So, there is some power involved, and the losses are due to ac losses in the conductor. A waveguide has fewer losses than a coaxial cable (see pictures) because
<- coaxial transmission line, with center conductor (the small black circle in the middle) and outer conductor
<- various waveguide shapes. Here, the losses are less since the losses of the power being transmitted are spread over just the outer conductor, so the current density (Power/area) is smaller than when ALL the current/power has to go down the small center conductor in a coax, which makes the current density high.
There are other reasons why a waveguide is better than coax (including skin depth and dielectric losses – see tutorial at the web site if you’re interested (and if their web site is working – not always the case.)
So, to our problem:
a.) Assume we have essentially a standing wave of our electromagnetic radiation in a rectangular wave guide (easy to visualize.) This is like getting a resonance in a wood instrument or an organ pipe. Since we want the power dissipated (=i2R) in the walls of the wave guide to be a minimum, what boundary condition for the electric field parallel to the boundary wall(s) do we want? (hint: big field at x=0 and x=width, and y=0 and y=height, or small field at those locations?)
b.) What is the maximum (beware factors of 2) wavelength, λ, in the x-direction that will match this boundary condition, if width=10 cm? (obviously in the perpendicular direction, we get another boundary condition for the perpendicular component of the E field.)
c.) what is the frequency of this radiation? (This essentially size-related limitation for waveguides limits their useful frequency range.)
3.) Using solar light ‘pressure’ to “sail” a spacecraft around the solar system:
a.) What is the total solar light force (units of Newtons) on a 3 km x 3 km light sail (assume total reflection) at the orbit of the earth, where the solar intensity is 1.4 kW/m2?
b.) Assuming the sail is made out of material 1 mm thick, density = 2 g/cm3. What does the sail mass? (no payload, just the sail material.)
c.) what is the maximum (radial to the sun in direction – no ‘tacking’) acceleration of the sail at Earth orbit? (use 3 sig figs and scientific notation)
d.)At what radius from the sun (distance of earth from sun is 1.5 108 km) would there be a radial acceleration of the light sail of 0.1 m/s2 ? (Mercury is 5.8 107 km from the Sun)
4.)According to that fount of all knowledge (Wikipedia), at low light levels the relative sensitivity (see green curve) of the human eye is different that at normal light levels (the black, right hand curve), see figure, where the x-axis is wavelength in nm and the y-axis is human eye sensitivity:
The energy of an electromagnetic wave, E, = hf, where h is Planck’s constant = 6.63 10-32 Js and f is the frequency in Hz. What is the range of energy that a human eye can ‘see’, i. e. what is E=400 nm– E=700 nm ?
5.)Using Snell’s law, if light in air impinges on a glass rectangle with an angle of 45o from the vertical or ‘normal’, what is the angle between the light in the glass from the normal if nglass=1.5? (See picture, credit Uni. Missouri, where the black region is the air and the blue is the glass)
6.)Using the data above in #5, at what angle from the normal for light coming from within the glass would there be total internal reflection?
7.) A galaxy is receding from the earth at v=0.6 c. If yellow light from a Sol-type star, =550 nm, is emitted in this receding galaxy, what color (wavelength) does an observer on Earth see?