Physics CS 33	Set # 7
Spring 2006
Due date	Wed. May 24th
Read HR&K:	None
Read K&K:	Chapter 13 Chapter 14
HR&K Problems:	None
K&K Problems:	Chapter 13  Problem  13.10,   13.11 Chapter 14  Problem  14.1,   14.2,   14.4
1.	Show by direct substitution that  E2 = p2c2 + m02c4  is consistent with  E = γm0c2 and  p = γm0u .
2.	Stars form when large clouds of matter condense due to their own gravitation.  Assume a cloud begins with all particles infinitely far away from each other (i.e. gravitational PE = 0).  A star of mass M and radius R (assume uniform density) then forms due to gravitational attraction.  Use Newtonian gravity in this problem. a) Calculate the drop in PE of the system caused by forming the star.  Hint: If all the mass condensed to one point, this would cause an infinite drop in PE.  Of course, the mass does not do this because pressure eventually resists the gravitational collapse.  Think of building the star one layer at a time. b) Calculate the answer to part a), using the solar values of M and R. c) The total power output (in radiation) from the sun is about  3.8 x 1026 Watts (known as the "solar luminosity").  Assuming this is constant throughout the life of the sun, what would be the lifetime (in years) of the sun if gravity were its only energy source? d) The answer to part c) is significantly less than the known age of the solar system, so the sun must have another energy source: nuclear fusion!  What fraction of the sun's total mass does it lose each year by radiating away energy?
3.	In this problem we will estimate how much of the sun's internal energy (what we commonly refer to as its "mass") is actually in the form of matter (as opposed to KE or photons).  Please look up and plug in actual numbers. a) Begin with the naive assumption that the sun's "mass" which exerts gravity on the Earth is entirely made up of an equal number of protons and electrons (no KE or photons).  By simple division, estimate the number of particles (protons plus electrons) in the sun.  In this view, what is the total "mass energy" of the sun? b) If the average temperature inside the sun is 106 Kelvin, estimate the total KE of protons and electrons in the sun, treating them as an ideal monatomic gas. c) To estimate the energy of photons, use the solar luminosity  3.8 x 1026 Watts  to estimate how much energy leaves the surface of the sun in 10-8 sec.  Now imagine that all this energy fills a thin spherical shell just outside the surface of the sun.  What is the energy density (energy per unit volume) inside this shell?  Hint: With what speed do the photons move outward? d) Assuming the average energy density of photons inside the sun is 10 times what you found in part c), estimate the total energy of photons in the sun. e) Compare your answers to parts a), b), and d).   Is most of the sun's "energy" actually matter?
4.	A high-energy photon has a glancing collision with an electron inside a material (the Compton Scattering experiment).  Solve for f , the frequency of the photon after the collision, in terms of f0 , the frequency of the incoming photon, and θ, the scattering angle of the photon.
5.	In the photoelectric effect, an electron inside a metal completely absorbs a photon's energy.  In this problem, we will see if "free" electrons, i.e. not bound to any nuclei, can exhibit this effect or not. a) A photon of frequency f approaches an electron.  Work in the reference frame where the electron begins at rest.  Assuming the electron completely absorbs the photon, write out the total 4-momentum of the system before and after the collision. b) From here, it will save some writing to introduce the variable x = ( hf0 / m0c2 ).  Write the equations which describe conservation of 4-momentum in terms of x. c) At first glance, these equations appear perfectly solvable.  Upon further inspection, show that this set of equations leads to a contradiction when solving for ( v / c ) of the electron after the collision.  Thus this process does not occur. d) Qualitatively describe how we could have reasoned this out (without calculations) by looking at the problem in the "center-of-momentum" frame.