Unformatted text preview: Physics 7c Final Exam (Muller) 21 May 1999 The following constants may be useful. They are all in MKS units:
nw = 1.33 nG = 1.5 e = 1.6x10‘19 coul rne = 9.1xlO'31 kg h = 6.7x10‘34 J sec 1. (10 pts) At what angle above the horizontal is the sun, if sunlight reﬂected from the surface
of a calm body of water is completely polarized? 2. (10 pts) Element X is radioactive. At t :1 hour, we have N1 atoms of element X present.
At t = 2 hours, we have N2 atoms. How many do we expect to ﬁnd at t = 5 hours? 3. (10 pts) Normally we "explain" alpha particle decay of a nucleus by assuming that the
alpha particle was bound inside the nucleus, but had positive energy and tunnelled out.
Now I would like you to consider the possibility that in beta decay, that the electron was
bound inside the nucleus prior to the decay, and it too tunneled out. Estimate the kinetic energy that the electron would have to have to be conﬁned inside
the nucleus. Assume that the volume of the nucleus is 10’39 cm3. Give your answer in
MeV and compare this to the typical electron kinetic energy found in beta decay. 4. (15 pts) John and Mary are twins. At age zero, Mary takes a trip to a star which is 5 light
years away. She travels at 0.6 times the speed of light. John leaves at the same time, but
travels at 0.8 times the speed of light. He arrives first, and waits for her to arrive. When
she does arrive, what are the ages of John and Mary? 5. This problem is about "resolving power" in microscopes. The numerical aperture NA of a
lens (optical or electron) is defined as the ratio f/D, where f is the focal length, and D is
the diameter. You may assume that the minimum possible value of NA is 1/2. (a) (8 pts) In an optical microscope, the objective lens is defined to be the small lens that is
close to the object. Let (1 = the distance between two small objects that can just barely
be resolved. Then I can show that d = x 7t NA, where x is a number. Find x. (b) (7 pts) An electron microscope accelerates the electron to an energy of 50 keV (1 keV
= 1000 eV). Estimate (get within a factor of 10) the size of the closest pair of objects
that can be resolved using an electron microscope with this energy. 6. (10 pts) An automobile is moving at a constant velocity v = 10 meters/sec. The wheels of
the automobile have radius R. A tack is stuck on the circumference of one wheel. If you
sketch the path of the tack with respect to the road, you will ﬁnd that it moves in a
complicated path known as a "cycloid." Let the acceleration of the tack, at any given
moment, be the vector a(t). The magnitude of this is la(t)|. What is (a) the maximum value for la(t)| as the car moves forward at constant velocity v?
(b) the minimum value for la(t)| as the car moves forward at constant velocity v?
Hint: this is a 7c problem. And you do not have to know anything about cycloids to solve it. 7. (10 pts) A pion is believed NOT to satisfy the Schrodinger equation. Instead, it satisfies the
82‘? h2 (92‘? _ 2 2 3x2 ' Z? 3:2 " m“ c ‘1’ Assuming that this equation is true, calculate the quantum mechanical phase velocity vp
for the pion in terms of its classical velocity vc. "Klein Gordon equation": 712 ...
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