01 - Problems

# 01 - Problems - C of the black hole. (a) Use (3) and (1) to...

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PHYSICS 7B QUIZ 1 SECTION 104 CORY SCHILLACI This problem investigates the thermodynamics of black holes from a very simplistic per- spective. Our black hole will have only one important property: its mass M . We will need several facts. The ﬁrst is the highly nontrivial result due to Hawking which tells us that a black hole radiates as if it had a temperature (1) kT = ~ c 3 8 πGM . Next, note that the characteristic size of a black hole is (2) R = 2 MG c 2 , determined by requiring that the escape velocity is equal to the speed of light (an easy 7A calculation). Finally, one of the most famous equations in physics tells us that the energy of an object at rest is given by (3) E = mc 2 . With these facts, let’s get to work. You should assume that the ﬁrst law for a black hole reads dE = ¯ dq and that the black hole is at rest. 1) First we will explore the heat capacity
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Unformatted text preview: C of the black hole. (a) Use (3) and (1) to nd the heat capacity of the black hole as a function of M . (Hint: Its easier to nd 1 /C rst.) (b) Apply (1) again to nd the heat capacity as a function of T . Notice anything strange about this heat capacity? 2) Now lets explore the entropy of this black hole and discover something else thats weird about black holes. (a) Assume that the entropy starts at zero for a black hole with mass zero, and integrate using the equation of state (1) and the denition of dS to nd S ( M ), the entropy as a function of mass. It will again be helpful to use E = mc 2 . (b) Use (2) to reason that the entropy scales with the area of the black hole.For com-parison, how does the entropy of an ideal gas sale with size? 1...
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