In this problem we will deal with simple thermodynamics model of a black hole. Assume that the properties of a black hole depend only on the mass of the black hole. (a) One characteristic of a black hole is the area of its event horizon. Roughly speaking, the event horizon is the boundary of the black hole. Inside this boundary, the gravity is so strong that even light cannot emerge from the region enclosed by the boundary. We shall write A = G°c®m°. Use dimensional analysis to find a, ß , and y. (b) One quantities in thermodynamics is entropy (S), that measure the disorder of a system. The relationship between entropy S, energy E and temperature 0 of a system is given as follow: = . For black holes, Bekenstein propose that the entropy is proportional the area of the black hole's event horizon S = nA, where n depends on c, k, G and h. Using dimensional analysis determine 7. (c) With the previous results, determine the temperature of the black hole, as a function of the mass. The energy of a black hole is E = mc². (d) The Stefan Boltzmann constant o is a can be written as k, = ah"®G*k{. Take a dimensionless constant a as one. Determine a, B, y and 8. (e) Assuming black hole is perfect blackbody , determine the rate of blackhole evapo- ration . dm (f) Determine how long for a black hole to evaporate completely if the initial mass is М.
In this problem we will deal with simple thermodynamics model of a black hole. Assume that the properties of a black hole depend only on the mass of the black hole. (a) One characteristic of a black hole is the area of its event horizon. Roughly speaking, the event horizon is the boundary of the black hole. Inside this boundary, the gravity is so strong that even light cannot emerge from the region enclosed by the boundary. We shall write A = G°c®m°. Use dimensional analysis to find a, ß , and y. (b) One quantities in thermodynamics is entropy (S), that measure the disorder of a system. The relationship between entropy S, energy E and temperature 0 of a system is given as follow: = . For black holes, Bekenstein propose that the entropy is proportional the area of the black hole's event horizon S = nA, where n depends on c, k, G and h. Using dimensional analysis determine 7. (c) With the previous results, determine the temperature of the black hole, as a function of the mass. The energy of a black hole is E = mc². (d) The Stefan Boltzmann constant o is a can be written as k, = ah"®G*k{. Take a dimensionless constant a as one. Determine a, B, y and 8. (e) Assuming black hole is perfect blackbody , determine the rate of blackhole evapo- ration . dm (f) Determine how long for a black hole to evaporate completely if the initial mass is М.
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:In this problem we will deal with simple thermodynamics model of a black hole. Assume
that the properties of a black hole depend only on the mass of the black hole.
(a) One characteristic of a black hole is the area of its event horizon. Roughly speaking,
the event horizon is the boundary of the black hole. Inside this boundary, the gravity
is so strong that even light cannot emerge from the region enclosed by the boundary.
We shall write A = G°c®m°. Use dimensional analysis to find a, ß , and y.
(b) One quantities in thermodynamics is entropy (S), that measure the disorder of a
system. The relationship between entropy S, energy E and temperature 0 of a
system is given as follow: = . For black holes, Bekenstein propose that the
entropy is proportional the area of the black hole's event horizon S = nA, where n
depends on c, k, G and h. Using dimensional analysis determine 7.
(c) With the previous results, determine the temperature of the black hole, as a function
of the mass. The energy of a black hole is E = mc².
(d) The Stefan Boltzmann constant o is a can be written as k, = ah"®G*k{. Take a
dimensionless constant a as one. Determine a, B, y and 8.
(e) Assuming black hole is perfect blackbody , determine the rate of blackhole evapo-
ration .
dm
(f) Determine how long for a black hole to evaporate completely if the initial mass is
М.
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