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(d) The Stefan Boltzmann constant o is a can be written as ki dimensionless constant a as one. Determine a, B, y and d. ah c®G"kg. Take a (e) Assuming black hole is perfect blackbody , determine the rate of blackhole evapo- ration dm dt : (f) Determine how long for a black hole to evaporate completely if the initial mass is M.

(d) The Stefan Boltzmann constant o is a can be written as ki dimensionless constant a as one. Determine a, B, y and d. ah c®G"kg. Take a (e) Assuming black hole is perfect blackbody , determine the rate of blackhole evapo- ration dm dt : (f) Determine how long for a black hole to evaporate completely if the initial mass is M.

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Parts d, e, f please 

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: entropy is proportional the area of the black hole's event horizon S = nA, where 7 depends on c, kip, G and h. Using dimensional analysis determine n. For black holes, Bekenstein propose that the || dE (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, = ahªc®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 m. dt (f) Determine how long for a black hole to evaporate completely if the initial mass is М.
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: entropy is proportional the area of the black hole's event horizon S = nA, where 7 depends on c, kip, G and h. Using dimensional analysis determine n. For black holes, Bekenstein propose that the || dE (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, = ahªc®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 m. dt (f) Determine how long for a black hole to evaporate completely if the initial mass is М.
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