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, B, and Y. (b) One quantities in thermodynamics is entropy (S), that measure the disorder of a system. The relationship bet ween 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 n depends on c, kip, G and h. Using dimensional analysis determine 7. dS dE For black holes, Bekenstein propose that the %3| (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 = mc?. (d) The Stefan Boltzmann constant o is a can be written as k, = ah cG^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 dt · (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, B, and Y. (b) One quantities in thermodynamics is entropy (S), that measure the disorder of a system. The relationship bet ween 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 n depends on c, kip, G and h. Using dimensional analysis determine 7. dS dE For black holes, Bekenstein propose that the %3| (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 = mc?. (d) The Stefan Boltzmann constant o is a can be written as k, = ah cG^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 dt · (f) Determine how long for a black hole to evaporate completely if the initial mass is М.

Similar questions

What are the two postulates of special relativity? (Choose two answers.) A In any closed system, the total momentum will not change. B F=ma (c) The entropy of an isolated system will always increase over time. The speed of light is the same regardless of the speed of the observer or the speed of the source. Light travels to us from the hands of the clock. If we could move away from the clock at the speed of light, time will appear to stand still. If you measure both the position of a particle and its momentum, then the product of the F uncertainties of these two measurements must always be at least half of the reduced Planck's constant. For all possible paths between two points, the actual trajectory is the one that results in the action being minimized.
Given: R(N)=f(N)g(N) where f(N) = 7.2N and g(N) = (1 - 7.9/N. If R(N) = H means that R'(N) = 0 and H = 10.5N, what is the value of N that causes R'(N) = 0?
A particle has γ=18,399. a)Calculate c-v in m/s. (I would have asked for 1 - v/c, making the answer dimensionless, but the system doesn't seem to take numbers that small. Gamma is chosen to make the particle extremely close to the speed of light.) If your calculator gives problems, you might want to solve the appropriate equation for c-v or c(1 - v/c) and use an approximation. b) In the previous problem, in a race to the moon, by 3/4ths the distance, light is one or ten meters ahead of the particle. We routinely approximate mass as zero, gamma as infinite, and speed as the speed of light. ("Massless particles" -- gamma and m have to be eliminated from the expressions. Light is a true massless particle.) If a massless particle has momentum 1,739 MeV/c, calculate its energy in MeV.
You are the pilot of a spacecraft intended for travel at very high speeds. Before leaving you measure the spacecraft to be 31.4 m long and have a mass of 5.28 x 104 kg. During your travel, you pass a planet and exchange information with an observer on the planet. You are told that your spacecraft has been measured to be 28.6 m long. a. How fast is your spacecraft travelling with respect to the planet? b. You are told there is a nearby planet that is stationary with respect to the first planet. The observer on the first planet says it will take you 25 s to reach the nearby planet. How far away does your co-pilot on the spacecraft say the nearby planet is? c. How much energy was required to accelerate your spacecraft to this speed?
Consider an electron gyrating in the magnetic field associated with a sunspot which has a magnetic field strength of 0.25 T. The typical temperature of a sunspot is 3500 K the equipartition equation for velocity v, temperature T and mass m of a particle, is, 3kgT V = m kB is Boltzmann's constant (1.38×10–23 J/K). a) The mass of the Sun is approximately 2 × 1030 kg. Use this information and the infor- mation given above to compute the acceleration g due to gravity at the surface of the Sun. b) Compute the drift velocity Ūg due to gravity for a proton and an electron located within a sunspot. c) Assuming a typical mass density of the Sun's photosphere of 10-3 kg/m³, compute the number density of protons and electrons in the photosphere (where sunspots reside) assuming that the Sun is comprised of pure ionized hydrogen. d) Calculate the total current density J (accounting for the contributions of both electrons and tons) within sunspots. You may assume that you can treat the ensemble of…
A spacecraft is in a 300 km circular parking orbit. It is desired to increase the altitude to 600 km and change the inclination by 20°. Find the total ΔV required if: Part A The plane change is made after insertion into the 600 km orbit (so there is a total of three ΔV burns).
Multiple-Concept Example 6 explores the approach taken in problems such as this one. Quasars are believed to be the nuclei of galaxies in the early stages of their formation. Suppose a quasar radiates electromagnetic energy at the rate of 1.5 × 104¹ W. At what rate (in kg/s) is the quasar losing mass as a result of this radiation? Number i A quasar at the center of a spiral galaxy Units
Doppler Effect: This problem looks at wavelength shifts instead of frequency shifts. In this problem, AX is defined as the difference between the wavelength in the observer's frame and the source frame, and you can pick 2 to be equal to the wavelength in the source frame. a. Show that, for speeds u<
An astronomer is measuring the light emitted by a binary star system. One of the stars appears to be much more massive than the other. The astronomer notices that the smaller star orbits the larger one in a nearly circular orbit at fr = 845 ×10º| a distance of r = 845 x 106 km. Through careful observation, she measures the smaller star's orbital speed to be v 12.0 km/s. Using Keppler's Laws she calculates the mass of the larger star. What is it? Use G = 6.67 x 10-¹1 N-m²/kg². - M = 30 x1030 kg (± 0.05 × 10³0 kg)
Helpful information: (1) An alpha particle is a helium nucleus, (2) e = 1.6 × 10-¹⁹ C, (3) k₂ = 9.0 × 10⁹ Nm² C-2, (4) 1nm = 1 × 10-⁹ m 1-An alpha particle lies on the x-axis, a distance of 1.0 nanometer from a proton (in this set-up, the alpha particle is at the origin while the proton is in the positive direction). Which of the following choices below represents the magnitude of the electric force on the alpha particle? (a) 2.3 × 10-10 N (b) 4.6 × 10-10 N (c) 2.3 x 10-19 N (d) 4.6 x 10-19 N cing the voltage so following insta choices below at a time! 1.00 s?
An asteroid of mass 1000 kg travels through space with speed a) What is the total energy of the asteroid? (You can leave your response as a math function, but please simplify as much as you can.) b) The asteroid has a rest length of 20 m. What length would an observer on Earth see? (You can leave your response as a math function, but please simplify as much as you can.)
2. An alternative derivation of the mass-energy formula E, = moc² also given by Einstein, is based on the principle that the location of the center of mass (CM) of an isolated system cannot be changed by any process that occurs inside the system. The Figure below shows a rigid box of length L that rests on a frictionless surface; the mass M of the box is equally divided between its two ends. A burst of electromagnetic radiation of energy E, is emitted by one end of the box. According to classical physics, the radiation has the momentum p = "0%c , and when it is emitted, the box recoils with the speed v z Eo/ so that the total momentum of Mc the system remains zero. After a time t z L/c the radiation reaches the other end of the box and is absorbed there, which brings the box to a stop after having moved the distance S. If the CM of the box is to remain in its original place, the radiation must have transferred mass from one end to the other. Show that this amount of mass is m, = c2.