# Homework Help: Evaporation of black hole

1. Sep 1, 2010

### hasan_researc

1. The problem statement, all variables and given/known data

"Given that a black hole emits black body radiation at a temperature (in degrees Kelvin) given by T = hc3/16pi2MGk, how long would it take a solar mass black hole to evaporate?

If small black holes were created during the big bang, what mass of black hole would now be evaporating? "

2. Relevant equations

This problem was set as part of my first year undergrad "Professional Skills Problem Solving" seminar. So I have to use first year undergrad physics concepts (or possibly more advanced tools) to solve this problem.

3. The attempt at a solution

No idea! That's why I'm here.

Thanks in advance for any help!

2. Sep 1, 2010

### betel

You should try to express the power of the emitted radiation in terms of the Temperature.
Then you can try to connect this energy loss to the time evolution of the mass of the black hole.

3. Sep 1, 2010

### hasan_researc

Energy (of the emitted radiation) is on the order of kT (is it not?).

So the (average) power is kT/t, where t is some time interval.

Therefore, using the formula in my previous post, Power = hc3/16pi2MGt.
Therefore, t = hc3/(16pi2MG*power).

But do I know what the power is?

4. Sep 1, 2010

### betel

Simply putting some time interval is not quite correct.

There is a certain law relating power and temperature of radiation of a black body.

5. Sep 1, 2010

### hasan_researc

Actually, I don't know what the formula is. I found Plankc's law on wikipedia, but that looks horribly complicated, so I omitted that.

6. Sep 1, 2010

### betel

Try Stefan-Boltzmann Law.

7. Sep 1, 2010

### hasan_researc

Okay, so the power radiated = sigma*T4
= sigma * (hc3/16pi2MGk)4

Now, how shall we assume that the mass radiated per unit time is a constant, and thus eliminate the time t in the equation?

8. Sep 1, 2010

### vela

Staff Emeritus
That's actually the power per unit area. You need to find the surface area of the black hole to get the total power radiated. Then try setting up a differential equation to work in the time dependence.

9. Sep 1, 2010