How long do black holes last?

1) The Schwarzchild radius of a Black Hole is the radius of the 'event horizon', past which even light cannot escape.

Find an expression for the Schwarzchild radius of a black hole using the notion of escape velocity (non-rigorous derivation :-o)

2) Over time, black holes lose energy and hence mass via "

Hawking Radiation". The power emitted by a Black hole is:

, where

, and

is

Planck's constant,

is

Boltzmann's constant, and

is the

speed of light,

is the surface area,

is emissivity, and

is temperature.

Since we may assume a black hole to be a perfect blackbody (perfectly absorbs and emits radiation),

.

Given that the temperature of a black hole is:

, where

is the

Gravitational constant, and the energy of a black hole is

, form a differential equation to find the lifetime of a black hole in terms of its initial mass.

3) How long would a black hole the mass of the sun last (

)? How about the mass of a proton (

)?