The age of the universe is roughly 14 billion years, which is also the approximate evaporation time for a black hole with a mass of 10¹⁰ kg. What is the Schwarzschild radius of such a black hole?

Question

Quizplus · Accepted Answer

The evaporation time for a black hole is proportional to the cube of its mass, and inversely proportional to the square of its Schwarzschild radius. Using this formula, we can find the Schwarzschild radius for a black hole with an evaporation time of 14 billion years and a mass of 10^110 solar masses:

t = (5120πG^2M^3)/(ℏc^4)

where t is the evaporation time, G is the gravitational constant, M is the mass of the black hole, ℏ is the reduced Planck constant, and c is the speed of light.

Solving for the Schwarzschild radius yields:

r = (2GM/c^2) = (2ℏc/GM)^(1/2)

Plugging in the values for t and M, we get:

r = 1.5 × 10^-17 m

Therefore, the answer is B.

The Age of the Universe Is Roughly 14 Billion Years