Question

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass\[ = 5.98 \times {10^{24}}\,{\text{kg}}\]) have to be compressed to be a black hole?
A. \[{10^{ - 9}}\,{\text{m}}\]
B. \[{10^{ - 6}}\,{\text{m}}\]
C. \[{10^{ - 2}}\,{\text{m}}\]
D. \[100\,{\text{m}}\]

Answer 1

Hint: First of all, we will apply the formula of escape velocity and compare this velocity with the escape velocity of earth at black hole condition. We will then substitute the required values in the expression and manipulate accordingly and find the radius.

Complete step by step answer:
The theory of general relativity predicts that to form a black hole, a sufficiently compact mass will distort space-time. The event horizon is called the limit of the area from which no escape is possible. It is predicted that black holes of stellar mass can develop when very large stars collapse at the end of their life cycle.
Escape Velocity is referred to as the minimum velocity needed to transcend the gravitational force of the planet earth by someone or object to be projected. In other words, escape velocity is the minimum velocity that one needs to escape the gravitational field.
The equation of escape velocity is given as:
\[{v_e} = \sqrt {\dfrac{{2GM}}{r}} \]
Here, \[G\] is the gravitational constant, \[M\] is the mass of the body to be escaped from, \[r\] is the radius of the body from the center.
For the earth to be black hole the escape velocity should be at least equal to the speed of light.
$
  \therefore {\text{escape}}\,{\text{velocity}}\,{\text{ = }}\,{\text{speed}}\,{\text{of}}\,{\text{light}}\, \\$
$ \sqrt {\dfrac{{2GM}}{r}} \,{\text{ = }}\,C \\$
$ r\,\,{\text{ = }}\,\dfrac{{{\text{2}}GM}}{{{C^{\text{2}}}}} \\$
$ r\,\,\,{\text{ = }}\,\dfrac{{{2 \times 6}{.67 \times 1}{{\text{0}}^{ - {\text{11}}}}{\times 5}{.98 \times 1}{{\text{0}}^{{\text{24}}}}}}{{{9 \times 1}{{\text{0}}^{{\text{16}}}}}} \\$
 $ r\,\,\,{\text{ = }}\,{\text{8}}{.86 \times 1}{{\text{0}}^{ - {\text{3}}}}{\text{m}} \\$
 $ r\,\,\, \approx {\text{1}}{{\text{0}}^{ - {\text{2}}}}\,{\text{m}} \\
$

Hence the correct option is (C) \[{10^{ - 2}}\,{\text{m}}\]

Note:
While solving this problem, most of the students take the escape velocity to be greater than the speed of the light itself. There is a fact that till now, nothing has been found which travels greater than the speed of light. In a black hole, no matter can survive its strong pull, not even light can escape. A hypothetical spaceship can survive, if it can travel at more than the speed of light.