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 = $5.98\times {10}^{24}\text{kg})$ has to be compressed to be a black hole?

• A. ${10}^{-9}\text{m}$
• B. ${10}^{-6}\text{m}$
• C. ${10}^{-2}\text{m}$
• D. ${10}^2\text{m}$

Answer 1

The correct option is C ${10}^{-2}\text{m}$

For a black hole, the escape velocity is greater than the speed of light.

$\therefore v_e\ge c$

$\Rightarrow \sqrt{\frac{2GM}{R}}\ge c(\because v_e=\sqrt{\frac{2GM}{R}})$

$\Rightarrow R\le \frac{2GM}{c^2}$

here, $G=(6.67\times {10}^{-11}{\text{Nm}}^2/{\text{kg}}^2$
$M=5.98\times {10}^{24}\text{kg}$
$c=3\times {10}^8\text{m/s}$

putting the value in the above expression, we get

$R\le \frac{2\times 6.67\times {10}^{-11}\times 5.98\times {10}^{24}}{{(3\times {10}^8)}^2}$

$\Rightarrow R\le 0.9\times {10}^{-2}\text{m}\approx {10}^{-2}\text{m}$

So, option $\left(c\right)$ is correct.

Key concept: For a black hole, the escape velocity is greater than the speed of light.