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 $$ = 5.98 \times {10^{24}}kg$$ ) have to be compressed to be a black hole?
A.
$${10^{ - 9}}\,m$$
B.
$${10^{ - 6}}\,m$$
C.
$${10^{ - 2}}\,m$$
D.
$$100\,m$$
Answer :
$${10^{ - 2}}\,m$$
Solution :
For the black hole, the escape speed is more than $$c$$ (speed of light). We should compare the escape speed with the $$c$$ (Note that the escape speed should be at least just greater than $$c$$).
$$\eqalign{
& {V_e} = \sqrt {\frac{{2GM}}{{R'}}} \,\,\left[ {R' = {\text{New radius of the earth}}} \right] \cr
& c = \sqrt {\frac{{2GM}}{{R'}}} \left[ {{v_e} \approx c} \right] \Rightarrow {c^2} = 2\frac{{GM}}{{R'}} \cr
& R' = \frac{{2GM}}{{{c^2}}} = \frac{{2 \times 6.67 \times {{10}^{ - 11}} \times 6 \times {{10}^{24}}}}{{9 \times {{10}^{16}}}} \cr
& = \frac{{4 \times 6.67}}{3} \times {10^{ - 3}} = 8.89 \times {10^{ - 3}} \cr
& = 0.889 \times {10^{ - 2}} \cr
& \simeq {10^{ - 2}}\;m \cr} $$
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