Question

The mass of sun is approximately $2 \times {10^{30}}kg$ .The Schwarzschild radius for the mass of a star that is ten times the mass of the sun is nearly
A) 3km
B) 30km
C) 300km
D) 0.3km

Answer 1

Hint: Any mass can become a black hole if it collapses down to the Schwarzchild radius. The gravitational field inside a black hole is so strong that even light cannot escape from it. The separation between the region where we know how things work and the region where we don’t know how things work is called the event horizon. This event horizon in a black hole is called Schwarzchild radius. Take $G = 6.67 \times {10^{ - 11}}$ and $C = 3 \times {10^8}$ .

Complete step by step answer:
Let mass of sun be $(M)$
Mass of sun is given as,
$M = 2 \times {10^{30}}kg$
Now, Schwarzchild radius is calculated by the following formula, in which $R$ is Schwarzschild radius, $M$ is mass of the sun, $G$ is universal gravitational constant and $C$ is Speed of light.
Formula used:
$R = \dfrac{{2GM}}{{{C^2}}}$
We will multiply by 10 to the mass of the sun as it is mentioned in the question that the star’s mass is ten times that of the sun.
Now, putting the values is the formula,
$R = \dfrac{{2 \times 6.67 \times {{10}^{ - 11}} \times 10 \times 2 \times {{10}^{30}}}}{{{{(3 \times {{10}^8})}^2}}}$
Now, simplifying the equation we get,
$R = 2.9644 \times {10^4}m$
Which is equal to,
$29.644km$
Which is approximately,
$ \simeq 30km$
Hence, the correct answer is B. 30km

Note: The universal gravitational constant is in ${m^3}k{g^{ - 1}}{s^{ - 2}}$ and the speed of light is in $m{s^{ - 1}}$. Make sure all the units are the same. Schwarzschild radius is the radius of the event horizon which is surrounding a non-rotating black hole. This quantity was first derived by German astronomer Karl Schwarzchild in 1916 for theory of relativity. It represents the ability of mass to cause curvature in space and time. The Schwarzchild radius of an object is directly proportional to the mass of the object as G and C are constants.