Question

Mini Black Holes.Cosmologists have speculatedthat black holes the size of a proton could have formed during theearly days of the Big Bang when the universe began. If we take thediameter of a proton to be $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{15}}$m, what would be the massof a mini black hole?

Short answer

The mass of a mini hole is$\mathbf{3}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{11}}\mathbf{kg}$

Step-by-step solution

Identification of the given data

The schwarzchild radius is given by,

Diameter = $1.0\times {10}^{-15}$m

Determination of the mass

$$\begin{gathered} R_s=\frac{2\mathrm{Gm}}{c^2} \\ m=\frac{c^2{R}_s}{2G} \\ \text{Here,} \\ {R}_s=0.5\times {10}^{-15}\mathrm{m}, \\ G=6.67\times {10}^{-11}{\mathrm{Nm}}^2/{\mathrm{kg}}^2, \\ m=\frac{c^2{R}_s}{2G} \\ =\frac{{\left(3\times {10}^8\right)}^2\times \left(0.5\times {10}^{-12}\right)}{2\times \left(6.67\times {10}^{-11}\right)} \\ =3.4\times {10}^{11}\mathrm{kg} \end{gathered}$$