Assuming that both the nucleus and the entire atom are perfectly spherical, if the radius of an atom

Question

Assuming that both the nucleus and the entire atom are perfectly spherical, if the radius of an atomic nucleus is approximately $10^{-13} \mathrm{~cm}$ and the radius of the atom is $10^{-8} \mathrm{~cm}$, what is the exact ratio of the surface area of the nucleus to the surface area of the atom?

Professor PC Thomas Classes · Answer

(a) : The surface area of a sphere is given by $A = 4\pi r^2$. Therefore, the surface area is directly proportional to the square of the radius ($A \propto r^2$). The ratio of the surface area of the nucleus to the surface area of the atom is $\frac{A_{	ext{nucleus}}}{A_{	ext{atom}}} = \left(\frac{r_{	ext{nucleus}}}{r_{	ext{atom}}}ight)^2$. Substituting the given values yields $\left(\frac{10^{-13}}{10^{-8}}ight)^2 = (10^{-5})^2 = 10^{-10}$.