# The Nucleus

The atomic nucleus contains most of the atom's mass and is made up of positively charged protons and neutral neutrons. The radius of a typical atomic nucleus is on the order of $10^{-15}$ meters, which is about 100,000 times smaller than the radius of the atom. It is related to the atomic number $A$ by \begin{align} R=R_0A^{1/3}, \end{align} where $R_0\approx 1.1\times10^{-15}\;\mathrm{m}$.

## Problems from IIT JEE

Problem (IIT JEE 1999): Order of magnitude of density of uranium nucleus is, (Given $R_0\approx {1.1\times{10}^{-15}}\mathrm{m}$ and $m_p = {1.67\times {10}^{-27}}\mathrm{kg}$.)

1. ${10^{20}}\;\mathrm{kg/m^3}$
2. ${10^{17}}\;\mathrm{kg/m^3}$
3. ${10^{14}}\;\mathrm{kg/m^3}$
4. ${10^{11}}\;\mathrm{kg/m^3}$

Solution: Let $A$ be mass number of the uranium. The nuclear mass and the nuclear radius are given by, \begin{align} & m=Am_p,\\ & R=R_0A^{1/3}\approx {1.1\times{10}^{-15} A^{1/3}}\;\mathrm{m}. \end{align} The nuclear density density is, \begin{align} \rho & =\frac{m}{\frac{4}{3}\pi R^3} \\ &=\frac{m_p}{\frac{4}{3}\pi R_0^3} \\ &=\frac{1.67\times{10}^{-27}}{\frac{4}{3}\times 3.14\times1.331\times{10}^{-45}} \\ &\approx {3\times{10}^{17}}\;\mathrm{kg/m^3}.\nonumber \end{align} Note that nuclear density is independent of $A$.