Question #42670

A hypothetical atomic nucleus of mass number $A$ and atomic number $Z$ is modeled as a sphere of radius $R = R_0 A^{1/3}$, where $R_0$ is a constant. The electric charge is distributed within the nucleus such that the volume charge density $\rho(r)$ at a distance $r$ from the center is given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)^2$ for $r \le R$, where $\rho_0$ is a constant. Let $U_E$ be the electrostatic self-energy of this nucleus. If $U_0$ is the electrostatic self-energy of a nucleus with the same total charge $Ze$ and radius $R$, but with a uniform volume charge distribution, find the ratio $\frac{U_E}{U_0}$ and the proportional dependence of $U_E$ on $Z$ and $A$.