Potential energy for uniform sphere doubt

Question

For a sphere of radius $R$, with electric charge distributed in uniform way $\rho=Q/V$, we have

$$U_e=\dfrac{3K_CQ^2}{5R}$$

However, when substituting the charge $Q=\rho V=\rho 4\pi R^3/3$, we get

$$U_e=\dfrac{16\pi^2 K_C\rho^2 R^5}{15}$$

My question is simple, which one gives the right electrostatic value as $R\rightarrow 0$ and $R\rightarrow \infty$? In the former case, the first expression provides $U_e=\infty$ but the latter gives $U_e=0$! In the case of big $R$, is the total energy inversely proportional to $R$ or directly proportional to $R^5$? Or both since, as $R\rightarrow \infty$, then $\rho\rightarrow 0$, and $R\rightarrow 0$ implies $\rho\rightarrow \infty$. But if this is the case, which is the correct power of proportionality for energy? Is energy directly proportional to $R^n$ or inversely proportional to $R$?

Answer 1

I think you need to define what are fixed during your limiting process. For example, maybe you want to ask the point charge limit. In that case, you will need to hold the total charge Q fixed, so you can not simply substitue Q in terms of R

Answer 2

Since your $Q(R)=\rho V(R)=\frac{\rho 4\pi R^3}{3}$ is function of $R$ so$$\displaystyle\lim_{R\to\infty} U_e=\displaystyle\lim_{R\to\infty}\dfrac{3K_C(Q(R))^2}{5R}=\displaystyle\lim_{R\to\infty}\dfrac{16\pi^2 K_C\rho^2 R^5}{15}=\infty$$.