Compare Potential Energy of Two Configurations of Point Charges

A question in my physics text is asking to compare the potential energy (relative to $\inf$) for two configurations, $C_1$, $C_2$ of $N$ point charges.

$C_1$:

$N$ point charges are uniformly distributed on a ring s.t. the distance between adjacent electrons is constant

$C_2$: $N-1$ point charges are uniformly distributed on a ring s.t. the distance between adjacent electrons is constant and one charge is placed in the center of the ring.

I need to determine the smallest value of $N$ s.t. $V_1>V_2$

My thinking is as follows:

1. If we consider a gaussian surface inside the ring, $E=0$. We know that the voltage at the center of the ring is $$V_\text{center}=\frac{Ne}{r}$$ and furthermore, because $V=\int E\circ ds$, $$V_\text{inside} = V_\text{center}$$
2. From this previous result, $$U_1 = eV_\text{center} = \frac{N(N-1)e^2}{r}$$
3. $C_2$ the configuration potential without the center electron is $$(1/2)(N-1)(N-2)\frac{e}{r}$$ The center electron adds $(N-1)\frac{e}{r}$ yielding $$U_2 = (1/2)(N-1)(N-2)\frac{e}{r} + (N-1)\frac{e}{r}$$
4. Let $k = \frac{e}{r}$, and, setting $(N^2-N)k = (N^2-3N+2)k+(2N-2)k$ $$\implies 0=0$$

Which is a ridiculous answer. So what is the correct way of answering this problem as I obviously have no direction.

• Are you sure that you were asked the potential and not the potential energy of the system? – Raziman T V Feb 13 '17 at 9:45
• Your right. I was asked potential energy – theideasmith Feb 13 '17 at 12:20