91Ó°ÊÓ

There are N equal point charges placed on or inside a circle of radius R.

The potential from a charge q at a distance R is

$\mathit{V}\mathbf{=}\frac{\mathbf{q}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}\mathbf{R}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

Here, ${\mathit{Î\mu }}_{\mathbf{0}}$is the permittivity of free space.

When n charges are equally spread on a circle of radius R, distance of the j-th charge from one reference charge is

${\mathit{r}}_{\mathbf{j}}\mathbf{=}\mathbf{2}\mathit{R}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\left(\frac{j\mathrm{Ï€}}{n}\right)$

From equation (1), the potential on the reference charge from the remainingcharges is

$$\begin{gathered} \mathit{V}\mathbf{=}\frac{\mathbf{q}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}}\underset{\mathbf{j}\mathbf{=}\mathbf{1}}{\overset{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{∑}}}\frac{\mathbf{1}}{{\mathbf{r}}_{\mathbf{j}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{q}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}}\underset{\mathbf{j}\mathbf{=}\mathbf{1}}{\overset{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{∑}}}\frac{\mathbf{1}}{\mathbf{2}\mathbf{R}\mathbf{s}\mathbf{i}\mathbf{n}\left({\displaystyle \frac{j\mathrm{Ï€}}{n}}\right)} \end{gathered}$$

All the charges have this same potential. The net energy of the configuration is

${\mathit{E}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathit{q}\mathit{V}$

The half factor is to avoid double counting. Substitute the value in the above equation and get

$$\begin{gathered} {\mathit{E}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathit{q}\frac{\mathbf{q}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}}\underset{\mathbf{j}\mathbf{=}\mathbf{1}}{\overset{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{∑}}}\frac{\mathbf{1}}{\mathbf{2}\mathbf{R}\mathbf{s}\mathbf{i}\mathbf{n}\left({\displaystyle \frac{j\mathrm{Ï€}}{n}}\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{q}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}\mathbf{R}}\underset{\mathbf{j}\mathbf{=}\mathbf{1}}{\overset{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{∑}}}\frac{\mathbf{n}}{\mathbf{4}\mathbf{s}\mathbf{i}\mathbf{n}\left({\displaystyle \frac{j\mathrm{Ï€}}{n}}\right)} \end{gathered}$$

If n - 1charges are kept at the circle and one is kept at the center, the net energy of the configuration is

${\mathit{E}}_{\mathbf{i}}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}\mathbf{R}}\underset{\mathbf{j}\mathbf{=}\mathbf{1}}{\overset{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{∑}}}\frac{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathbf{4}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\left({\displaystyle \frac{j\mathrm{Ï€}}{n-1}}\right)}\mathbf{+}\frac{\left(n-1\right){\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}\mathbf{R}}$

The values for n = 10, 11, 12are obtained from Mathematica as follows

role="math" localid="1657634953289" $$\begin{gathered} {\mathit{E}}_{\mathbf{∘}}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{10}\mathbf{)}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}{\mathbf{Ï€Î\mu }}_{\mathbf{0}}\mathbf{R}}\mathbf{×}\mathbf{38}\mathbf{.}\mathbf{6245} \\ {\mathit{E}}_{\mathbf{∘}}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{11}\mathbf{)}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}{\mathbf{Ï€Î\mu }}_{\mathbf{0}}\mathbf{R}}\mathbf{×}\mathbf{48}\mathbf{.}\mathbf{5757} \\ {\mathit{E}}_{\mathbf{∘}}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{12}\mathbf{)}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}{\mathbf{Ï€Î\mu }}_{\mathbf{0}}\mathbf{R}}\mathbf{×}\mathbf{59}\mathbf{.}\mathbf{8074} \\ {\mathit{E}}_{\mathbf{i}}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{11}\mathbf{)}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}{\mathbf{Ï€Î\mu }}_{\mathbf{0}}\mathbf{R}}\mathbf{×}\mathbf{48}\mathbf{.}\mathbf{6245} \\ {\mathit{E}}_{\mathbf{i}}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{12}\mathbf{)}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{4}{\mathbf{Ï€Î\mu }}_{\mathbf{0}}\mathbf{R}}\mathbf{×}\mathbf{59}\mathbf{.}\mathbf{5757} \end{gathered}$$

Thus, for n = 12, the configuration with 11charges at the circle and one charge at the center has lower energy but for n = 11 the configuration with all charges at the center has lower energy.