Net Coulomb Force on a charge in a system

Question

Suppose that there are $n$ charges each at rest and positioned at the vertices of an $n$-sided regular polygon with side length $l$. Each charge has a magnitude of $q$ $C$.

Can a general formula be derived which gives the net Coulomb force on any charge $q$ in an $n$-sided regular polygon?

For example (these are the one's I've been able to calculate myself), if 4 charges of magnitude $q$ are placed at the vertices of a square, $$F_{net} = \frac{kq^2}{l^2}\left(\sqrt{2}+\frac{1}{2}\right)$$ For an equilateral triangle, $$F_{net} = \frac{kq^2}{l^2}$$ For a regular hexagon, $$F_{net} = \frac{kq^2}{l^2}\left(\frac{5+2\sqrt{5}}{3+\sqrt{5}}\right).$$

Answer 1

You can achieve it by a few steps:

1. let's try to find the force on one of the charges $q_o$
2. assume the n-sided regular polygon is positioned such that the force on it has only x component.
3. find the side and angles of a n-sided regular polygon (see here https://en.wikipedia.org/wiki/Regular_polygon)
4. sum all the (n-1) forces acting on $q_o$
5. test your result, take the limit that n goes to infinity and recover the answer for a circle.