# Coulomb's choreography

**Electricity and Magnetism**Level 5

Exact solutions for many-body problems are rarely encountered in physics. The following problem deals with a non-trivial motion of four charges. Due to the symmetry of the problem it is possible to determine the trajectories of the charges analytically.

Four identical particles with mass \(m\) and charge \(+q\), orbit a charge \(-q\) as shown in the figure. The four positive charges always form a square of side \(l(t)\) while the negative charge stays at rest at the center of the square. The motion of the charges is periodic with period \(T\). That is, if the vectors \(\vec{r}_k(t)\), \(k=\{1,2,3,4\}\), describe the position of the charges then we have that
\[\vec{r}_{k}(t+T)=\vec{r}_{k}(t).\]
It is also known that the side of the square oscillates between \(l_{\textrm{min}}=\frac{1}{4}L_{0}\) and \(l_{\textrm{max}}=L_{0}\). Determine the period T **in seconds** if the parameters \(q, m\) and \(L_{0}\) satisfy the relation
\[\frac{k q^{2}}{m L_{0}^{3}}=10^{4}~\mbox{s}^{-2}, \quad \textrm{where} \quad k=\frac{1}{4\pi \epsilon_{0}}.\]

**Details and assumptions**

- Ignore any radiation from the charges.