Two isolated metallic spheres $S_1$ and $S_2$ of radii $1 \, \mathrm{cm}$ and $3 \, \mathrm{cm}$ respectively are charged such that both have the same charge density \[ \sigma = \frac{2}{\pi} \times 10^{-9} \, \mathrm{C/m^2}. \] They are placed far away from each other and connected by a thin wire. Calculate the new charge on sphere $S_1$.