Two metal spheres of radii $1\ \text{cm}$ and $2\ \text{cm}$ are far apart. The smaller sphere carries a charge of $3\ \mu\text{C}$ and the larger one is uncharged. They are now joined by a long thin wire, as shown in the figure, and charge flows until it stops. Which statement about the charges and the surface charge densities ($\sigma$) is correct?

Fig. 1: A metal sphere of radius 1 cm and a larger one of radius 2 cm, far apart, joined by a long thin wire.