For the wire to float on the liquid surface without sinking, the upward force due to surface tension must balance the downward force due to the weight of the wire.

1. Upward Force ($F_{ST}$): Surface tension acts along the two lengths of the wire in contact with the surface. $F_{ST} = T \times 2L$

2. Downward Force ($W$): This is the weight of the wire. $W = mg = (\text{Volume} \times \text{Density}) \times g$ $W = (\pi r^2 L) \times \rho \times g$ (Note: In this context, $\rho$ represents the density of the wire for the weight equation. If the text specifies liquid density, it is likely assuming the wire density is comparable or $\rho$ refers to the object's density in the standard formula derivation).

3. Equilibrium Condition: For the maximum radius before sinking, the forces are balanced: $F_{ST} = W$ $2TL = \pi r^2 L \rho g$

4. Solve for $r$: $2T = \pi r^2 \rho g$ $r^2 = \frac{2T}{\pi \rho g}$ $r = \sqrt{\frac{2T}{\pi \rho g}}$