Breaking stress of a steel wire is $p$ and the density of steel is $\rho$. The greatest length of steel wire that can hang vertically without breaking is

Maximum stress $=\frac{ \mathrm{Maximum} \mathrm{weight} }{ \mathrm{Cross}-\mathrm{sectional} \mathrm{area} }$

Now, maximum weight of steel wire

$=$ Volume $\times$ Density $\times g$$=Al\rho g$

where, $l$ is the maximum length of steel wire that can hang vertically without breaking. $\rho$ is the density of steel and $A$ is the cross-sectional area of steel wire.
$\therefore$ Maximum stress, $p=\frac{Al\rho g}{A}=l\rho g$

$\therefore l=\frac{p}{\rho g}$