In the circuit shown below, the safe maximum value for the current I is _______

The maximum current that can flow through each resistor is,

$I_{100\mathrm{\Omega }}^2=\frac{1}{100}\Rightarrow I_{100}=0.1A$

$I_{2\mathrm{\Omega }}^2=\frac{0.5}{2}\Rightarrow I_{2\mathrm{\Omega }}=0.5A$

$I_{1\mathrm{\Omega }}^2=\frac{0.25}{1}\Rightarrow I_{1\mathrm{\Omega }}=0.5A$

As all the resistors are connected in series, the current flows through all the resistors must be the same.

So, the maximum safe current that can flow in the circuit = min (0.1, 0.5, 0.5) = 0.1 A