Under steady state condition the potential difference across the...

In steady state, the capacitor acts as an open circuit, meaning no current flows through the middle branch.

The circuit simplifies to a single loop containing the $2 \text{ V}$ battery, the $6 \Omega$ resistor, and the $2 \Omega$ resistor.

The total resistance of this loop is $R_{\text{eq}} = 2 + 6 = 8 \Omega$.

The current flowing through the loop is $I = \dfrac{2}{8} = 0.25 \text{ A}$.

The potential difference across the parallel branches is equal to the voltage drop across the $2 \Omega$ resistor:
$V = I \times 2 = 0.25 \times 2 = 0.5 \text{ V}$

Since there is no current in the middle branch, the voltage drop across the $4 \Omega$ resistor is zero. Thus, the entire potential difference of $0.5 \text{ V}$ appears across the capacitor.

Answer: $0.5$