$\begin {array} {1 1} (A)\;V & \quad (B)\;\large\frac{V}{2} \\ (C)\;\large\frac{V}{3} & \quad (D)\;\large\frac{2V}{3} \end {array}$

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Let the voltages at the two nodes be 0 and X volts as shown

Writing the KVL equations for the three branches,

$0 + V – I_1R = X$

$0 + V – \Delta V = X$

$0 + 2V – I_2 ^*2R = X$

At steady state, the net current flowing in the branch at potential X is zero, hence

$I_1 + I_2 = 0$

On solving the above equations, $ \Delta V = \large\frac{V}{3}$

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