A wire of length $1\,\mathrm{m}$ is moving at a speed of $2\,\mathrm{ms}^{-1}$ perpendicular to its length and a homogeneous magnetic field of $0.5\,\mathrm{T}$. The ends of the wire are joined to a circuit of resistance $6\,\Omega$. The rate at which work is being done to keep the wire moving at constant speed is

The motional emf induced across the wire is $$e = Blv$$ $$e = 0.5 \times 1 \times 2 = 1\,\mathrm{V}$$ So the current in the circuit is $$I = \frac{e}{R}$$ $$I = \frac{1}{6}\,\mathrm{A}$$ The mechanical power needed to keep the wire moving at constant speed equals the electrical power developed: $$P = eI$$ $$P = 1 \times \frac{1}{6} = \frac{1}{6}\,\mathrm{W}$$ This matches option $\mathrm{(B)}$.