A wire of length \(2 \mathrm{~m}\) is moving at a speed \(2 \mathrm{~ms}^{-1}\) keep its length perpendicular to uniform magnetic field of \(0.5 \mathrm{~T}\). The resistance of circuit joined with this wire is \(6 \Omega\). The rate at which work is being done to keep the wire moving at constant speed is ........... (a) \((1 / 3) \infty\) (b) \((2 / 3) \omega\) (c) \((1 / 6) \propto\) (d) \(2 \mathrm{c}\)

Let's first find the induced EMF in the moving wire using Faraday's law of electromagnetic induction, which states that the induced EMF is equal to the rate of change of the magnetic flux. In this case, as the wire moves perpendicular to the magnetic field and has a constant speed, the induced EMF can be calculated as: \(EMF = B \cdot L \cdot v\) Here, B is the magnetic field strength, L is the length of the wire, and v is the speed of the wire. Plug in the given values: \(EMF = (0.5 \mathrm{T}) \cdot (2 \mathrm{m}) \cdot (2 \mathrm{ms^{-1}})\)

Now that we have the induced EMF, we can calculate the induced current I using Ohm's law, which states that the current is equal to the voltage (EMF) divided by the resistance R: \(I = \frac{EMF}{R}\) Given that the resistance of the circuit is 6 Ω, plug in the previously calculated EMF: \(I = \frac{(0.5 \mathrm{T}) \cdot (2 \mathrm{m}) \cdot (2 \mathrm{ms^{-1}})}{6 \Omega}\)

Finally, we'll find the rate at which work is being done, which is represented by the power P. The power in an electrical circuit can be calculated using the following formula: \(P = I^2 \cdot R\) Plug in the values we've calculated for I and R: \(P = (\frac{(0.5 \mathrm{T}) \cdot (2 \mathrm{m}) \cdot (2 \mathrm{ms^{-1}})}{6 \Omega})^2 \cdot (6 \Omega)\) Now, simplify and calculate the power: \(P = (\frac{4}{6})^2 \cdot (6)\) \(P = (\frac{2}{3})^2 \cdot (6)\) \(P = \frac{4}{3} \cdot 2\) \(P = \frac{8}{3}\) Thus, the rate at which work is being done to keep the wire moving at a constant speed is \(\frac{8}{3} \, W\). This answer does not match any of the given answer choices; it seems the choices contain errors or symbols unrelated to the problem. The correct answer should be \(\frac{8}{3}\, W\).