A copper rod of mass \(m\) slides under gravity on two smooth parallel rails, with separation \(l\) and set at an angle of \(\theta\) with the horizontal. At the bottom, rails are joined by a resistance \(R\). There is a uniform magnetic field \(B\) normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is

A \(\frac{{mgR\cos \,\theta }}{{{B^2}{l^2}}}\)
B \(\frac{{mgR\sin \,\theta }}{{{B^2}{l^2}}}\)
C \(\frac{{mgR\tan \,\theta }}{{{B^2}{l^2}}}\)
D \(\frac{{mgR\cot \,\theta }}{{{B^2}{l^2}}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{{mgR\sin \,\theta }}{{{B^2}{l^2}}}\)

Step-by-step Solution

Detailed explanation

From Faraday's law of electomagnetic induction, \(e=\frac{d \phi}{d t}=\frac{d(B A)}{l t}=\frac{d(B l l)}{d t}\) \( = \frac{{Bdl \times l}}{{dt}} = BVl\) Also, \(F=i l B=\left(\frac{B V}{R}\right)\left(l^{2} B\right)=\frac{B^{2} l^{2} V}{R}\) At equilibrium…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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