MHT CET · Physics · Electromagnetic Induction

A long, rectangular conducting loop of width \(l\) mass \(m\) and resistance \(R\) is placed partly in a perpendicular magnetic field \(B\). It is pushed downwards with velocity \(v\) so that it may continue to fall freely. The velocity \(v\) is \((g=\) acceleration due to gravity)

A \(\frac{m g R^2}{B l}\)
B \(\frac{m g R}{B^2 l^2}\)
C \(\frac{m g l}{B^2 R^2}\)
D \(\frac{B^2 l^2 R}{m g}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{m g R}{B^2 l^2}\)

Step-by-step Solution

Detailed explanation

The motional emf is given by: \(V=B v l\)
Current in the loop,
\(i=\frac{V}{R}=\frac{B v l}{R}\)
During free fall the electromagnetic force felt by the loop balances its weight, and since the loop is long so we assume its falling with a terminal velocity.
\(\begin{aligned}
& \therefore i l B=m g \\
& \Rightarrow B\left(\frac{B v l}{R}\right) l=m g \\
& \Rightarrow v=\frac{m g R}{B^2 l^2}
\end{aligned}\)

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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