\(l\) લંબાઇ ધરાવતા એક ધાતુના સળિયાને \(2l\) જેટલી લંબાઇ ધરાવતી દોરી વડે બાંધી,\(\omega \) જેટલી કોણીય ઝડપથી સમક્ષિતિજ ટેબલ પર દોરીનો એક છેડો જડિત રાખી પરિભ્રમણ કરવામાં આવે છે, જો આ દિશામાં શિરોલંબ ( ઊર્ધ્વ ) દિશામાં \(B\) જેટલું ચુંબકીયક્ષેત્ર હોય,તો સળિયાના છેડે ઉદ્‍ભવતું પ્રેરિત \(emf\) ______ થશે.

A \(\frac{{2B\omega {l^2}}}{2}\)
B \(\;\frac{{3B\omega {l^2}}}{2}\)
C \(\;\frac{{4B\omega {l^2}}}{2}\)
D \(\;\frac{{5B\omega {l^2}}}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\;\frac{{5B\omega {l^2}}}{2}\)

Step-by-step Solution

Detailed explanation

Here, induced \(e.m.f.\) \(e=\int_{2 f}^{3 \ell}(\omega x) B d x=B \omega \frac{\left[(3 \ell)^{2}-(2 \ell)^{2}\right]}{2}=\frac{5 B \ell^{2} \omega}{2}\)

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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