\('r'\) त्रिज्या के धातु वृत्तीय-तार-लूप का पृष्ठ, \(B=B_{0} e ^{\frac{{ - t}}{\tau }}\) द्वारा बदलते हुए चुम्बकीय-क्षेत्र के लम्बवत रखा है। जहाँ समय \(t=0\) पर \(B_{0}\) तथा \(\tau\) अचर हैं। यदि लूप का प्रतिरोध \(R\) है, तब काफी ज्यादा समय \((t \rightarrow \infty)\) गुजरने के बाद उस लूप में पैदा हुई ऊर्जा है

A \(\frac{{{\pi ^2}{r^4}B_0^4}}{{2\tau R}}\)
B \(\frac{{{\pi ^2}{r^4}B_0^2}}{{2\tau R}}\)
C \(\frac{{{\pi ^2}{r^4}B_0^2R}}{\tau }\)
D \(\frac{{{\pi ^2}{r^4}B_0^2}}{{\tau R}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{{{\pi ^2}{r^4}B_0^2}}{{2\tau R}}\)

Step-by-step Solution

Detailed explanation

Electric flux is given by \(\phi=\mathrm{B} . \mathrm{A}\) \(\phi=\mathrm{B}_{0} \pi \mathrm{r}^{2} \mathrm{e}^{-1 / \tau}\) \(\left(\because B=B_{0} e^{-t / \tau}\right)\) Induced \(E.m.f.\)…

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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