एक संधारित्र C को पहले \( V_{0} \) के विभवांतर से पूर्णतः आवेशित किया जाता है और बैटरी से विच्छेदित किया जाता है। आवेशित संधारित्र को L प्रेरकत्व वाले एक प्रेरक से जोड़ा जाता है। t समय में, संधारित्र में संचित प्रारंभिक ऊर्जा का 25% प्रेरक को स्थानांतरित हो जाता है। t का मान है

A \(\frac{\pi \sqrt{ LC }}{3}\)
B \(\frac{\pi \sqrt{ LC }}{6}\)
C \(\frac{\pi \sqrt{ LC }}{2}\)
D \(\pi \sqrt{\frac{ LC }{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{\pi \sqrt{ LC }}{6}\)

Step-by-step Solution

Detailed explanation

\( U_{c_{f}}=75\%U_{c_{i}} \) \( Q_{F}^{2}=\frac{3}{4}Q_{1}^{2} \) \( Q_{i}cos~\omega t=\frac{\sqrt{3}}{2}Q_{i}\Rightarrow t=\frac{T}{12} \) \( t=\frac{\pi}{6}\sqrt{LC} \)

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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