अण्डाकार आकार की एक कुण्डली के अर्धप्रमुख (semi major) अक्ष की लम्बाई \(a\) तथा अर्धलघु (semi minor) अक्ष की लम्बाई \(b\) है तथा इसका प्रतिरोध \(R\) है। इसे एक चुम्कीय क्षेत्र में चित्रानुसार रखा गया है। यदि \(x\)-अक्ष के चारों ओर इसे \(\omega\) कोणीय वेग से घुमाया जाता है तो कुण्डली में होने वाली जूल हीटिंग से इसमें शक्ति की औसत क्षति होगी।

A \(\frac{\pi^{2} a ^{2} b ^{2} B ^{2} \omega^{2}}{2 R }\)
B Zero
C \(\frac{\pi^{2} a^{2} b^{2} B^{2} \omega^{2}}{R}\)
D \(\frac{\pi abB \omega}{ R }\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{\pi^{2} a ^{2} b ^{2} B ^{2} \omega^{2}}{2 R }\)

Step-by-step Solution

Detailed explanation

\(\in= NAB \omega \cos \omega t \quad[ N =1]\) \(P _{ avg }=<\frac{\epsilon^{2}}{ R }>=<\frac{( AB \omega \cos \omega t )^{2}}{ R }>\) \(=\frac{ A ^{2} B ^{2} \omega^{2}}{ R } \frac{1}{2}=\frac{\pi^{2} a ^{2} b ^{2} B ^{2} \omega^{2}}{2 R }\)

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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