धारा \(I\) वाले एक वृत्ताकार पाश का द्विध्रुव आघूर्ण \(m\) तथा उसके केन्द्र पर चुम्बकीय क्षेत्र \(B_{1}\) है। धारा स्थिर रखते हुए द्विध्रुव आघूर्ण को दोगुना करने पर पाश के केन्द्र पर चुम्बकीय क्षेत्र \(B_{2}\) हो जाता है। अनुपात \(\frac{B_{1}}{B_{2}}\) होगा

A \(\sqrt 3 \)
B \(\sqrt 2 \)
C \(\frac{1}{{\sqrt 2 }}\)
D \(2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\sqrt 2 \)

Step-by-step Solution

Detailed explanation

Magnetic field at the centre of \(\operatorname{loop}, B_{1}=\frac{\mu_{0} I}{2 R}\) Dipole moment of circular loop is \(m=I A\) \(\mathrm{m}_{1}=\mathrm{I} \cdot \mathrm{A}=\mathrm{I} \pi \mathrm{R}^{2} \quad\{\mathrm{R}=\text { Radius of the loop }\}\) If moment is doubled…

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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