विभवान्तर \(V\) से एक प्रोटॉन तथा एक \(\alpha\)-कण (जिनके द्रव्यमान का अनुपात \(1: 4\) तथा आवेशों का अनुपात \(1: 2\) है) को स्थिरावस्था से त्वरित करते हैं। यदि उनके वेगों के लम्बवत् एक एकसमान चुम्बकीय क्षेत्र \((B)\) लगाया जाये तो इन कणों के वृत्ताकार पथों की त्रिज्याओं का अनुपात \(r _{ p }: r _{\alpha}\) होगा ।

A \(1: \sqrt 2\)
B \(1 : 2\)
C \(1 : 3\)
D \(1: \sqrt 3\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(1: \sqrt 2\)

Step-by-step Solution

Detailed explanation

\({{\text{m}}_p} = {\text{m}}\) \({{\text{q}}_{\text{p}}} = {\text{q}}\) \({{\text{k}}_{\text{p}}} = {\text{qV}} = {\text{k}}\) \({{\text{m}}_\alpha } = {\text{4m}}\) \({q_\alpha } = 2q\) \({k_\alpha } = 2qV = 2k\) Radius of circular path,…

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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