બે તારા પૃથ્વીથી \(10\) પ્રકાશવર્ષ દૂર છે. જેને \(30\, cm\) વ્યાસવાળા ટેલિસ્કોપ વડે જોવામાં આવે છે. આ બંને તારાને ટેલિસ્કોપ વડે અલગ જોવા માટે તેમની વચ્ચેનું ઓછામાં ઓછું અંતર ક્યાં ક્રમનું હોવું જોઈએ? (\(1\) પ્રકાશવર્ષ \(= 9.46 \times 10^{15}\, m\))

A \(10^8\,km\)
B \(10^{10}\,km\)
C \(10^{11}\,km\)
D \(10^6\,km\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(10^8\,km\)

Step-by-step Solution

Detailed explanation

We know that \(\Delta \theta=\frac{1.22 \lambda}{D}=\frac{l}{R}\) The minimum distance between them \(l=\frac{9.46 \times 10^{15} \times 10 \times 1.22 \times 600 \times 10^{-9}}{0.3}\) \(=2308.24 \times 10^{8}\, \mathrm{m}\) \(\Rightarrow 2.308 \times 10^{8}\, \mathrm{km}\)

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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