બહિર્ગોળ લેન્સની કેન્દ્રલંબાઈ \((f)\) માપવાના એક પ્રયોગમાં, વસ્તુના સ્થાન \((u)\) અને પ્રતિબિંબના સ્થાન \((v)\) માટે માપવાના માપપટ્ટીના લઘુતમ માપ અનુક્રમે \(\Delta u\) અને \(\Delta v\) છે. બહિર્ગોળ લેન્સની કેન્દ્રલંબાઈના માપનમાં ત્રુટિ _______ હશે.

A \(\frac{\Delta \mathrm{u}}{\mathrm{u}}+\frac{\Delta \mathrm{v}}{\mathrm{v}}\)
B \(f^2\left[\frac{\Delta u}{u^2}+\frac{\Delta v}{v^2}\right]\)
C \(2 \mathrm{f}\left[\frac{\Delta \mathrm{u}}{\mathrm{u}}+\frac{\Delta \mathrm{v}}{\mathrm{v}}\right]\)
D \(f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(f^2\left[\frac{\Delta u}{u^2}+\frac{\Delta v}{v^2}\right]\)

Step-by-step Solution

Detailed explanation

\(f^{-1}=v^{-1}-u^{-1}\) \(-f^{-2} d f=-v^{-2} d v-u^{-2} d u\) \(\frac{d f}{f^2}=\frac{d v}{v^2}+\frac{d u}{u^2}\) \(d f=f^2\left[\frac{d v}{v^2}+\frac{d u}{u^2}\right]\)

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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