વર્ણપટ રેખાઓની પાશ્ચન શ્રેણીમાં ટૂંકામાં ટૂંકી કઈ તરંગલંબાઈ હાજર છે ?

A 840 nm
B 320 nm
C 720 nm
D 820 nm

Verified Solution

Answer & Solution

Correct Answer

(D) 820 nm

Step-by-step Solution

Detailed explanation

D
પાશ્ચન શ્રેણી માટે,
\(\frac{1}{\lambda}=R\left[\frac{1}{3^2}-\frac{1}{n^2}\right]\)
\(n=4,5,6 \ldots\) પરથી,
ટૂંકામાં ટૂંકી તરંગલંબાઈ માટે
\(n=\infty\) લેતા,
\(\frac{1}{\lambda_{\min }}=\mathrm{R}\left[\frac{1}{3^2}-\frac{1}{\infty^2}\right]\)
\(\lambda_{\text {min }}=\frac{9}{R}\)
\(\therefore \lambda_{\min }=\frac{9}{1.097 \times 10^{-7}}\)
\(\begin{aligned} & \therefore \lambda_{\min } \approx 820.41 \times 10^{-9} \mathrm{~m} \\ & \therefore \lambda_{\min } \approx 820 \mathrm{~nm}\end{aligned}\)

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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