હાઇડ્રોજન પરમાણુની બીજી ઉત્તેજિત અવસ્થામાં રહેલા ઇલેક્ટ્રોનની દ’બ્રોગલી તરંગલંબાઈ\(({\lambda _B})\) અને તેની ધરા અવસ્થામાં રહેલા ઇલેક્ટ્રોનની દ’બ્રોગલી તરંગલંબાઈ\(({\lambda _G})\) વચ્ચેનો સંબંધ શું હશે?

A \({\lambda _B}\, = \,{\lambda _{G}}/3\)
B \({\lambda _B}\, = \,{\lambda _{G}}/2\)
C \({\lambda _B}\, = \,{2\lambda _{G}}\)
D \({\lambda _B}\, = \,{3\lambda _{G}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \({\lambda _B}\, = \,{3\lambda _{G}}\)

Step-by-step Solution

Detailed explanation

de-Broglie wavelength, \(\lambda=\frac{h}{P}\) \(\frac{\lambda_{\mathrm{B}}}{\lambda_{\mathrm{G}}}=\frac{\mathrm{P}_{\mathrm{G}}}{\mathrm{P}_{\mathrm{B}}}=\frac{\mathrm{mv}_{\mathrm{G}}}{\mathrm{mv}_{\mathrm{B}}}\) Speed of electron \(v \propto \frac{z}{n}\) so…

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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