जब \(\lambda_{1}\) तरंगदैर्ध्य के फोटान एक विलगित गोले को प्रदीप्त करते हैं, तो संगत 'निरोधी-विभव' का मान \(V\) पाया जाता है। जब \(\lambda_{2}\) तरंगदैर्ध्य के फोटान उपयोग में लाये जाते हैं तो निरोधी-विभव का मान तिगुना \((3\;V)\) हो जाता है। अगर \(\lambda_{3}\) तरंगदैर्ध्य के फोटान से गोले को प्रदीप्त किया जाए तो निरोधी-विभव का मान होगा

A \(\frac{{hc}}{e}\left[ {\frac{1}{{{\lambda _3}}} + \frac{1}{{{\lambda _2}}} - \frac{1}{{{\lambda _1}}}} \right]\)
B \(\frac{{hc}}{e}\left[ {\frac{1}{{{\lambda _3}}} + \frac{1}{{{2\lambda _2}}} - \frac{1}{{{\lambda _1}}}} \right]\)
C \(\frac{{hc}}{e}\left[ {\frac{1}{{{\lambda _3}}} - \frac{1}{{{\lambda _2}}} - \frac{1}{{{\lambda _1}}}} \right]\)
D \(\frac{{hc}}{e}\left[ {\frac{1}{{{\lambda _3}}} + \frac{1}{{{2\lambda _2}}} - \frac{3}{{{2\lambda _1}}}} \right]\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{{hc}}{e}\left[ {\frac{1}{{{\lambda _3}}} + \frac{1}{{{2\lambda _2}}} - \frac{3}{{{2\lambda _1}}}} \right]\)

Step-by-step Solution

Detailed explanation

From Einstein's photoelectric equation, we have \(\frac{h c}{\lambda_{1}}=\frac{h c}{\lambda_{0}}+e V\) ..... \((1)\) \(\frac{\mathrm{hc}}{\lambda_{2}}=\frac{\mathrm{hc}}{\lambda_{0}}+3 \mathrm{eV}\) ..... \((2)\)…

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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