प्रकाशिक इलेक्ट्रॉन से सम्बन्धित डि-ब्रॉग्ली तरंगदैर्घ्य \((\lambda)\), आपतित विकिरण की आवृत्ति \((v)\) के साथ इस प्रकार परिवर्तित होती है, \(\left(v_{0}=\right.\) देहली आवृत्ति \()\)

A \(\lambda \propto \frac{1}{{\left( {v - {v_0}} \right)}}\)
B \(\lambda \propto \frac{1}{{{{\left( {v - {v_0}} \right)}^{\frac{1}{4}}}}}\)
C \(\lambda \propto \frac{1}{{{{\left( {v - {v_0}} \right)}^{\frac{3}{2}}}}}\)
D \(\lambda \propto \frac{1}{{{{\left( {v - {v_0}} \right)}^{\frac{1}{2}}}}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\lambda \propto \frac{1}{{{{\left( {v - {v_0}} \right)}^{\frac{1}{2}}}}}\)

Step-by-step Solution

Detailed explanation

\(\lambda = \frac{h}{{mv}}\) According to Einstein's theory of photoelectric effect: \(hv = h{v_0} + KE\) \(hv = h{v_0} + \frac{1}{2}m{v^2}\) \(2h(v - {v_0}) = m{v^2}\) \(\frac{{2h(v - {v_0})}}{m} = {v^2}\) \(v \propto {(v - {v_0})^{\frac{1}{2}}}\)…

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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