वायु में \(\lambda_1\) तरंगदैर्ध्य एवं \(v_1\) आवृत्ति की एक एकवर्णी प्रकाश तरंग दूसरे माध्यम में प्रवेश करती है। यदि अन्तः सतह पर आपतन कोण तथा अपवर्तन कोण क्रमशः \(45^{\circ}\) व \(30^{\circ}\) हैं। तब अपवर्तित तरंग की तरंगदैर्ध्य \(\lambda_2\) व आवृत्ति \(v_2\) है:

A \(\lambda_2=\lambda_1, v_2=\sqrt{2} v_1\)
B \(\lambda_2=\frac{1}{\sqrt{2}} \lambda_1, v_2=v_1\)
C \(\lambda_2=\sqrt{2} \lambda_1, v_2=v_1\)
D \(\lambda_2=\lambda_1, v_2=\frac{1}{\sqrt{2}} v_1\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\lambda_2=\frac{1}{\sqrt{2}} \lambda_1, v_2=v_1\)

Step-by-step Solution

Detailed explanation

Snell's law \(\mu_1 \sin 45^{\circ}=\mu_2 \sin 30^{\circ}\) \(\frac{\mu_1}{\mu_2}=\frac{1}{\sqrt{2}}\) \(\Rightarrow \frac{\mu_1}{\mu_2}=\frac{\lambda_2}{\lambda_1}=\frac{1}{\sqrt{2}}\) \(\Rightarrow \lambda_2=\frac{\lambda_1}{\sqrt{2}}\) Frequency doesn't change on change in…

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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