\(d\) બાજુ અને \(\mu_2\) વક્રીભવનાંક ધરાવતા પારદર્શક ઘનને \(\mu_1(\mu_1 < \mu_2)\) વક્રીભવનાંક ધરાવતા પ્રવાહીમાં મુકેલ છે આકૃતિમાં દર્શાવ્યા પ્રમાણે \(AB\) બાજુ પરથી એક પ્રકાશનું કિરણ \(\theta \) ખૂણે આપત કરવામાં આવે છે જે \(BC\) બાજુ પર \(E\) બિંદુ આગળ પૂર્ણ આંતરિક પરાવર્તન પામે છે. આ માટે \(\theta \) નું મૂલ્ય કેટલું હોવું જોઈએ?

A \(\theta > {\sin ^{ - 1}}\,\frac{{{\mu _1}}}{{{\mu _2}}}\)
B \(\theta > {\sin ^{ - 1}}\,\sqrt {\frac{{\mu _2^2}}{{\mu _1^2}} - 1} \)
C \(\theta < {\sin ^{ - 1}}\,\frac{{{\mu _1}}}{{{\mu _2}}}\)
D \(\theta < {\sin ^{ - 1}}\,\sqrt {\frac{{\mu _2^2}}{{\mu _1^2}} - 1} \)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\theta < {\sin ^{ - 1}}\,\sqrt {\frac{{\mu _2^2}}{{\mu _1^2}} - 1} \)

Step-by-step Solution

Detailed explanation

\(\sin \mathrm{c}=\frac{\mu_{1}}{\mu_{2}}\) \(\mu_{1} \sin \theta=\mu_{2} \sin \left(90^o-\mathrm{C}\right)\) \(\sin \theta=\frac{\mu_{2} \sqrt{1-\frac{\mu_{1}^{2}}{\mu_{2}^{2}}}}{\mu_{1}}\) \(\theta=\sin ^{-1} \sqrt{\frac{\mu_{1}^{2}-\mu_{2}^{2}}{\mu_{1}^{2}}}\) For \(TIR\)…

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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