MHT CET · Physics · Wave Optics

In a diffraction pattern due to single slit of width 'a', the first minimum is observed at an angle of \(30^{\circ}\) when the light of wavelength \(5400 Å\) is incident on the slit. The first secondary maximum is observed at an angle of \(\left(\sin 30^{\circ}=\frac{1}{2}\right)\)

A \(\sin ^{-1}\left(\frac{3}{4}\right)\)
B \(\sin ^{-1}\left(\frac{2}{3}\right)\)
C \(\sin ^{-1}\left(\frac{1}{2}\right)\)
D \(\sin ^{-1}\left(\frac{1}{4}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\sin ^{-1}\left(\frac{3}{4}\right)\)

Step-by-step Solution

Detailed explanation

For \(\mathrm{n}^{\text {th }}\) secondary minimum, path difference \(=a \sin \theta_{\mathrm{n}}=\mathrm{n} \lambda\) For \(\mathrm{n}^{\text {th }}\) secondary maximum, path difference \(=a \sin \theta_n=(2 n+1) \frac{\lambda}{2}\)
\(\therefore \quad\) For \(1^{\text {st }}\) minimum, a \(\sin 30^{\circ}=\lambda\)
For \(2^{\text {nd }}\) maximum, a \(\sin \theta_{\mathrm{n}}=(2+1) \frac{\lambda}{2}=\frac{3 \lambda}{2}\)
\(\therefore \quad\) Dividing equation (i) by equation (ii),
\(\frac{\left(\frac{1}{2}\right)}{\sin \theta_n}=\frac{2}{3} \Rightarrow \theta_n=\sin ^{-1}\left(\frac{3}{4}\right)\)

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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