MHT CET · Physics · Wave Optics

A single slit of width \(d\) is illuminated by violet light of wavelength 400 nm and the width of the diffraction pattern is measured as ' \(Y\) '. When half of the slit width is covered and illuminated by yellow light of wavelength 600 nm , the width of the diffraction pattern is

A zero
B \(\frac{\mathrm{Y}}{3}\)
C 3 Y
D 4 Y

Verified Solution

Answer & Solution

Correct Answer

(C) 3 Y

Step-by-step Solution

Detailed explanation

Width of diffraction pattern,
\(\mathrm{W}=\frac{2 \lambda \mathrm{D}}{\mathrm{~d}}\)
When half the slit is covered, \(\mathrm{d}^{\prime}=\frac{\mathrm{d}}{2}\) and \(\lambda^{\prime}=600 \mathrm{~nm}\)
\(\begin{array}{ll}
\therefore & \mathrm{W}^{\prime}=\frac{2 \lambda^{\prime}}{\mathrm{d} / 2} \\
\therefore & \frac{\mathrm{~W}^{\prime}}{\mathrm{W}}=\frac{2 \lambda^{\prime}}{\lambda}=\frac{2 \times 600}{400} \\
\therefore & \frac{\mathrm{~W}^{\prime}}{\mathrm{W}}=3 \\
\therefore & \mathrm{~W}^{\prime}=3 \mathrm{~W}=3 \mathrm{Y}
\end{array}\)
...(given, \(\mathrm{W}=\mathrm{Y}\) )

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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