b
(b)

The path difference when transparent sheet is introduced \(\Delta x=(\mu-1) t\)

If the central maxima occupies position of nth fringe, then \((\mu-1) t=n \lambda=d \sin \theta\)

\(\Rightarrow \sin \theta  =\frac{(\mu-1) t }{ d }\)

\(=\frac{(1.17-1) \times 1.5 \times 10^{-7}}{3 \times 10^{-7}}=0.085\)

Therefore, angular position of central maxima \(\theta=\sin ^{-1}(0.085)=4.88^{\circ} \approx 4.9\)

For small angles, \(\sin \theta \approx \theta \approx \tan \theta\)

\(\Rightarrow \quad \tan \theta=\frac{y}{D}\)

\(\therefore \quad \frac{y}{D}=\frac{(\mu-1) t}{d} \Rightarrow y=\frac{D(\mu-1) t}{d}\)