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(A) When a glass plate of thickness $t$ and refractive index $\mu$ is introduced in the path of one of the beams,the path difference introduced is $\D

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$2\lambda$

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(A) When a glass plate of thickness $t$ and refractive index $\mu$ is introduced in the path of one of the beams,the path difference introduced is $\Delta x = (\mu - 1)t$.The shift in the central maximum is given by $\Delta y = \frac{D}{d}(\mu - 1)t$.For the intensity at the original central position to remain unchanged,the new path difference at that point must correspond to a condition of constructive interference,i.e.,$\Delta x = n\lambda$,where $n$ is an integer.Since the intensity remains the same,the path difference must be an integer multiple of the wavelength. For the minimum thickness,we take the smallest non-zero integer $n = 1$.$(\mu - 1)t = 1 \cdot \lambda$Given $\mu = 1.5$,we have $(1.5 - 1)t = \lambda$.$0.5t = \lambda \implies t = 2\lambda$.

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