Explanation

(a) When a mica sheet is introduced in the path of one of the two interfering beams, then entire fringe pattern is displaced towards the beam is the path of which plate is introduce, but fringe width is not changed.
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$x_{0} = \frac{D}{d}(\mu - 1)t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(i)$
Also fringe width is
$W = \frac{D\lambda}{d}$
$\therefore\ \frac{W}{\lambda} = \frac{D}{d}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(ii)$
Using Eq. (ii) we get Eq. (i) as
$x_{0} = \frac{W}{\lambda}(\mu - 1)t$
Given, $x_{0} = 1.89 \times 10^{- 3}m,\ W = 0.431 \times 10^{- 3}m,$
$\mu = 1.59,\ \ \ \ \ \lambda = 5.89 \times 10^{- 7}m.$
$\ \ 1.89 \times 10^{- 3} = \frac{0.431 \times 10^{- 3}}{5.89 \times 10^{- 7}}(1.59 - 1)t$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t = \frac{5.89 \times 10^{- 7} \times 1.89 \times 10^{- 3}}{0.431 \times 0.59 \times 10^{- 3}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t = 4.38 \times 10^{- 6}m$