Answer
a) $3.0\;\rm mm$
b) $T/4$
c) $\pi/2\;\rm rad$
d) $0.75\;\rm mm$ toward the glass slit
Work Step by Step
$$\color{blue}{\bf [a]}$$
In a double slit experiment, the bright fringes position in the screen is given by
$$y_m=\dfrac{m\lambda L}{d}$$
At $m=1$,
$$y_1=\dfrac{ \lambda L}{d}$$
Plugging the known;
$$y_1=\dfrac{ (600\times 10^{-9})(1.0)}{(0.20\times 10^{-3})}=\bf 0.003\;\rm m$$
$$y_1=\color{red}{\bf 3.0}\;\rm mm$$
$$\color{blue}{\bf [b]}$$
First, we need to find the normal period of the 600 nm light beam then we can find the delay of its period when it passes through the glass.
We know that
$$c=\lambda f$$
where $f$ is the frequency and is given by $1/T$, so
$$c=\dfrac{\lambda }{T}$$
Thus,
$$T=\dfrac{\lambda }{c}=\dfrac{600\times 10^{-9}}{3\times 10^8}=\bf 2 \times 10^{-15}\;\rm s$$
The delay is then given by
$$\Delta t=\dfrac{\Delta t}{T}\cdot T=\dfrac{ 5 \times 10^{-16}}{2 \times 10^{-15}}T$$
Thus,
$$ \Delta t=\color{red}{\bf \frac{1}{4}}T $$
$$\color{blue}{\bf [c]}$$
We know that the wave that passes through the glass will be delayed by $\frac{1}{4}T$ which means that the phase difference is then
$$\Delta \phi_0=\frac{1}{4}(2\pi)=\color{red}{\bf \dfrac{\pi}{2}}\;\rm rad$$
$$\color{blue}{\bf [d]}$$
To have the central maximum, the two rays from the two slits have to reach the screen on phase.
We have here a $\frac{1}{4}T$ delaying due to the glass-slit. So the central maximum will be shifted by some distance $x$ in one direction so the two rays reach the screen in phase. But at what direction should it shift? toward the empty slit or toward the glass slit?
It will shift toward the glass slit since it will travel a shorter distance so both rays will reach the screen in phase and interfere constructively.
Now we need to find this distance $x$,
We know if there is $2\pi$ phase difference between the two rays, then the central maximum will move a distance of 3 mm since there is a bright fringe there. In other words, the central maximum will be just at $m=1$ before putting the glass.
Thus,
$$x=\frac{1}{4}\cdot 3=\color{red}{\bf 0.75}\;\rm mm\tag{Toward the glass slit}$$
