A microscope is focussed on a coin lying at the bottom of a beaker. The microscope is now raised up by 1cm. To what depth should the water be poured into the beakerso that coin is again in focus? (refractive index of water is 4/3)

Suppose that real depth of water poured into the beaker be $h$ and the microscope is raised upon to 1 cm to focus the coin placed in the beaker. Thus, the apparent depth of the coin becomes $h\text{'}=h-1$

Now, the refractive index in terms of real and apparent depth is given as,

 $$\begin{gathered} n=\frac{real depth}{apperant depth} \\ n=\frac{h}{h\text{'}} \\ n=\frac{h}{h-1} \\ nh-n=h \\ nh-h=n \\ h=\frac{n}{n-1} \end{gathered}$$
on substituting the values in the above equation we get,
 
$$\begin{gathered} h=\frac{{\displaystyle \frac{4}{3}}}{{\displaystyle \frac{4}{3}}-1} \\ h=\frac{{\displaystyle \frac{4}{3}}}{{\displaystyle \frac{1}{3}}} \\ h=\frac{4}{3}\times \frac{3}{1} \\ h=4 cm \end{gathered}$$

Thus, water is poured up to 4 cm in the beaker.