1. A transparent cube of 12 cm edge contains a small air bubble. Its apparent depth when viewed through one face of the cube is 6 cm and when viewed through the opposite face is 2 cm. What is the actual distance of the bubble from the first face?
   Given,
   
   Sides of cube = 12 cm
   Apparent depth from first face = 6 cm
   Apparent depth from second face = 2 cm
   Suppose, actual depth = x cm from first face.
   So, actual depth = (12 – x) cm from the second face.
   Now, for 1^st face,
   Refractive index [latex](\mu) = \frac{Actual\ depth}{Apparent\ depth}[/latex]
         = [latex]\frac{x}{6}[/latex] ……………… (i)
   Similarly, for 2^nd face,
   Refractive index [latex](\mu) = \frac{Actual\ depth}{Apparent\ depth}[/latex]
         = [latex]\frac{12-x}{2}[/latex] ………………. (ii)
   So, from eq^ns. (i) & (ii), we get,
   [latex]\frac{x}{6} = \frac{12 – x}{2}[/latex]
   Or, 2x = 72 – 6x
   Or, 8x = 72
   Or, x = [latex]\frac{72}{8}[/latex]
         = 9 cm.
   So, actual depth of the air bubble = 9 cm from the first face.

2. Calculate the critical angle of (i) glass – water and (ii) water – air interfaces if object lies in the denser medium.
   Solution:
   Given,
   (i) Critical angle of glass – water interface (c) =?
   We know that,
   [latex]\mu_g^w = \frac{\mu_g^a}{\mu_w^a}[/latex]
         = [latex]\frac{1.5}{1.33} = 1.127[/latex]
   Now, [latex]\mu_g^w = \frac{1}{sinc}[/latex]
   Or, 1.127 = [latex]\frac{1}{sinc}[/latex]
   Or, sinc = [latex]\frac{1}{1.127} = 0.887[/latex]
   Or, c = sin^-1(0.887) = 62.53^o
   (ii) Critical angle of water – air interface (c) =?
   We have,
   [latex]\mu_w^a = \frac{1}{sinc}[/latex]
   Or, 1.33 = [latex]\frac{1}{sinc}[/latex]
   Or, sinc = [latex]\frac{1}{1.33} = 0.752[/latex]
   Or, c = sin^-1(0.752)
               = 48.76^o.