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 1st face, Refractive index [latex](\mu) = \frac{Actual\ depth}{Apparent\ depth}[/latex] = [latex]\frac{x}{6}[/latex] ……………… (i) Similarly, for 2nd face, Refractive index [latex](\mu) = \frac{Actual\ depth}{Apparent\ depth}[/latex] = [latex]\frac{12-x}{2}[/latex] ………………. (ii) So, from eqns. (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.
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) = 62.53o (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) = 48.76o.
