A parallel beam of light travelling in water (refractive index = 4/3) is refracted by a spherical air bubble of radius 2 mm situated in water. Assuming the light rays to be paraxial, (a) find the position of the image due to refraction at the first surface and the position of the final image and (b) draw a ray diagram showing the positions of both the images.

Refractive index of water = 4/3 
Radius of spherical air bubble, r = 2 mm 

(a) For refraction at the first surface, using the formula,
                  $\frac{{\mathrm{\mu }}_2}{\mathrm{v}}-\frac{{\mathrm{\mu }}_1}{\mathrm{u}} = \frac{{\mathrm{\mu }}_2-{\mathrm{\mu }}_1}{\mathrm{R}}$
where${\mathrm{\mu }}_1 = 4/3, {\mathrm{\mu }}_2 = 1 \mathrm{and} \mathrm{u} = \infty \mathrm{and} \mathrm{R} = 2 \mathrm{mm}.$

Thus,

                 $\frac{1}{{\mathrm{v}}_1}-\frac{4/3}{\infty } = \frac{1-4/3}{2}$
which gives,  v₁ = –6 mm.
Negative sign indicates that the image I₁ is virtual and is on the same side as the object at a distance of 6 mm from the first surface. 

For refraction at the second surface, the image I₁ serves as the virtual object which is at a distance of 6 mm + 4 mm = 10 mm from the second surface.

For refraction to take place, we use
                $\frac{{\mathrm{\mu }}_1}{{\mathrm{v}}_2}-\frac{{\mathrm{\mu }}_2}{\mathrm{u}} = \frac{{\mathrm{\mu }}_1-{\mathrm{\mu }}_2}{\mathrm{R}}$
where,
$$\begin{gathered} \mathrm{object} \mathrm{distance}, \mathrm{u} = -10 \mathrm{mm} \mathrm{and} \\ Radius of curvature, R = -2 \mathrm{mm}. \end{gathered}$$ 

Thus, substituting the values in the formula, we get

                 $\frac{4/3}{{\mathrm{v}}_2}-\frac{1}{-10} = \frac{4/3-1}{(-2)}$ 

which given v₂ = –5 mm.  

The final image I₂ is virtual and is formed at a distance of 5 mm from the second surface to the left of the second surface, i.e., the final image is formed at a distance of 1 mm from the first surface.

(b) Figure given below shows the ray diagram.
                                 $\frac{{\mathrm{\mu }}_1}{{\mathrm{v}}_2}-\frac{{\mathrm{\mu }}_2}{\mathrm{u}} = \frac{{\mathrm{\mu }}_1-{\mathrm{\mu }}_2}{\mathrm{R}}$