Question

Figure shows a transparent hemisphere of radius 3.0 cm made of a material of refractive index 2.0. (a) A narrow beam of parallel rays is incident on the hemisphere as shown in the figure. Are the rays totally reflected at the plane surface? (b) Find the image formed by the refraction at the first surface. (c) Find the image formed by the reflection or by the refraction at the plane surface. (d) Trace qualitatively the final rays as they come out of the hemisphere.

Answer 1

Given,
The radius of the transparent hemisphere (R) = 3.0 cm
Refractive index of the material (μ₂) = 2.0
Let the critical angle be θ_c
∴ critical angle is given by θ_c = \[\sin^{- 1} \left( \frac{1}{\mu_2} \right) =  \sin^{- 1} \left( \frac{1}{2} \right) = 30^\circ\] 
(a)
From the figure it is seen that the angle of incidence is greater than the critical angle, so the rays are totally reflected at the plane surface.

(b) Using the lens equation:

\[\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}\] 

\[ \Rightarrow \frac{2}{v} - \left( - \frac{1}{\infty} \right) = \frac{2 - 1}{3}  (\text{ for  parallel  rays  u =  \infty })\] 

\[ \Rightarrow \frac{2}{v} = \frac{1}{3}\] 

\[ \Rightarrow v =   6  cm\]

If we complete the sphere then the image will be formed diametrically opposite to A.

(c)
By internal reflection, the image is formed in front of A.