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.
Figure

Answer 1

The image given in the question is attached as image 1.

(a) Yes, the rays are totally reflected at the plane surface.

From question, the refractive index of material = μ₂ = 0.2

$\text { critical angle }=\sin ^{-1}\left(\frac{1}{\mu_{2}}\right)$

$\text { critical angle }=\sin ^{-1}\left(\frac{1}{2}\right)$

$\therefore \text { critical angle }=30^{\circ}$

Since, the critical angle is less than incidence angle. The rays undergo total internal reflection.

(b) The image formed by the refraction at the first surface is 6 cm.

The formula used is given below:

$\frac{\mu_{2}}{v}-\frac{\mu_{1}}{u}=\frac{\mu_{2}-\mu_{1}}{R}$

For parallel rays, u = ∞

$\frac{2}{v}-\left(-\frac{1}{\infty}\right)=\frac{2-1}{3}$

$\frac{2}{v}=\frac{1}{3}$

$\therefore v = 6 \ cm$

(c) The image formed by the reflection or by the refraction at the plane surface is in front of the A.

Since, the radius of the hemisphere is 3 cm and the distance of image is 6 cm which is exactly opposite. So, the image is formed in front of the incidence.

(d) Trace qualitatively the final rays as they come out of the hemisphere is given in image 2.

Answer 2

(a) The rays are totally internally mirrored at the plane surface.

(b) The image formed by the refraction at the first surface is diametrically opposite to A.

(c) The image formed by the internal reflection at the mirror.

(d) Trace of the diagram.

Explanation :              

Given data in the question  :      

(a). As the angle of incidence is greater than that of critical angle, the rays are entirely internally mirrored.

Refractive index $\mu_{2}$ = 2.0      

                             = -1

For parallel ray u = ∞

R = 3 radius of curvature  

So critical angle $=\sin ^{-1} \frac{1}{\mu_{2}}$

                          $=\sin ^{-1} \frac{1}{2}$

                          = 30°

(b). Here

We know that,  

                     $\frac{\mu_{2}}{v}-\frac{\mu_{1}}{u}=\frac{\mu_{2}-\mu_{1}}{R}$

                  $\frac{2}{v}-\left(\frac{-1}{\infty}\right)=\frac{2-1}{3}$    

                  $\frac{2}{v}-\left(\frac{-1}{\infty}\right)=\frac{1}{3}$

                                 $\frac{2}{v}=\frac{1}{3}$                  

                                  v = 6          

When the sphere is full, picture is shaped diametrically opposite to A.

(c) The picture is created by internal reflection at the mirror before A.

(d) Refer the diagram for better understanding.