(a) Describe any two characteristic features which distinguish between interference and diffraction phenomena. Derive the expression for the intensity at a point of the interference pattern in Young’s double slit experiment.
(b) In the diffraction due to a single slit experiment, the aperture of the slit is 3 mm. If monochromatic light of wavelength 620 nm is incident normally on the slit, calculate the separation between the first order minima and the 3rd order maxima on one side of the screen. The distance between the slit and the screen is 1.5 m.
OR 
(a) Under what conditions is the phenomenon of total internal reflection of light observed?
Obtain the relation between the critical angle of incidence and the refractive index of the medium.
(b) Three lenses of focal length +10 cm, -10 cm and +30 cm are arranged coaxially as in the figure given below. Find the position of the final image formed by the combination.

Separation between them.

A sources of monochromatic light illuminates two narrow slits S₁ and S₂.The two illuminated slits act as the two coherent sources. The two slits is very close to each other and at equal distance from source. The wave front S₁ and S₂ spread in all direction and superpose and produces dark and bright fringe on screen. Let the displacement of waves from Si and S₂ at point P on screen at time t is

$\begin{array}{l}{\mathrm{y}}_1={\mathrm{a}}_1\sin \mathrm{\omega t}\\ {\mathrm{y}}_2={\mathrm{a}}_2\sin (\mathrm{\omega t}+\mathrm{\varphi })\end{array}$
The resultant displacement at point P is given by
$\begin{array}{l}\mathrm{y}={\mathrm{y}}_1+{\mathrm{y}}_2\\ ={\mathrm{a}}_1\sin \mathrm{\omega t}+{\mathrm{a}}_2\sin (\mathrm{\omega t}+\mathrm{\varphi })\\ ={\mathrm{a}}_1,\sin \mathrm{\omega t}+{\mathrm{a}}_2\sin \mathrm{\omega t}\\ \cos \mathrm{\varphi }+{\mathrm{a}}_2\cos \mathrm{\omega tsin}\mathrm{\varphi }\\ =\left({\mathrm{a}}_1+{\mathrm{a}}_2\cos \mathrm{\varphi }\right)\sin \mathrm{\omega t}\\ +{\mathrm{a}}_2\sin \mathrm{\varphi cos}\mathrm{\omega t}\dots .\left(\mathrm{i}\right)\end{array}$
Let 
$\begin{array}{l}{\mathrm{a}}_1+{\mathrm{a}}_2\cos \mathrm{\varphi }=\mathrm{Acos}\mathrm{\varphi } ....\left(\mathrm{ii}\right)\\ {\mathrm{a}}_2\sin \mathrm{\varphi }=\mathrm{Asin}\mathrm{\varphi } ...\left(\mathrm{iii}\right)\end{array}$
Therefore, equation (i) becomes
$\begin{array}{l}\mathrm{y}=\mathrm{Acos}\mathrm{\theta sin}\mathrm{\omega t}+\mathrm{Asin}\mathrm{\theta }\\ =\mathrm{Asin}(\mathrm{\omega t}+\mathrm{\theta })\end{array}$
This is the resultant displacement.
Now, squaring and adding equations (ii) and (iii)
$\begin{array}{l}{\mathrm{A}}^2{\cos }^2\mathrm{\theta }+{\mathrm{A}}^2{\sin }^2\mathrm{\theta }={\left({\mathrm{a}}_1+{\mathrm{a}}_2\cos \mathrm{\varphi }\right)}^2+{\mathrm{a}}_2^2{\sin }^2\mathrm{\varphi }\\ {\mathrm{A}}^2={\mathrm{a}}_1^2+{\mathrm{a}}_2^2+2{\mathrm{a}}_1{\mathrm{a}}_2\cos \mathrm{\varphi }\end{array}$
The is intensity of light is directly proportional to the square of the amplitude
ie., $\mathrm{I}={\mathrm{a}}_1^2+{\mathrm{a}}_2^2+2{\mathrm{a}}_1{\mathrm{a}}_2\cos \mathrm{\varphi }$
This is the expression for intensity at a point of interference pattern.
or   $\mathrm{I}={\mathrm{I}}_1+{\mathrm{I}}_2+2\sqrt{{\mathrm{I}}_1{\mathrm{I}}_2}\cos \mathrm{\varphi }$
(b) Here, $\mathrm{\lambda }=620\mathrm{nm}=620\times {10}^{-9}\mathrm{m}$
$\mathrm{a}=3\times {10}^{-3}\mathrm{m},\mathrm{D}=1.5\mathrm{m}$
Distance of first order minima from the centre,
$\begin{array}{l}{\mathrm{y}}_1=\frac{\mathrm{D\lambda }}{\mathrm{a}}=\frac{1.5\times 620\times {10}^{-9}}{3\times {10}^{-3}}\\ =3.1\times {10}^{-4}\mathrm{m}\end{array}$
Distance of third order maxima on the same side, 
$\begin{array}{r}{\mathrm{y}}_2=\frac{7\mathrm{D\lambda }}{2\mathrm{a}}=\frac{7\times 1.5\times 620\times {10}^{-9}}{2\times 3\times {10}^{-3}}\\ =10.85\times {10}^{-4}\mathrm{m}\end{array}$

OR

Total internal reflection is the phenomenon of reflection of light into a denser medium from an interface of the denser medium and the rarer medium. Two essential conditions for total internal reflection: Incident ray should travel in the denser medium and refracted ray should travel in the rarer medium. 

The angle of incidence (i) should be greater than the critical angle for the pair of media in contact. 

The relation between refractive index and critical angle (C): 

 $$\begin{gathered} \text{ When }\mathrm{i}=\mathrm{C}\text{ and }\mathrm{r}=90 \\ \frac{{\mu }_{\mathrm{b}}}{{\mu }_{\mathrm{a}}}=\frac{1}{\sin \mathrm{C}} \\ {a}_{{\mu }_b}=\frac{1}{\sin C} \end{gathered}$$