(i) Deduce the conditions for a) constructive and b) destructive interference Graph showing the variation of intensity (ii) Three distinguishing features (i)From figure Path difference $= (S_2 P - S_1 P)$ $\; (S_2 P)^2 - (S_1 P)^2 = \left[D^2 + \left(x + \frac{d}{2}\right)^2\right]$ $- \left[D^2 + \left(x - \frac{d}{2}\right)^2\right]$ $\; (S_2 P + S_1 P) (S_2 P - S_1 P) = 2 x d$ $S_2 P - S_1 P = \frac{2 x d}{S_2 P + S_1 P}$ For $x, d \ll D$ $\;S_2 P + S_1 P = 2 D$ $\therefore \;\; S_2 P - S_1 P = \frac{2 x d}{2 D} = \frac{x d}{D}$ For constructive interference $S_2 P - S_1 P = n \lambda$ , $n = 0, 1, 2 \dots$ $\Rightarrow \frac{x d}{D} = n \lambda$ $\Rightarrow x = \frac{n \lambda D}{d}$ For destructive interference $S_2 P - S_1 P$ $= (2n+1) \frac{\lambda}{2}, n = 0, 1, 2 \dots$ $\frac{x d}{D} = (2n+1) \frac{\lambda}{2}$ $\Rightarrow x = (2n+1) \frac{\lambda D}{2 d}$ (ii) (a) The Interference pattern has number of equally spaced bright and dark bands, while in the diffraction pattern the width of the central maximum is twice the width of other maxima. (b) In Interference all bright fringes are of equal intensity, whereas in the diffraction pattern the intensity falls as order of maxima increases. (c) In Interference pattern, maxima occurs at angle $\frac{\lambda}{a}$ , where $a$ is the slit width, whereas in diffraction pattern, at the same angle, first minimum occurs. (Here ' $a$ ' is the size of the slit) OR(i) Plot showing the variation of the angle of deviation as a function of angle of incidence Derivation of expression of refractive index (ii) Definition of dispersion and its cause (iii) Calculation of minimum value of refractive index (i) From figure $\delta = D_m, i = e$ which implies $r_1 = r_2$ $2 r = A, \text{ or } r = \frac{A}{2}$Using $\delta = i + e - A$ $D_m = 2 i - A$ $i = \frac{A + D_m}{2}$ $\mu = \frac{\sin i}{\sin r} = \frac{\sin \left(\frac{A + D_m}{2}\right)}{\sin \frac{A}{2}}$ (ii) The phenomenon of splitting of white light into its constituent colours. Cause : Refractive index of the material is different for different colours. According to the equation, $\delta \text{ not } (\mu - 1) A$ , where $A$ is the angle of prism, different colours will deviate through different amount. For total internal reflection, $\angle i \geq \angle i_c \text{ (critical angle) }$ $\Rightarrow 45^{\circ} \geq \angle i_c, \text{ i.e.}, \angle i_c \leq 45^{\circ}$ $\sin i_c \leq \sin 45^{\circ}$ $\leq \frac{1}{\sqrt{2}}$ $\frac{1}{\sin i_c} \geq \sqrt{2}$ $\Rightarrow \mu \geq \sqrt{2}$ Hence, the minimum value of refractive index must be $\sqrt{2}$