(a) (i) A ray of light passes through a triangular prism. Show graphically, how the angle of deviation varies with the angle of incidence? Hence…

Minimum deviation angle is defined as the angle at which angle of incidence is equal to the angle of emergence.

Alternatively

At minimum deviation refracted ray inside the prism becomes parallel to the base of the prism.

At the face XZ :-

$$\mu \sin i = 1 \times \sin r \qquad \text{-----(1)}$$

$$r = i + \delta \qquad [\text{from diagram}] \qquad \text{-----(2)}$$

In $\displaystyle \Delta XMN$ ; $\displaystyle A + (90 - i) + 90 = 180$

$$\Rightarrow A = i \qquad \text{-----(3)}$$

Putting eq. ($\displaystyle 3$) & ($\displaystyle 2$) in eq. ($\displaystyle 1$)

$$\mu \sin A = \sin (A + \delta)$$

$$\mu = \frac{\sin (A + \delta)}{\sin A}$$

(1)

$$\mu = \frac{\sin\left(\dfrac{A + \delta_m}{2}\right)}{\sin \dfrac{A}{2}}$$

$$\sqrt{2} = \frac{\sin\left(\dfrac{60 + \delta_m}{2}\right)}{\sin 30^\circ}$$

$$\Rightarrow \sin\left(\frac{60 + \delta_m}{2}\right) = \frac{1}{\sqrt{2}} = \sin 45^\circ$$

$$\frac{60 + \delta_m}{2} = 45^\circ \Rightarrow \delta_m = 30^\circ$$

$$i = \frac{A + \delta_m}{2}$$

$$\Rightarrow i = \frac{60 + 30}{2}$$

$$i = 45^\circ$$

Each point of the wavefront is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the spread of the wave. Each point of the wavefront is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the speed of the wave. These wavelets emanating from the wavefront are usually referred to as secondary wavelets and if we draw a common tangent to all these spheres, we obtain the new position of the wavefront at a later time.

$\displaystyle \Delta EAC$ is congruent to $\displaystyle \Delta BAC$; so $\displaystyle \angle i = \angle r$

Two sources are said to be coherent if the phase difference between them does not change with time.

No, two independent sodium lamps cannot be coherent.

Two independent sodium lamps cannot be coherent as the phase between them does not remain constant with time.

$$4\beta_2 = 5\beta_1$$

$$4 \times \frac{\lambda D}{d} = 5 \times \frac{\lambda_{known} D}{d}$$

$$\Rightarrow \lambda = \frac{5}{4} \times \lambda_{known}$$

$$= \frac{5}{4} \times 520$$

$$= 650 \text{ nm}$$