Question

A light ray is incident on the surface of a sphere of refractive index n at an angle of incidence $\theta_0$. The ray partially refracts into the sphere with angle of refraction $\phi$ and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is $a$. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

List-I		List-II
P	If $n = 2$ and $\alpha = 180°$, then all the possible values of $\theta_0$ will be	I	$30\degree$ or $0\degree$
Q	If $n = √3$ and $\alpha= 180°$, then all the possible values of $\theta_0$ will be	II	$60\degree$ or $0\degree$
R	If $n = √3$ and $\alpha= 180°$, then all the possible values of $\phi_0$ will be	III	$45\degree$ or $0\degree$
S	If $n = \sqrt2$ and $\theta_0 = 45°$, then all the possible values of  $\alpha$ will be	IV	$150\degree$
			$$0\degree$$

• A. P → 5; Q → 2; R→ 1; S→ 4
• B. P → 5; Q → 1; R→ 2; S→ 4
• C. P → 3; Q → 2; R→ 1; S→ 4
• D. P → 3; Q → 1; R→ 2; S→ 5

Answer 1

The Correct Option is A

Step 1: Calculate for P 

Given $n = 2$ and $\alpha = 180^\circ$,

$$\delta = 2(\theta_0 - \phi_0) + \pi - 2\phi_0 = 2.$$

From the above equation, $\phi_0 = 0^\circ$.

$$\theta_0 = 2\phi_0 = 0^\circ.$$

Thus, $\theta_0 = 0^\circ$ for P.

Step 2: Calculate for Q

Given $n = \sqrt{3}$ and $\alpha = 180^\circ$,

$$1 \cdot \sin \theta_0 = \sqrt{3} \cdot \sin \phi_0.$$

For $\phi_0 = 0^\circ$:

$$\theta_0 = 0^\circ \quad \text{or} \quad \theta_0 = 60^\circ.$$

Step 3: Calculate for R

Given $n = \sqrt{3}$ and $\alpha = 180^\circ$, using Snell’s law:

$$n \cdot \sin \phi_0 = \sin \theta_0.$$

Solving gives $\phi_0 = 0^\circ$ or $\phi_0 = 30^\circ$.

Step 4: Calculate for S

Given $n = \sqrt{2}$ and $\theta_0 = 45^\circ$:

$$\sin \phi_0 = \frac{\sin 45^\circ}{\sqrt{2}} = \frac{1}{2}.$$

The total angle $\alpha$ is:

$$\alpha = 150^\circ.$$

Conclusion

Based on the calculations:

• P → 5
• Q → 2
• R → 1
• S → 4

Final Answer:

The correct option is (A).