A ray of light is incident at `60^(@)` on one face of a prism of angle `30^(@)` and the emergent ray makes `30^(@)` with the incident ray. The refractive index of the prism is

Options

A. `1.732`
B. `1.414`
C. `1.5`
D. `1.33`

Explanation

To find the refractive index of the prism, we can follow these steps:

### Step 1: Identify Given Values
- Angle of the prism (A) = 30°
- Angle of incidence (I) = 60°
- Angle of deviation (Δ) = 30° (since the emergent ray makes 30° with the incident ray)

### Step 2: Use the Relation Between Angles
We know the relation:
\[ I + E = A + \Delta \]
Where:
- I = angle of incidence
- E = angle of emergence
- A = angle of the prism
- Δ = angle of deviation

### Step 3: Rearrange to Find Angle of Emergence (E)
Rearranging the equation gives:
\[ E = A + \Delta - I \]
Substituting the known values:
\[ E = 30° + 30° - 60° \]
\[ E = 0° \]

### Step 4: Understand the Implication of E = 0°
If the angle of emergence (E) is 0°, it implies that the ray emerges parallel to the base of the prism. Therefore, the angle of refraction at the second face (R2) is also 0°.

### Step 5: Use the Relation for R1 and R2
From the geometry of the prism, we know:
\[ A = R1 + R2 \]
Since R2 = 0°, we have:
\[ A = R1 \]
Thus,
\[ R1 = 30° \]

### Step 6: Calculate the Refractive Index (μ)
Using Snell's law at the first face of the prism:
\[ \mu = \frac{\sin I}{\sin R1} \]
Substituting the values:
\[ \mu = \frac{\sin 60°}{\sin 30°} \]

### Step 7: Substitute the Values of Sine
We know:
- \(\sin 60° = \frac{\sqrt{3}}{2}\)
- \(\sin 30° = \frac{1}{2}\)

So,
\[ \mu = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \]
\[ \mu = \sqrt{3} \]

### Step 8: Final Answer
The refractive index of the prism is:
\[ \mu \approx 1.732 \]