One of the refracting surfaces of a prism of angle of `30^(@)` is silvered. A ray of light incident at an angle of `60^(@)` retraces its path. The refractive index of the material of prism is

Options

A. `sqrt(2)`
B. `sqrt(3)`
C. `3//2`
D. 2

Answer

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

### Step 1: Understand the Geometry of the Prism
We have a prism with an apex angle \( A = 30^\circ \). One of its refracting surfaces is silvered, which means it acts as a mirror. A ray of light is incident on the prism at an angle of \( 60^\circ \).

### Step 2: Analyze the Path of the Ray
Since the ray retraces its path after reflecting off the silvered surface, we can conclude that the angle of incidence at the silvered surface must be such that the angle of refraction is \( 90^\circ \). This means that the angle of incidence at the silvered surface must be equal to the angle of refraction.

### Step 3: Determine the Angles
Let's denote:
- The angle of incidence at the first surface of the prism as \( i = 60^\circ \).
- The angle of refraction at the first surface as \( r \).
- The angle of incidence at the silvered surface as \( r' \).

Since the prism has an apex angle of \( 30^\circ \), we can use the relationship:
\[ r + r' = A = 30^\circ \]
From the geometry, we know that if the ray retraces its path, then:
\[ r' = 90^\circ \]
Thus, we can find \( r \):
\[ r + 90^\circ = 30^\circ \implies r = 30^\circ - 90^\circ = -60^\circ \]
This indicates that the ray is actually refracted back at the silvered surface.

### Step 4: Apply Snell's Law
Using Snell's law at the first surface of the prism:
\[ \sin(i) = \mu \cdot \sin(r) \]
Substituting the known values:
\[ \sin(60^\circ) = \mu \cdot \sin(30^\circ) \]
Using the values:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \]
We can substitute these into the equation:
\[ \frac{\sqrt{3}}{2} = \mu \cdot \frac{1}{2} \]
### Step 5: Solve for the Refractive Index \( \mu \)
To find \( \mu \):
\[ \mu = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \]

### Conclusion
The refractive index of the material of the prism is \( \sqrt{3} \).

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