For an equilateral prism, it is observed that when a ray strikes grazingly at one face it emerges grazingly at the other. Its refractive index will be

Options

A. `sqrt(3)`
B. `(2)/(sqrt(3))`
C. 2
D. Data not sufficient

Answer

To find the refractive index of an equilateral prism when a ray strikes grazingly at one face and emerges grazingly at the other, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry of the Prism**:
- An equilateral prism has an angle of \( A = 60^\circ \).

2. **Identify the Incident and Emergent Angles**:
- When a ray strikes the prism grazingly, the angle of incidence \( i \) at the first face is \( 90^\circ \).
- Since it emerges grazingly at the other face, the angle of emergence \( e \) is also \( 90^\circ \).

3. **Apply Snell's Law at the First Face**:
- According to Snell's Law:
\[ n_1 \sin i = n_2 \sin r_1 \]
- Here, \( n_1 = 1 \) (the refractive index of air), \( i = 90^\circ \), and \( r_1 \) is the angle of refraction at the first face.
- Since \( \sin 90^\circ = 1 \), we have:
\[ 1 \cdot 1 = n \sin r_1 \implies n = \sin r_1 \]

4. **Apply Snell's Law at the Second Face**:
- At the second face, we apply Snell's Law again:
\[ n \sin r_2 = 1 \cdot \sin e \]
- Here, \( e = 90^\circ \), so:
\[ n \sin r_2 = 1 \cdot 1 \implies n = \frac{1}{\sin r_2} \]

5. **Use the Prism Property**:
- For a prism, the sum of the angles of refraction at both faces is equal to the angle of the prism:
\[ r_1 + r_2 = A \]
- Since \( A = 60^\circ \), we have:
\[ r_1 + r_2 = 60^\circ \]

6. **Since \( r_1 = r_2 \)**:
- Let \( r_1 = r_2 = r \). Therefore:
\[ r + r = 60^\circ \implies 2r = 60^\circ \implies r = 30^\circ \]

7. **Calculate the Refractive Index**:
- From the earlier equations, we can substitute \( r \):
\[ n = \sin r_1 = \sin 30^\circ = \frac{1}{2} \]
- Thus, the refractive index \( n \) is:
\[ n = 2 \]

### Final Answer:
The refractive index of the equilateral prism is \( n = 2 \).