The angle of incidance for an equilateral prism is `60^(@)`. What should be the refractive index of prism so that the ray is parallel to the base inside the prism ?

Options

A. `sqrt(2)`
B. `sqrt(3)`
C. `(4)/(3)`
D. `(9)/(8)`

Answer

To solve the problem, we need to find the refractive index (μ) of an equilateral prism when the angle of incidence is 60 degrees, and the ray inside the prism is parallel to the base.

### Step-by-Step Solution:

1. **Understand the Geometry of the Prism**:
- An equilateral prism has angles of 60 degrees at each vertex.
- The angle of incidence (i) is given as 60 degrees.

2. **Identify the Angles**:
- Since the prism is equilateral, the angle at the base (angle of the prism) is also 60 degrees.
- When the ray is refracted inside the prism and is parallel to the base, the angle of refraction (r) at the second face of the prism will also be 30 degrees (because the total angle at the vertex of the prism is 60 degrees).

3. **Apply Snell's Law**:
- Snell's Law states that \( n_1 \sin(i) = n_2 \sin(r) \).
- Here, \( n_1 \) is the refractive index of air (approximately 1), \( i \) is the angle of incidence (60 degrees), \( n_2 \) is the refractive index of the prism (μ), and \( r \) is the angle of refraction (30 degrees).

4. **Substitute the Values into Snell's Law**:
- We can write the equation as:
\[ 1 \cdot \sin(60^\circ) = \mu \cdot \sin(30^\circ) \]

5. **Calculate the Sine Values**:
- We know that:
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)

6. **Substitute the Sine Values into the Equation**:
- The equation now becomes:
\[ \frac{\sqrt{3}}{2} = \mu \cdot \frac{1}{2} \]

7. **Solve for the Refractive Index (μ)**:
- Multiply both sides by 2 to eliminate the fraction:
\[ \sqrt{3} = \mu \]
- Thus, the refractive index of the prism is:
\[ \mu = \sqrt{3} \]

### Final Answer:
The refractive index of the prism should be \( \sqrt{3} \).

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