What is the angle of incidence for an equilateral prism of refractive index `sqrt(3)` so that the ray si parallel to the base inside the prism?

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. Either `30^(@)` or `60^(@)`

Explanation

To find the angle of incidence for an equilateral prism with a refractive index of \( \sqrt{3} \) such that the ray is parallel to the base inside the prism, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the angle of incidence \( i \) such that the light ray inside the prism is parallel to the base. This condition occurs when the light ray experiences the angle of minimum deviation \( \delta_m \).

2. **Use the Formula for Minimum Deviation**: The formula relating the refractive index \( \mu \), the angle of the prism \( A \), and the angle of minimum deviation \( \delta_m \) is given by:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Identify Known Values**:
- The refractive index \( \mu = \sqrt{3} \)
- For an equilateral prism, the angle of the prism \( A = 60^\circ \).

4. **Substitute Values into the Formula**:
\[ \sqrt{3} = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \]
\[ \sqrt{3} = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{\sin(30^\circ)} \]

5. **Calculate \( \sin(30^\circ) \)**:
\[ \sin(30^\circ) = \frac{1}{2} \]
Thus, the equation simplifies to:
\[ \sqrt{3} = 2 \cdot \sin\left(\frac{60^\circ + \delta_m}{2}\right) \]

6. **Rearranging the Equation**:
\[ \sin\left(\frac{60^\circ + \delta_m}{2}\right) = \frac{\sqrt{3}}{2} \]

7. **Finding the Angle**:
The value of \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \) corresponds to \( 60^\circ \):
\[ \frac{60^\circ + \delta_m}{2} = 60^\circ \]
Therefore,
\[ 60^\circ + \delta_m = 120^\circ \implies \delta_m = 60^\circ \]

8. **Using the Relation for Angle of Incidence**:
For minimum deviation, the relationship between angle of incidence \( i \) and angle of emergence \( e \) is:
\[ i + e = A + \delta_m \]
Since \( i = e \) at minimum deviation, we have:
\[ 2i = A + \delta_m \]
Substituting the known values:
\[ 2i = 60^\circ + 60^\circ = 120^\circ \]
Thus,
\[ i = \frac{120^\circ}{2} = 60^\circ \]

### Final Answer:
The angle of incidence \( i \) is \( 60^\circ \).